has characteristic
type: ObjectProperty
definition: 'has characteristic' links algorithms to the characteristics, they possess.
domain: modeling and simulation algorithm
range: modeling and simulation algorithm characteristic

is hybrid of
type: ObjectProperty
definition: The basic idea of hybrid simulation methods is to combine the advantages of complementary simulation approaches: the whole system is subdivided into appropriate parts and different simulation methods operate on these parts at the same time. 'is hybrid of' is a relation between the complex simulation method and algorithms it applies to the different parts of the system.
range: modeling and simulation algorithm
sub property of: uses
domain: Class
  intersection of: modeling and simulation algorithm
  intersection of: Restriction
    on property: has characteristic
    some values from: hybridity

Axiom
annotated target: http://identifiers.org/doi/10.1093/bib/bbn050
comment: Biochemical simulations: stochastic, approximate stochastic and hybrid approaches. Pahle J. Brief Bioinform. 2009 Jan;10(1):53-64. Epub 2009 Jan 16.
annotated source: is hybrid of
annotated property: seeAlso

is parameter of
type: ObjectProperty
definition: 'is parameter of' is an inverse of 'has parameter' and links parameters to the algorithms which use them.
range: modeling and simulation algorithm
domain: modeling and simulation algorithm parameter

has parameter
type: ObjectProperty
definition: 'has parameter' links algorithms to the parameters they use.
domain: modeling and simulation algorithm
range: modeling and simulation algorithm parameter
inverse of: is parameter of

is characteristic of
type: ObjectProperty
definition: 'is characteristic of' is an inverse of 'has characteristic' and links characteristics to the algorithms which possess them.
range: modeling and simulation algorithm
domain: modeling and simulation algorithm characteristic
inverse of: has characteristic

is similar to
type: ObjectProperty
type: ReflexiveProperty
type: SymmetricProperty
created: 2011-06-10
definition: A relationship indicating that two entities are similar to each other.
creator: AZ

uses
type: ObjectProperty
created: 2011-06-10
definition: A relation between algorithms indicating that one algorithm uses another one (for example, if the algorithm switches between several algorithms).
creator: AZ
domain: modeling and simulation algorithm
range: modeling and simulation algorithm

is generalization of
type: ObjectProperty
created: 2011-06-10
definition: A relation between kinetic simulation algorithms, indicating that one is a generalization of another (e.g.if one extends another to suit systems with more general characteristics ).
creator: AZ
range: modeling and simulation algorithm
domain: modeling and simulation algorithm
sub property of: is similar to

is used by
type: ObjectProperty
created: 2011-07-19
definition: A relation between algorithms indicating that one algorithm is used by another one (for example, if the algorithm switches between several algorithms). 'is used by' is inverse of 'uses'.
creator: AZ
inverse of: uses

has type
type: DatatypeProperty
type: FunctionalProperty
definition: Indicates the type of algorithm parameter value, such as, for example, xsd:double for 'absolute tolerance'.
domain: modeling and simulation algorithm parameter

has Runge-Kutta method order
type: DatatypeProperty
definition: Indicates the order of a 'Runge-Kutta based method' [urn.miriam.biomodels.kisao:KISAO_0000064]. A 'Runge-Kutta based method' has order p if for sufficiently smooth initial value problems (y'=f(x,y), y(x0)=y0), ||y(x0 + h)-y1||<=Kh^(p+h).
domain: Runge-Kutta based method
range: int

modeling and simulation algorithm
type: Class
created: 2008-05-26
is organizational: true
definition: Algorithm used to instantiate a simulation from a mathematical model.
creator: dk

weighted stochastic simulation algorithm
type: Class
subclass of: accelerated stochastic simulation algorithm
subclass of: Class
  complement of: Restriction
    on property: has characteristic
    some values from: spatial description
subclass of: Restriction
  on property: has characteristic
  some values from: exact solution
created: 24JAN2009
creator: NLN
definition: The weighted Stochastic Simulation Algorithm manipulates the probabilities measure of biochemical systems by sampling, in order to increase the fraction of simulation runs exhibiting rare events.
alternative label: weighted SSA

Axiom
annotated target: http://identifiers.org/pubmed/19045316
comment: Kuwahara H, Mura I. (2008) An efficient and exact stochastic simulation method to analyse rare events in biochemical systems. J Chem Phys. 129(16):165101.
annotated source: weighted stochastic simulation algorithm
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: weighted SSA
annotated source: weighted stochastic simulation algorithm
annotated property: altLabel

Gillespie first reaction algorithm
type: Class
subclass of: Gillespie-like method
subclass of: Class
  complement of: Restriction
    on property: has characteristic
    some values from: spatial description
subclass of: Restriction
  on property: has characteristic
  some values from: exact solution
created: 2007-11-09
is implemented in: Cain
alternative label: Gillespie's first reaction method
creator: NLN
definition: Stochastic simulation algorithm using the reaction probability density function (next-reaction density function), giving the probability that the next reaction will happen in a given time interval. To choose the next reaction to fire, the algorithm calculates a tentative reaction time for each reaction and then select the smallest.

Axiom
annotated target: http://identifiers.org/doi/10.1021/j100540a008
comment: Gillespie DT. Exact stochastic simulation of coupled chemical reactions. Journal of Physical Chemistry, Vol. 81, No. 25. (1977), pp. 2340-2361.
annotated source: Gillespie first reaction algorithm
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: Gillespie's first reaction method
annotated source: Gillespie first reaction algorithm
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1016/0021-9991(76)90041-3
comment: Gillespie DT. A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions. Journal of Computational Physics, Volume 2 , pages 403-434 (1976).
annotated source: Gillespie first reaction algorithm
annotated property: seeAlso

multi-state agent-based simulation method
type: Class
subclass of: rule-based simulation method
subclass of: Restriction
  on property: has characteristic
  some values from: stochastic system behaviour
subclass of: Class
  complement of: Restriction
    on property: has characteristic
    some values from: spatial description
subclass of: Restriction
  on property: has characteristic
  some values from: progression with adaptive time step
alternative label: Morton-Firth
is implemented in: StochSim
definition: The agent-based simulation method instantiates each molecule as an individual software object. The interactions between those objects are determined by interaction probabilities according to experimental data. The probability is depended on the state the molecule is in at that specific time (molecules have multiple-state). Additionally, ''pseudo-molecules'' are introduced to the system in order to simulate unimolecular reactions. For simulation, continuous time is broken down into discrete, independent ''slices''. During each time slice one molecule is selected randomly, a second molecule or pseudo-molecule is selected afterwards (leading to either a unimolecular or a bimolecular reaction). The reaction will only take place if a produced random number exceeds the reaction probability calculated beforehand. In that case, the system is updated after that reaction.

Axiom
annotated target: http://identifiers.org/pubmed/9628844
comment: Morton-Firth, C.J, Bray, D. Predicting temporal fluctuations in an intracellular signalling pathway. Journal of Theoretical Biology, volume 192, pages 117-128 (1998).
annotated source: multi-state agent-based simulation method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: Morton-Firth
annotated source: multi-state agent-based simulation method
annotated property: altLabel

CVODE
type: Class
subclass of: CVODE-like method
subclass of: Restriction
  on property: is similar to
  some values from: PVODE
disjoint with: DASPK
created: 2007-11-30
alternative label: code value ordinary differential equation solver
is implemented in: BioNetGen
is implemented in: JSim
is implemented in: SBML-SAT
is implemented in: SUNDIALS
definition: The CVODE is a package written in C that solves ODE initial value problems, in real N-space, written as y'=f(t,y), y(t0)=y0. It is capable for stiff and non-stiff systems and uses two different linear multi-step methods, namely the Adam-Moulton [http://identifiers.org/biomodels.kisao/KISAO_0000280] method and the backward differentiation formula [http://identifiers.org/biomodels.kisao/KISAO_0000288].
is implemented in: VCell
alternative label: VODE
alternative label: VODEPK
creator: dk

Axiom
comment: EXACT
annotated target: code value ordinary differential equation solver
annotated source: CVODE
annotated property: altLabel

Axiom
annotated target: citeulike:1832863
comment: Cohen S, Hindmarsh C. Cvode, A Stiff/nonstiff Ode Solver In C. Computers in Physics, Vol. 10 (2), pages 138-143 (1996).
annotated source: CVODE
annotated property: seeAlso

Axiom
annotated target: http://identifiers.org/doi/10.1145/1089014.1089020
comment: Hindmarsh, A. C., et al., SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers, ACM Trans. Math. Softw., 31:363-396, 2005.
annotated source: CVODE
annotated property: seeAlso

Axiom
comment: RELATED
annotated target: VODE
annotated source: CVODE
annotated property: altLabel

Axiom
comment: RELATED
annotated target: VODEPK
annotated source: CVODE
annotated property: altLabel

PVODE
type: Class
subclass of: CVODE-like method
subclass of: Restriction
  on property: is similar to
  some values from: CVODE
subclass of: Class
  union of: Class
    intersection of: Restriction
      on property: uses
      some values from: backward differentiation formula
    intersection of: Restriction
      on property: uses
      some values from: generalized minimal residual algorithm
  union of: Restriction
    on property: uses
    some values from: Adams-Moulton method
disjoint with: IDA
disjoint with: DASSL
definition: PVODE is a general-purpose solver for ordinary differential equation (ODE) systems that implements methods for both stiff and nonstiff systems. [...] In the stiff case, PVODE uses a backward differentiation formula method [http://identifiers.org/biomodels.kisao/KISAO_0000288] combined with preconditioned GMRES [http://identifiers.org/biomodels.kisao/KISAO_0000253] iteration. Parallelism is achieved by distributing the ODE solution vector into user-specified segments and parallelizing a set of vector kernels accordingly. For PDE-based ODE systems, we provide a module that generates a band block-diagonal preconditioner for use with the GMRES [http://identifiers.org/biomodels.kisao/KISAO_0000253] iteration. PVODE is based on CVODE [http://identifiers.org/biomodels.kisao/KISAO_0000019].
is implemented in: SUNDIALS
alternative label: parallel code value ordinary differential equation solver

Axiom
comment: EXACT
annotated target: parallel code value ordinary differential equation solver
annotated source: PVODE
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1177/109434209901300405
comment: Byrne GD, Hindmarsh AC. PVODE, an ODE Solver for Parallel Computers. International Journal of High Performance Computing Applications, Vol. 13 (4), pages 354-365 (1999).
annotated source: PVODE
annotated property: seeAlso

Axiom
annotated target: http://identifiers.org/doi/10.1145/1089014.1089020
comment: Hindmarsh, A. C., et al., SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers, ACM Trans. Math. Softw., 31:363-396, 2005.
annotated source: PVODE
annotated property: seeAlso

StochSim nearest-neighbour algorithm
type: Class
subclass of: rule-based simulation method
subclass of: Restriction
  on property: is similar to
  some values from: cellular automata update method
subclass of: Restriction
  on property: has characteristic
  some values from: spatial description
subclass of: Restriction
  on property: has characteristic
  some values from: stochastic system behaviour
subclass of: Restriction
  on property: uses
  some values from: multi-state agent-based simulation method
is implemented in: Stochsim 1.2 and more recent versions
definition: The nearest-neighbour algorithm allows for the representation of spatial information, by adding a two-dimensional lattice in the form of a probabilistic cellular automata. That way, nearest neighbour interactions do additionally influence reactions taking place in the systems. Reactions between entities are calculated using the agent-based simulation algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000017].

Axiom
annotated target: http://identifiers.org/pubmed/11395441
comment: Le Novere N, Shimizu TS. STOCHSIM: modelling of stochastic biomolecular processes. Bioinformatics, volume 17 (6), pages 575-576 (2001).
annotated source: StochSim nearest-neighbour algorithm
annotated property: seeAlso

Elf and Ehrenberg method
type: Class
subclass of: sub-volume stochastic reaction-diffusion algorithm
subclass of: Restriction
  on property: uses
  some values from: Gibson-Bruck next reaction algorithm
subclass of: Restriction
  on property: uses
  some values from: Gillespie direct algorithm
disjoint with: Stundzia and Lumsden method
alternative label: Elf algorithm
is implemented in: MesoRD
alternative label: NSM
is implemented in: SmartCell
definition: Sub-volume stochastic reaction-diffusion method that is a combination of the Direct Method [http://identifiers.org/biomodels.kisao/KISAO_0000029] for sampling the time for a next reaction or diffusion event in each subvolume, with Gibson and Bruck's Next Reaction Method [http://identifiers.org/biomodels.kisao/KISAO_0000027], which is used to keep track of in which subvolume an event occurs next. The subvolumes are kept sorted in a queue, implemented as a binary tree, according to increasing time of the next event. When an event has occurred in the subvolume at the top of the queue, new event times need to be sampled only for one (the event is a chemical reaction) or two (the event is a diffusion jump) subvolume(s).
alternative label: next-subvolume method

Axiom
comment: EXACT
annotated target: Elf algorithm
annotated source: Elf and Ehrenberg method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: next-subvolume method
annotated source: Elf and Ehrenberg method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: NSM
annotated source: Elf and Ehrenberg method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1049/sb:20045021
comment: Elf J, Ehrenberg M. Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases. Systems Biology, IEE Proceedings, volume 1 (2), pages 230-236 (2004).
annotated source: Elf and Ehrenberg method
annotated property: seeAlso

Gibson-Bruck next reaction algorithm
type: Class
subclass of: accelerated stochastic simulation algorithm
subclass of: Restriction
  on property: is similar to
  some values from: Gillespie first reaction algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: exact solution
subclass of: Class
  complement of: Restriction
    on property: has characteristic
    some values from: spatial description
created: 2007-11-10
definition: As with the first reaction method [http://identifiers.org/biomodels.kisao/KISAO_0000015], a putative reaction time is calculated for each reaction, and the reaction with the shortest reaction time will be realized. However, the unused calculated reaction times are not wasted. The set of reactions is organized in a priority queue to allow for the efficient search for the fastest reaction. In addition, by using a so-called dependency graph only those reaction times are recalculated in each step, that are dependent on the reaction, which has been realized.
is implemented in: Cain
is implemented in: E-Cell
alternative label: Gibson and Bruck algorithm
alternative label: Gibson-Bruck's next reaction algorithm
alternative label: Gillespie-Gibson stochastic simulation algorithm
alternative label: SSA-GB
is implemented in: SmartCell
is implemented in: VCell
creator: dk
alternative label: next reaction method

Axiom
comment: EXACT
annotated target: next reaction method
annotated source: Gibson-Bruck next reaction algorithm
annotated property: altLabel

Axiom
comment: EXACT
annotated target: Gibson-Bruck's next reaction algorithm
annotated source: Gibson-Bruck next reaction algorithm
annotated property: altLabel

Axiom
comment: EXACT
annotated target: Gibson and Bruck algorithm
annotated source: Gibson-Bruck next reaction algorithm
annotated property: altLabel

Axiom
comment: EXACT
annotated target: Gillespie-Gibson stochastic simulation algorithm
annotated source: Gibson-Bruck next reaction algorithm
annotated property: altLabel

Axiom
comment: EXACT
annotated target: SSA-GB
annotated source: Gibson-Bruck next reaction algorithm
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1021/jp993732q
comment: Gibson MA, Bruck J. Efficient Exact Stochastic Simulation of Chemical Systems with Many Species and Many Channels. Journal of Physical Chemistry A, Vol. 104, pages 1876-1889 (2000).
annotated source: Gibson-Bruck next reaction algorithm
annotated property: seeAlso

slow-scale stochastic simulation algorithm
type: Class
subclass of: accelerated stochastic simulation algorithm
subclass of: Class
  complement of: Restriction
    on property: has characteristic
    some values from: spatial description
subclass of: Restriction
  on property: has characteristic
  some values from: approximate solution
definition: Attempt to overcome the problem of stiff systems by developing an ''approximate theory that allows one to stochastically advance the system in time by simulating the firings of only the slow reaction events''.
alternative label: slow-scale stochastic SSA
alternative label: ssSSA

Axiom
annotated target: http://identifiers.org/pubmed/15638651
comment: Cao Y, Gillespie DT, Petzold LR. The slow-scale stochastic simulation algorithm. Journal of Chemical Physics, Vol. 122, No. 1. (1 January 2005).
annotated source: slow-scale stochastic simulation algorithm
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: slow-scale stochastic SSA
annotated source: slow-scale stochastic simulation algorithm
annotated property: altLabel

Axiom
comment: EXACT
annotated target: ssSSA
annotated source: slow-scale stochastic simulation algorithm
annotated property: altLabel

Gillespie direct algorithm
type: Class
subclass of: Gillespie-like method
subclass of: Restriction
  on property: is similar to
  some values from: Bortz-Kalos-Lebowitz algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: exact solution
subclass of: Class
  complement of: Restriction
    on property: has characteristic
    some values from: spatial description
created: 2007-11-10
is implemented in: BetaWB
is implemented in: BioNetGen
is implemented in: ByoDyn
is implemented in: Cain
alternative label: DM
alternative label: Doob-Gillespie method
alternative label: Gillespie's direct method
alternative label: SSA
definition: Stochastic simulation algorithm using the reaction probability density function (next-reaction density function), giving the probability that the next reaction will happen in a given time interval. To choose the next reaction to fire, the algorithm directly and separately calculates the identity of the reaction and the time it will fire.
creator: dk
is implemented in: iBioSim
alternative label: stochastic simulation algorithm

Axiom
annotated target: DM
comment: EXACT
annotated source: Gillespie direct algorithm
annotated property: altLabel

Axiom
annotated target: Doob-Gillespie method
comment: EXACT
annotated source: Gillespie direct algorithm
annotated property: altLabel

Axiom
comment: EXACT
annotated target: SSA
annotated source: Gillespie direct algorithm
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1021/j100540a008
comment: Gillespie DT. Exact stochastic simulation of coupled chemical reactions. Journal of Physical Chemistry, Vol. 81, No. 25. (1977), pp. 2340-2361.
annotated source: Gillespie direct algorithm
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: stochastic simulation algorithm
annotated source: Gillespie direct algorithm
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1016/0021-9991(76)90041-3
comment: Gillespie DT. A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions. Journal of Computational Physics, Volume 2 , pages 403-434 (1976).
annotated source: Gillespie direct algorithm
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: Gillespie's direct method
annotated source: Gillespie direct algorithm
annotated property: altLabel

Euler forward method
type: Class
subclass of: Euler method
subclass of: Restriction
  on property: has characteristic
  some values from: explicit method type
disjoint with: Euler backward method
created: 2007-11-10
definition: The Euler method is an explicit one-step method for the numerical integration of ODES with a given initial value. The calculation of the next integration step at time t+1 is based on the state of the system at time point t.
is implemented in: VCell
creator: dk
alternative label: explicit Euler method
alternative label: explicit Gaussian first order Runge-Kutta
is implemented in: iBioSim

Axiom
comment: EXACT
annotated target: explicit Euler method
annotated source: Euler forward method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/isbn/052143064X
comment: Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes. Cambridge University Press, New York, 2nd edition (1992).
annotated source: Euler forward method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: explicit Gaussian first order Runge-Kutta
annotated source: Euler forward method
annotated property: altLabel

Euler backward method
type: Class
subclass of: Euler method
subclass of: Restriction
  on property: has characteristic
  some values from: implicit method type
created: 2007-11-02
is implemented in: GSL
definition: The Euler backward method is an implicit one-step method for the numerical integration of ODES with a given initial value. The next state of a system is calculated by solving an equation that considers both, the current state of the system and the later one.
creator: dk
alternative label: implicit Euler method
alternative label: implicit Gaussian first order Runge-Kutta

Axiom
comment: EXACT
annotated target: implicit Euler method
annotated source: Euler backward method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: implicit Gaussian first order Runge-Kutta
annotated source: Euler backward method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/isbn/052143064X
comment: Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes in Fortran 77. Cambridge University Press (2001).
annotated source: Euler backward method
annotated property: seeAlso

explicit fourth-order Runge-Kutta method
type: Class
subclass of: Runge-Kutta based method
subclass of: Restriction
  on property: has Runge-Kutta method order
  has value: 4
subclass of: Restriction
  on property: has characteristic
  some values from: explicit method type
subclass of: Restriction
  on property: has characteristic
  some values from: progression with fixed time step
created: 2007-11-12
alternative label: ERK4
is implemented in: GSL
is implemented in: JSim
alternative label: RK4
alternative label: Runge-Kutta method
definition: The Runge-Kutta method is a method for the numerical integration of ODES with a given initial value. The calculation of the next integration step at time t+1 is based on the state of the system at time point t, plus the product of the size of the interval and an estimated slope. The slope is a weighted average of 4 single slope points (beginning of interval-midpoint-midpoint-end of interval).
is implemented in: VCell
creator: dk

Axiom
annotated target: ERK4
comment: EXACT
annotated source: explicit fourth-order Runge-Kutta method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/isbn/0-471-91046-5
comment: Butcher JC. The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods (1987).
annotated source: explicit fourth-order Runge-Kutta method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: Runge-Kutta method
annotated source: explicit fourth-order Runge-Kutta method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: RK4
annotated source: explicit fourth-order Runge-Kutta method
annotated property: altLabel

Rosenbrock method
type: Class
subclass of: Runge-Kutta based method
subclass of: Restriction
  on property: has characteristic
  some values from: implicit method type
subclass of: Restriction
  on property: has characteristic
  some values from: progression with fixed time step
created: 2007-11-12
is implemented in: E-Cell
alternative label: Kaps-Rentrop method
definition: Some general implicit processes are given for the solution of simultaneous first-order differential equations. These processes, which use successive substitution, are implicit analogues of the (explicit) Runge-Kutta processes. They require the solution in each time step of one or more set of simultaneous linear equations, usually of a special and simple form. Processes of any required order can be devised, and they can be made to have a wide margin of stability when applied to a linear problem.
creator: dk
alternative label: generalized fourth order Runge-Kutta method

Axiom
comment: EXACT
annotated target: Kaps-Rentrop method
annotated source: Rosenbrock method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: generalized fourth order Runge-Kutta method
annotated source: Rosenbrock method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/isbn/052143064X
comment: Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes. Cambridge University Press, New York, 2nd edition, pages 742-746 (1992).
annotated source: Rosenbrock method
annotated property: seeAlso

Axiom
annotated target: http://identifiers.org/doi/10.1093/comjnl/5.4.329
comment: Rosenbrock HH. Some general implicit processes for the numerical solution of differential equations. The Computer Journal, pages 329-330 (1963).
annotated source: Rosenbrock method
annotated property: seeAlso

sorting stochastic simulation algorithm
type: Class
subclass of: accelerated stochastic simulation algorithm
subclass of: Class
  complement of: Restriction
    on property: has characteristic
    some values from: spatial description
subclass of: Restriction
  on property: has characteristic
  some values from: exact solution
definition: In order to overcome the problem of high complexity of the Stochastic Simulation Algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000029] when simulating large systems, the sorting direct method maintains a loosely sorted order of the reactions as the simulation executes.
alternative label: sorting SSA
alternative label: sorting direct method

Axiom
comment: EXACT
annotated target: sorting direct method
annotated source: sorting stochastic simulation algorithm
annotated property: altLabel

Axiom
comment: EXACT
annotated target: sorting SSA
annotated source: sorting stochastic simulation algorithm
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1016/j.compbiolchem.2005.10.007
comment: McCollum J M, Peterson G D, Cox C D, Simpson M L, Samatova N F. The sorting direct method for stochastic simulation of biochemical systems with varying reaction execution behaviour. Computational Biology and Chemistry, Volume 30 (1), pages 39-49 (2006).
annotated source: sorting stochastic simulation algorithm
annotated property: seeAlso

tau-leaping method
type: Class
subclass of: accelerated stochastic simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: approximate solution
subclass of: Restriction
  on property: has parameter
  on class: tau-leaping epsilon
  qualified cardinality: 1
created: 2008-07-08
definition: Approximate acceleration procedure of the Stochastic Simulation Algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000029] that divides the time into subintervals and ''leaps'' from one to another, firing all the reaction events in each subinterval.
is implemented in: ByoDyn
is implemented in: Cain
creator: NLN
is implemented in: SmartCell
alternative label: tauL

Axiom
annotated target: http://identifiers.org/doi/10.1063/1.1378322
comment: Gillespie DT. Approximate accelerated stochastic simulation of chemically reacting systems. The Journal of Chemical Physics, Vol. 115 (4), pages 1716-1733 (2001). Section V.
annotated source: tau-leaping method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: tauL
annotated source: tau-leaping method
annotated property: altLabel

Poisson tau-leaping method
type: Class
subclass of: tau-leaping method
subclass of: Restriction
  on property: has characteristic
  some values from: explicit method type
alternative label: explicit tau-leaping method with basic pre-leap check
is implemented in: ByoDyn
definition: Explicit tau-leaping method with basic pre leap check.
alternative label: explicit tau-leaping
alternative label: explicit tau-leaping method with basic preleap check
alternative label: poisson tau-leaping

Axiom
annotated target: http://identifiers.org/doi/10.1063/1.1378322
comment: Gillespie DT. Approximate accelerated stochastic simulation of chemically reacting systems. The Journal of Chemical Physics, Vol. 115 (4), pages 1716-1733 (2001). Section V.
annotated source: Poisson tau-leaping method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: explicit tau-leaping method with basic preleap check
annotated source: Poisson tau-leaping method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: explicit tau-leaping
annotated source: Poisson tau-leaping method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: poisson tau-leaping
annotated source: Poisson tau-leaping method
annotated property: altLabel

implicit tau-leaping method
type: Class
subclass of: tau-leaping method
subclass of: Class
  complement of: Restriction
    on property: has characteristic
    some values from: spatial description
subclass of: Restriction
  on property: has characteristic
  some values from: implicit method type
subclass of: Restriction
  on property: has parameter
  on class: tau-leaping delta
  qualified cardinality: 1
created: 2007-10-12
definition: Contrary to the explicit tau-leaping [http://identifiers.org/biomodels.kisao/KISAO_0000039 and http://identifiers.org/biomodels.kisao/KISAO_0000245 some http://identifiers.org/biomodels.kisao/KISAO_0000239] , the implicit tau-leaping allows for much larger time-steps when simulating stiff systems.
creator: dk

Axiom
annotated target: http://identifiers.org/doi/10.1063/1.1627296
comment: Rathinam M, Petzold L R, Cao Y, Gillespie D T. Stiffness in stochastic chemically reacting systems: The implicit tau-leaping method. The Journal of Chemical Physics, Volume 119 (24), pages 12784-12794 (2003).
annotated source: implicit tau-leaping method
annotated property: seeAlso

trapezoidal tau-leaping method
type: Class
subclass of: tau-leaping method
subclass of: Restriction
  on property: has characteristic
  some values from: implicit method type
subclass of: Class
  complement of: Restriction
    on property: has characteristic
    some values from: spatial description
created: 2007-10-16
alternative label: trapezoidal implicit tau-leaping method
definition: Formula for accelerated discrete efficient stochastic simulation of chemically reacting system [which] has better accuracy and stiff stability properties than the explicit and implicit [http://identifiers.org/biomodels.kisao/KISAO_0000045] tau-leaping formulas for discrete stochastic systems, and it limits to the trapezoidal rule in the deterministic regime.
creator: dk

Axiom
annotated target: citeulike:1755561
comment: Cao Y, Petzold L. Trapezoidal tau-leaping formula for the stochastic simulation of biochemical systems. Foundations of Systems Biology in Engineering (FOSBE), pages 149-152 (2005).
annotated source: trapezoidal tau-leaping method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: trapezoidal implicit tau-leaping method
annotated source: trapezoidal tau-leaping method
annotated property: altLabel

adaptive explicit-implicit tau-leaping method
type: Class
subclass of: tau-leaping method
subclass of: Class
  complement of: Restriction
    on property: has characteristic
    some values from: spatial description
subclass of: Restriction
  on property: has characteristic
  some values from: explicit method type
subclass of: Restriction
  on property: has characteristic
  some values from: implicit method type
definition: Modification of the original tau-selection strategy [http://identifiers.org/biomodels.kisao/KISAO_0000040], designed for explicit tau-leaping, is modified to apply to implicit tau-leaping, allowing for longer steps when the system is stiff. Further, an adaptive strategy is proposed that identifies stiffness and automatically chooses between the explicit and the (new) implicit tau-selection methods to achieve better efficiency.

Axiom
annotated target: http://identifiers.org/doi/10.1063/1.2745299
comment: Cao Y, Gillespie DT, Petzold LR. Adaptive explicit-implicit tau-leaping method with automatic tau selection. The Journal of Chemical Physics, Vol. 126 (22) (2007).
annotated source: adaptive explicit-implicit tau-leaping method
annotated property: seeAlso

Bortz-Kalos-Lebowitz algorithm
type: Class
subclass of: Gillespie-like method
subclass of: Restriction
  on property: is similar to
  some values from: Gillespie direct algorithm
subclass of: Class
  complement of: Restriction
    on property: has characteristic
    some values from: spatial description
alternative label: BKL
alternative label: DMC
alternative label: KMC
definition: The Bortz-Kalos-Lebowitz (or: kinetic Monte-Carlo-) method is a stochastic method for the simulation of time evolution of processes using (pseudo-)random numbers.
alternative label: dynamic Monte Carlo
alternative label: dynamic Monte Carlo method
alternative label: kinetic Monte Carlo
alternative label: kinetic Monte Carlo method
alternative label: n-fold way

Axiom
annotated target: BKL
comment: EXACT
annotated source: Bortz-Kalos-Lebowitz algorithm
annotated property: altLabel

Axiom
comment: EXACT
annotated target: kinetic Monte Carlo method
annotated source: Bortz-Kalos-Lebowitz algorithm
annotated property: altLabel

Axiom
comment: EXACT
annotated target: KMC
annotated source: Bortz-Kalos-Lebowitz algorithm
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1016/0021-9991(75)90060-1
comment: Bortz AB, Kalos MH, Lebowitz JL. A new algorithm for Monte Carlo simulation of Ising spin systems. Journal of Computational Physics, Vol. 17 (1), pages 10-18 (1975).
annotated source: Bortz-Kalos-Lebowitz algorithm
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: n-fold way
annotated source: Bortz-Kalos-Lebowitz algorithm
annotated property: altLabel

Axiom
comment: EXACT
annotated target: kinetic Monte Carlo
annotated source: Bortz-Kalos-Lebowitz algorithm
annotated property: altLabel

Axiom
comment: EXACT
annotated target: dynamic Monte Carlo
annotated source: Bortz-Kalos-Lebowitz algorithm
annotated property: altLabel

Axiom
comment: EXACT
annotated target: dynamic Monte Carlo method
annotated source: Bortz-Kalos-Lebowitz algorithm
annotated property: altLabel

Axiom
annotated target: DMC
comment: EXACT
annotated source: Bortz-Kalos-Lebowitz algorithm
annotated property: altLabel

Smoluchowski equation based method
type: Class
subclass of: modeling and simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: stochastic system behaviour
created: 2007-10-29
is organizational: true
definition: Method based on the Smoluchowski equation.
creator: dk

Axiom
annotated target: urn:issn:0942-9352
comment: Smoluchowski M. Mathematical theory of the kinetics of the coagulation of colloidal solutions. Z. Phys. Chem, Vol. 92, No. 129. (1917).
annotated source: Smoluchowski equation based method
annotated property: seeAlso

Brownian diffusion Smoluchowski method
type: Class
subclass of: Smoluchowski equation based method
subclass of: Restriction
  on property: has parameter
  on class: Brownian diffusion accuracy
  qualified cardinality: 1
subclass of: Restriction
  on property: has parameter
  on class: surface-bound epsilon
  qualified cardinality: 1
subclass of: Restriction
  on property: has characteristic
  some values from: discrete variable
subclass of: Restriction
  on property: has parameter
  on class: virtual box size
  qualified cardinality: 1
subclass of: Restriction
  on property: has parameter
  on class: neighbour distance
  qualified cardinality: 1
subclass of: Restriction
  on property: has characteristic
  some values from: progression with fixed time step
subclass of: Restriction
  on property: has characteristic
  some values from: spatial description
disjoint with: Greens function reaction dynamics
definition: In the Brownian diffusion Smoluchowski method, ''each molecule is treated as a point-like particle that diffuses freely in three-dimensional space. When a pair of reactive molecules collide, such as an enzyme and its substrate, a reaction occurs and the simulated reactants are replaced by products. [..] Analytic solutions are presented for some simulation parameters while others are calculated using look-up tables''. Supported chemical processes include molecular diffusion, treatment of surfaces, zeroth-order-, unimolecular-, and bimolecular reactions.
is implemented in: Smoldyn

Axiom
annotated target: http://identifiers.org/pubmed/16204833
comment: Andrews SS, Bray D. Stochastic simulation of chemical reactions with spatial resolution and single molecule detail. Phys Biol, volume 1 (3-4), pages 137-151 (December 2004).
annotated source: Brownian diffusion Smoluchowski method
annotated property: seeAlso

Greens function reaction dynamics
type: Class
subclass of: Smoluchowski equation based method
subclass of: Restriction
  on property: has characteristic
  some values from: discrete variable
subclass of: Restriction
  on property: has characteristic
  some values from: approximate solution
subclass of: Restriction
  on property: has characteristic
  some values from: progression with adaptive time step
subclass of: Restriction
  on property: has characteristic
  some values from: spatial description
alternative label: GFRD
alternative label: Green's function reaction dynamics
definition: Method that simulates biochemical networks on particle level. It considers both changes in time and space by ''exploiting both the exact solution of the Smoluchowski Equation to set up an event-driven algorithm'' which allows for large jumps in time when the considered particles are far away from each other [in space] and thus cannot react. GFRD combines the propagation of particles in space with the reactions taking place between them in one simulation step.

Axiom
comment: EXACT
annotated target: Green's function reaction dynamics
annotated source: Greens function reaction dynamics
annotated property: altLabel

Axiom
comment: EXACT
annotated target: GFRD
annotated source: Greens function reaction dynamics
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1063/1.2137716
comment: Van Zon JS, Ten Wolde PR. Green's-function reaction dynamics: A particle-based approach for simulating biochemical networks in time and space. Journal of Chemical Physics, Volume 123 (23) (2005).
annotated source: Greens function reaction dynamics
annotated property: seeAlso

Runge-Kutta based method
type: Class
subclass of: one-step method
subclass of: Restriction
  on property: has characteristic
  some values from: progression with fixed time step
subclass of: Restriction
  on property: is generalization of
  some values from: Euler method
subclass of: Restriction
  on property: has characteristic
  some values from: deterministic system behaviour
subclass of: Restriction
  on property: has Runge-Kutta method order
  some values from: integer
subclass of: Restriction
  on property: has characteristic
  some values from: ordinary differential equation problem
created: 2007-11-12
is organizational: true
definition: A method of numerically integrating ordinary differential equations, which uses a sampling of slopes through an interval and takes a weighted average to determine the right end point. This averaging gives a very accurate approximation.
is implemented in: ByoDyn
creator: dk
alternative label: modified Euler method

Axiom
comment: RELATED
annotated target: modified Euler method
annotated source: Runge-Kutta based method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/isbn/0-471-91046-5
comment: Butcher JC. The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods (1987).
annotated source: Runge-Kutta based method
annotated property: seeAlso

deterministic cellular automata update algorithm
type: Class
subclass of: cellular automata update method
subclass of: Restriction
  on property: has characteristic
  some values from: deterministic system behaviour
created: 2007-11-30
definition: A cellular automaton is a discrete model of a regular grid of cells with a finite number of dimensions. Each cell has a finite number of defined states. The automaton changes its state in a discrete manner, meaning that the state of a cell at time t is determined by a function of the states of its neighbours at time t - 1. These neighbours are a selection of cells relative to the specified cell. Famous examples for deterministic cellular automata are Conway's game of life or Wolfram's elementary cellular automata.
creator: dk

Axiom
annotated target: http://identifiers.org/isbn/978-0252000232
comment: Burks, Arthur W (Editor). Essays on Cellular Automata. University of Illinois Press (1970).
annotated source: deterministic cellular automata update algorithm
annotated property: seeAlso

LSODE
type: Class
subclass of: Livermore solver
subclass of: Restriction
  on property: uses
  some values from: backward differentiation formula
subclass of: Restriction
  on property: uses
  some values from: Adams predictor-corrector method
subclass of: Restriction
  on property: has characteristic
  some values from: explicit method type
created: 2007-11-16
definition: LSODE solves stiff and nonstiff systems of the form dy/dt = f. In the stiff case, it treats the Jacobian matrix sf/dy as either a dense (full) or a banded matrix, and as either user-supplied or internally approximated by difference quotients. It uses Adams methods (predictor-corrector) [http://identifiers.org/biomodels.kisao/KISAO_0000364] in the nonstiff case, and Backward Differentiation Formula (BDF) methods (the Gear methods) [http://identifiers.org/biomodels.kisao/KISAO_0000288] in the stiff case.
alternative label: Livermore solver for ordinary differential equations
creator: dk

Axiom
annotated target: http://identifiers.org/doi/10.1145/1218052.1218054
comment: Hindmarsh AC. LSODE and LSODI, two new initial value ordinary differential equation solvers. SIGNUM Newsletter, Volume 15 (4), pages 10-11 (1980).
annotated source: LSODE
annotated property: seeAlso

Axiom
annotated target: citeulike:1774586
comment: Radhakrishnan K, Hindmarsh AC. Description and Use of LSODE, the Livermore Solver for Ordinary Differential Equations. Lawrence Livermore National Laboratory Report, Vol. UCRL-ID-113855 (1993).
annotated source: LSODE
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: Livermore solver for ordinary differential equations
annotated source: LSODE
annotated property: altLabel

binomial tau-leaping method
type: Class
subclass of: tau-leaping method
subclass of: Restriction
  on property: is similar to
  some values from: sub-volume stochastic reaction-diffusion algorithm
subclass of: Restriction
  on property: has parameter
  on class: coarse-graining factor
  qualified cardinality: 1
created: 2007-10-16
alternative label: binomial tau-leap spatial stochastic simulation algorithm
alternative label: BtauL
definition: Coarse grained modified version of the next subvolume method [http://identifiers.org/biomodels.kisao/KISAO_0000022] that allows the user to consider both diffusion and reaction events in relatively long simulation time spans as compared with the original method and other commonly used fully stochastic computational methods.
creator: dk

Axiom
comment: EXACT
annotated target: binomial tau-leap spatial stochastic simulation algorithm
annotated source: binomial tau-leaping method
annotated property: altLabel

Axiom
annotated target: BtauL
comment: EXACT
annotated source: binomial tau-leaping method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1063/1.2771548
comment: Marquez-Lago T., Burrage K. Binomial tau-leap spatial stochastic simulation algorithm for applications in chemical kinetics. The Journal of Chemical Physics, Vol. 127 (10) (2007).
annotated source: binomial tau-leaping method
annotated property: seeAlso

Gillespie multi-particle method
type: Class
subclass of: Gillespie-like method
subclass of: Restriction
  on property: has characteristic
  some values from: spatial description
subclass of: Restriction
  on property: uses
  some values from: Gillespie direct algorithm
subclass of: Restriction
  on property: uses
  some values from: multiparticle lattice gas automata
definition: Combination of the multiparticle method for diffusion [http://identifiers.org/biomodels.kisao/KISAO_0000334] and the SSA [http://identifiers.org/biomodels.kisao/KISAO_0000029].
alternative label: GMP
alternative label: Gillespie's multi-particle method
alternative label: particle-based spatial stochastic method

Axiom
comment: EXACT
annotated target: particle-based spatial stochastic method
annotated source: Gillespie multi-particle method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/pubmed/16731694
comment: Rodriguez VJ, Kaandorp JA, Dobrzynski M, Blom JG. Spatial stochastic modelling of the phosphoenolpyruvate-dependent phosphotransferase (PTS) pathway in Escherichia coli. Bioinformatics, Vol. 22 (15), pages 1895-1901 (2006).
annotated source: Gillespie multi-particle method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: GMP
annotated source: Gillespie multi-particle method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: Gillespie's multi-particle method
annotated source: Gillespie multi-particle method
annotated property: altLabel

Stundzia and Lumsden method
type: Class
subclass of: sub-volume stochastic reaction-diffusion algorithm
alternative label: RD SSA
definition: Sub-volume stochastic reaction-diffusion method that using Green's function to link the bulk diffusion coefficient D in Fick's differential law to the corresponding transition rate probability for diffusion of a particle between finite volume elements. This generalized stochastic algorithm enables to numerically calculate the time evolution of a spatially inhomogeneous mixture of reaction-diffusion species in a finite volume. The time step is stochastic and is generated by a probability distribution determined by the intrinsic reaction kinetics and diffusion dynamics.
alternative label: reaction-diffusion stochastic simulation algorithm

Axiom
comment: EXACT
annotated target: RD SSA
annotated source: Stundzia and Lumsden method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: reaction-diffusion stochastic simulation algorithm
annotated source: Stundzia and Lumsden method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1006/jcph.1996.0168
comment: Stundzia AB, Lumsden CJ. Stochastic simulation of coupled reaction-diffusion processes. J Comput Phys, Vol. 127 (1), pages 196-207 (1996).
annotated source: Stundzia and Lumsden method
annotated property: seeAlso

estimated midpoint tau-leaping method
type: Class
subclass of: tau-leaping method
subclass of: Restriction
  on property: has characteristic
  some values from: explicit method type
definition: Estimated-Midpoint tau-Leap Method: For the selected leaping time tau which satisfies the Leap Condition, compute the expected state change lambda' = tau sumj( aj(x)vj ) during [t, t + tau). Then, with x' =x + [lambda'/2], generate for each j = 1,...,M a sample value kj of the Poisson random variable P(aj(x'), tau). Compute the actual state change, lambda = sumj( kjvj ), and effect the leap by replacing t by t + tau and x by x + lambda.
alternative label: explicit tau-leaping method with estimated-mid point technique

Axiom
annotated target: http://identifiers.org/doi/10.1063/1.1378322
comment: Gillespie DT. Approximate accelerated stochastic simulation of chemically reacting systems. The Journal of Chemical Physics, Vol. 115 (4), pages 1716-1733 (2001). Section VI.
annotated source: estimated midpoint tau-leaping method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: explicit tau-leaping method with estimated-mid point technique
annotated source: estimated midpoint tau-leaping method
annotated property: altLabel

k-alpha leaping method
type: Class
subclass of: accelerated stochastic simulation algorithm
subclass of: Restriction
  on property: is similar to
  some values from: tau-leaping method
subclass of: Restriction
  on property: has characteristic
  some values from: approximate solution
definition: Alternative to the tau-leaping [http://identifiers.org/biomodels.kisao/KISAO_0000039], where one leaps a fixed number of reaction-events.

Axiom
annotated target: http://identifiers.org/doi/10.1063/1.1378322
comment: Gillespie DT. Approximate accelerated stochastic simulation of chemically reacting systems. The Journal of Chemical Physics, Vol. 115 (4), pages 1716-1733 (2001). Section VIII.
annotated source: k-alpha leaping method
annotated property: seeAlso

nonnegative Poisson tau-leaping method
type: Class
subclass of: tau-leaping method
subclass of: Restriction
  on property: is similar to
  some values from: Poisson tau-leaping method
subclass of: Restriction
  on property: has characteristic
  some values from: explicit method type
subclass of: Restriction
  on property: has parameter
  on class: critical firing threshold
  qualified cardinality: 1
definition: The explicit tau-leaping procedure attempts to speed up the stochastic simulation of a chemically reacting system by approximating the number of firings of each reaction channel during a chosen time increment Tau as a Poisson random variable. Since the Poisson random variable can have arbitrarily large sample values, there is always the possibility that this procedure will cause one or more reaction channels to fire so many times during Tau that the population of some reactant species will be driven negative. Two recent papers have shown how that unacceptable occurrence can be avoided by replacing the Poisson random variables with binomial random variables, whose values are naturally bounded. This paper describes a modified Poisson tau-leaping procedure that also avoids negative populations, but is easier to implement than the binomial procedure. The new Poisson procedure also introduces a second control parameter, whose value essentially dials the procedure from the original Poisson tau-leaping at one extreme to the exact stochastic simulation algorithm at the other; therefore, the modified Poisson procedure will generally be more accurate than the original Poisson procedure [http://identifiers.org/biomodels.kisao/KISAO_0000040].
alternative label: modified poisson tau-leaping

Axiom
annotated target: http://identifiers.org/doi/10.1063/1.1992473
comment: Cao Y, Gillespie DT, Petzold LR. Avoiding negative populations in explicit Poisson tau-leaping. Journal of Chemical Physics, Vol. 123, 4104 (2005).
annotated source: nonnegative Poisson tau-leaping method
annotated property: seeAlso

Fehlberg method
type: Class
subclass of: embedded Runge-Kutta method
subclass of: Restriction
  on property: uses
  some values from: Class
    intersection of: Runge-Kutta based method
    intersection of: Restriction
      on property: has Runge-Kutta method order
      has value: 4
subclass of: Restriction
  on property: uses
  some values from: Class
    intersection of: Runge-Kutta based method
    intersection of: Restriction
      on property: has Runge-Kutta method order
      has value: 5
is implemented in: E-Cell
is implemented in: GSL
is implemented in: JSim
alternative label: RKF45
alternative label: Runge-Kutta-Fehlberg method
definition: The method was developed by the German mathematician Erwin Fehlberg and is based on the class of Runge-Kutta methods. The Runge-Kutta-Fehlberg method uses an O(h4) method together with an O(h5) method that uses all of the points of the O(h4) method, and hence is often referred to as an RKF45 method. Similar schemes with different orders have since been developed. By performing one extra calculation that would be required for an RK5 method, the error in the solution can be estimated and controlled and an appropriate step size can be determined automatically, making this method efficient for ordinary problems of automated numerical integration of ordinary differential equations.
is implemented in: VCell
is implemented in: iBioSim

Axiom
annotated target: http://identifiers.org/pubmed/14990450
comment: Takahashi K, Kaizu K, Hu B, Tomita M. A multi-algorithm, multi-timescale method for cell simulation. Bioinformatics volume 20(4), pages 538-46 (2004).
annotated source: Fehlberg method
annotated property: seeAlso

Axiom
annotated target: http://identifiers.org/doi/10.1007/BF02241732
comment: Erwin Fehlberg (1970). Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Wärmeleitungsprobleme, Computing (Arch. Elektron. Rechnen), vol. 6, pp. 61-71. @de
annotated source: Fehlberg method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: RKF45
annotated source: Fehlberg method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: Runge-Kutta-Fehlberg method
annotated source: Fehlberg method
annotated property: altLabel

Dormand-Prince method
type: Class
subclass of: embedded Runge-Kutta method
created: 2007-11-12
definition: Dormand-Prince is an explicit method for the numerical integration of ODES with a given initial value.
is implemented in: ECell3
is implemented in: GSL
is implemented in: JSim
is implemented in: Matlab
alternative label: Prince-Dormand method
creator: dk
is implemented in: iBioSim

Axiom
annotated target: http://identifiers.org/pubmed/14990450
comment: Takahashi K, Kaizu K, Hu B, Tomita M. A multi-algorithm, multi-timescale method for cell simulation. Bioinformatics volume 20(4), pages 538-46 (2004).
annotated source: Dormand-Prince method
annotated property: seeAlso

Axiom
annotated target: http://identifiers.org/doi/10.1016/0771-050X(80)90013-3
comment: Dormand J, Prince P. A family of embedded Runge-Kutta formulae. Journal of Computational and Applied Mathematics, Volume 6, pages 19-26 (1980).
annotated source: Dormand-Prince method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: Prince-Dormand method
annotated source: Dormand-Prince method
annotated property: altLabel

LSODA
type: Class
subclass of: Livermore solver
subclass of: Restriction
  on property: has parameter
  on class: LSODA maximum non-stiff order
  qualified cardinality: 1
subclass of: Restriction
  on property: uses
  some values from: Adams method
subclass of: Restriction
  on property: has parameter
  on class: relative tolerance
  qualified cardinality: 1
subclass of: Restriction
  on property: has parameter
  on class: LSODA maximum stiff order
  qualified cardinality: 1
subclass of: Restriction
  on property: uses
  some values from: backward differentiation formula
subclass of: Restriction
  on property: is similar to
  some values from: LSODE
subclass of: Restriction
  on property: has parameter
  on class: absolute tolerance
  qualified cardinality: 1
created: 2007-11-30
definition: LSODA solves systems dy/dt = f with a dense or banded Jacobian when the problem is stiff, but it automatically selects between non-stiff (Adams [http://identifiers.org/biomodels.kisao/KISAO_0000289]) and stiff (BDF [http://identifiers.org/biomodels.kisao/KISAO_0000288]) methods. It uses the non-stiff method initially, and dynamically monitors data in order to decide which method to use.
alternative label: Livermore solver for ordinary differential equations with automatic method switching
creator: dk

Axiom
comment: EXACT
annotated target: Livermore solver for ordinary differential equations with automatic method switching
annotated source: LSODA
annotated property: altLabel

Axiom
annotated target: http://www.nea.fr/abs/html/uscd1227.html
comment: Petzold LR, Hindmarsh, AC. LSODAR: Livermore solver of ordinary differential equations with automatic method switching and root finding, Computing and Mathematics Research Division, 1-316 Lawerence Livermore National Laboratory (1987).
annotated source: LSODA
annotated property: seeAlso

Axiom
annotated target: http://identifiers.org/isbn/978-0444866073
comment: A. C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman et al. (eds.), North-Holland, Amsterdam, 1983 (vol. 1 of IMACS Transactions on Scientific Computation), pp. 55-64.
annotated source: LSODA
annotated property: seeAlso

Axiom
annotated target: http://identifiers.org/doi/10.1137/0904010
comment: Petzold LR. Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations. SIAM Journal on Scientific and Statistical Computing, Vol. 4 (1), pages 136-148 (1983).
annotated source: LSODA
annotated property: seeAlso

LSODAR
type: Class
subclass of: Livermore solver
subclass of: Restriction
  on property: is similar to
  some values from: LSODA
created: 2007-10-27
definition: LSODAR is a variant of LSODA [http://identifiers.org/biomodels.kisao/KISAO_0000088] with a root finding capability added. Thus it solves problems dy/dt = f with dense or banded Jacobian and automatic method selection, and at the same time, it finds the roots of any of a set of given functions of the form g(t,y). This is often useful for finding stop conditions, or for finding points at which a switch is to be made in the function f.
alternative label: Livermore solver for ordinary differential equations with automatic method switching and root finding
creator: dk
alternative label: ordinary differential equation solver for stiff or non-stiff systems with root finding

Axiom
annotated target: http://www.nea.fr/abs/html/uscd1228.html
comment: Petzold LR, Hindmarsh, AC. LSODAR: Livermore solver of ordinary differential equations with automatic method switching and root finding, Computing and Mathematics Research Division, 1-316 Lawerence Livermore National Laboratory (1987).
annotated source: LSODAR
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: ordinary differential equation solver for stiff or non-stiff systems with root finding
annotated source: LSODAR
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/isbn/978-0444866073
comment: A. C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman et al. (eds.), North-Holland, Amsterdam, 1983 (vol. 1 of IMACS Transactions on Scientific Computation), pp. 55-64.
annotated source: LSODAR
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: Livermore solver for ordinary differential equations with automatic method switching and root finding
annotated source: LSODAR
annotated property: altLabel

LSODI
type: Class
subclass of: Livermore solver
subclass of: Restriction
  on property: has characteristic
  some values from: implicit method type
subclass of: Restriction
  on property: is similar to
  some values from: LSODE
definition: LSODI solves systems given in linearly implicit form, including differential-algebraic systems.
alternative label: Livermore solver for ordinary differential equations, implicit version

Axiom
comment: EXACT
annotated target: Livermore solver for ordinary differential equations, implicit version
annotated source: LSODI
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/isbn/978-0444866073
comment: A. C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman et al. (eds.), North-Holland, Amsterdam, 1983 (vol. 1 of IMACS Transactions on Scientific Computation), pp. 55-64.
annotated source: LSODI
annotated property: seeAlso

Axiom
annotated target: http://identifiers.org/doi/10.1145/1218052.1218054
comment: Hindmarsh AC. LSODE and LSODI, two new initial value ordinary differential equation solvers. SIGNUM Newsletter, Volume 15 (4), pages 10-11 (1980).
annotated source: LSODI
annotated property: seeAlso

LSODIS
type: Class
subclass of: Livermore solver
subclass of: Restriction
  on property: has characteristic
  some values from: implicit method type
subclass of: Restriction
  on property: is similar to
  some values from: LSODE
created: 2007-11-30
definition: LSODIS is a set of general-purpose FORTRAN routines solver for the initial value problem for ordinary differential equation systems. It is suitable for both stiff and nonstiff systems. LSODIS treat systems in the linearly implicit form A(t,y) dy/dt = g(t,y), A = a square matrix, i.e. with the derivative dy/dt implicit, but linearly so.
alternative label: Livermore solver for ordinary differential equations, implicit sparse version
creator: dk

Axiom
comment: EXACT
annotated target: Livermore solver for ordinary differential equations, implicit sparse version
annotated source: LSODIS
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/isbn/978-0444866073
comment: A. C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman et al. (eds.), North-Holland, Amsterdam, 1983 (vol. 1 of IMACS Transactions on Scientific Computation), pp. 55-64.
annotated source: LSODIS
annotated property: seeAlso

Axiom
annotated target: http://www.nea.fr/abs/html/uscd1225.html
comment: Seager M, Balsdon S. LSODIS - A sparse implicit ODE solver. 10th World Congress on System Simulation and Scientific Computation, pages 437-439 (1983).
annotated source: LSODIS
annotated property: seeAlso

LSODPK
type: Class
subclass of: Livermore solver
subclass of: Restriction
  on property: is similar to
  some values from: LSODE
subclass of: Restriction
  on property: uses
  some values from: Krylov subspace projection method
created: 2008-07-08
definition: LSODPK is a set of FORTRAN subroutines for solving the initial value problem for stiff and nonstiff systems of ordinary differential equations. In solving stiff systems, LSODPK uses a corrector iteration composed of Newton iteration and one of four preconditioned Krylov subspace iteration methods [http://identifiers.org/biomodels.kisao/KISAO_0000354]. The user must select the desired Krylov method and supply a pair of routine to evaluate, preprocess, and solve the (left and/or right) preconditioner matrices. Aside from preconditioning, the implementation is matrix-free, meaning that explicit storage of the Jacobian (or related) matrix is not required. The method is experimental because the scope of problems for which it is effective is not well-known, and users are forewarned that LSODPK may or may not be competitive with traditional methods on a given problem. LSODPK also includes an option for a user-supplied linear system solver to be used without Krylov iteration.
alternative label: Livermore solver for ordinary differential equations for stiff and nonstiff systems with krylov corrector iteration
creator: NLN

Axiom
comment: EXACT
annotated target: Livermore solver for ordinary differential equations for stiff and nonstiff systems with krylov corrector iteration
annotated source: LSODPK
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/isbn/978-0444866073
comment: A. C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman et al. (eds.), North-Holland, Amsterdam, 1983 (vol. 1 of IMACS Transactions on Scientific Computation), pp. 55-64.
annotated source: LSODPK
annotated property: seeAlso

Livermore solver
type: Class
subclass of: modeling and simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: continuous variable
subclass of: Class
  complement of: Restriction
    on property: has characteristic
    some values from: spatial description
subclass of: Restriction
  on property: has characteristic
  some values from: deterministic system behaviour
subclass of: Restriction
  on property: has characteristic
  some values from: ordinary differential equation problem
subclass of: Restriction
  on property: has characteristic
  some values from: progression with adaptive time step
created: 2008-07-08
is organizational: true
definition: Method to solve ordinary differential equations developed at the Lawrence Livermore National Laboratory.
creator: NLN

sub-volume stochastic reaction-diffusion algorithm
type: Class
subclass of: generalized stochastic simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: spatial description
disjoint with: D-leaping method
created: 2008-07-08
is organizational: true
creator: NLN
definition: Stochastic method using a combination of discretisation of compartment volumes into voxels and Gillespie-like algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000241] to simulate the evolution of the system.

modeling and simulation algorithm characteristic
type: Class
is organizational: true
creator: AZ
definition: Simulation algorithm property, which can, for example, describe the model, such as the type of variables (discrete or continuous), and information on the treatment of spatial descriptions, or can be a numerical characteristic, such as the system's behaviour (deterministic or stochastic) as well as the progression mechanism (fixed or adaptive time steps).

type of variable
type: Class
subclass of: modeling and simulation algorithm characteristic
is organizational: true
creator: AZ
definition: Type of variables used for the simulation.

type of system behaviour
type: Class
subclass of: modeling and simulation algorithm characteristic
is organizational: true
definition: A characteristic describing the rules the algorithm uses to simulate the temporal evolution of a system, specifically whether or not the final state is uniquely determined from a precise initial state.
creator: AZ

type of progression time step
type: Class
subclass of: modeling and simulation algorithm characteristic
is organizational: true
creator: AZ
definition: Type of time steps used by the algorithm.

spatial description
type: Class
subclass of: modeling and simulation algorithm characteristic
created: 2008-07-08
definition: Algorithm, possessing this characteristic, takes into account the location of the reacting components.
creator: NLN

deterministic system behaviour
type: Class
subclass of: type of system behaviour
disjoint with: stochastic system behaviour
created: 2008-07-08
definition: Algorithm, possessing this characteristic, simulates the temporal evolution of a system deterministically, so that from a precise initial state the algorithm will always end up in the same final state.
creator: NLN

stochastic system behaviour
type: Class
subclass of: type of system behaviour
created: 2008-07-08
definition: Algorithm, possessing this characteristic, simulates the temporal evolution of a system using probabilistic rules, so that between two simulations, the same precise initial state may result in a different final state.
creator: NLN

discrete variable
type: Class
subclass of: type of variable
disjoint with: continuous variable
created: 2008-07-08
definition: Algorithm, possessing this characteristic, allows values of system's variables to change by discrete (integral) amounts.
creator: NLN

continuous variable
type: Class
subclass of: type of variable
created: 2008-07-08
definition: Algorithm, possessing this characteristic, allows the values of a system's variables to change by continuous (non-integral) amounts.
creator: NLN

progression with adaptive time step
type: Class
subclass of: type of progression time step
disjoint with: progression with fixed time step
created: 2008-07-08
definition: Algorithm, possessing this characteristic, does not use fixed time steps to update the state of a system during the whole simulation, but on the contrary adapts the length of the time steps to the local situation.
creator: NLN

progression with fixed time step
type: Class
subclass of: type of progression time step
created: 2008-07-08
definition: Algorithm, possessing this characteristic, uses time steps of constant length to update the state of a system during the whole simulation.
creator: NLN

modeling and simulation algorithm parameter
type: Class
subclass of: Restriction
  on property: is parameter of
  some values from: modeling and simulation algorithm
subclass of: Restriction
  on property: is parameter of
  all values from: modeling and simulation algorithm
is organizational: true
creator: AZ
definition: Parameter that can be used in the simulation experiment settings.

particle number lower limit
type: Class
subclass of: partitioning control parameter
subclass of: Restriction
  on property: has type
  some values from: double
subclass of: Restriction
  on property: is parameter of
  all values from: Pahle hybrid method
definition: This parameter of 'Pahle hybrid method' [http://identifiers.org/biomodels.kisao/KISAO_0000231] is a double value specifying the lower limit for particle numbers. Species with a particle number below this value are considered as having a low particle number. The 'particle number lower limit' cannot be higher than the 'particle number upper limit' [http://identifiers.org/biomodels.kisao/KISAO_0000204].

particle number upper limit
type: Class
subclass of: partitioning control parameter
subclass of: Restriction
  on property: has type
  some values from: double
subclass of: Restriction
  on property: is parameter of
  all values from: Pahle hybrid method
definition: This parameter of 'Pahle hybrid method' [http://identifiers.org/biomodels.kisao/KISAO_0000231] is a double value specifying the upper limit for particle numbers. Species with a particle number above this value are considered as having a high particle number. The 'particle number upper limit' cannot be lower than the 'particle number lower limit' [http://identifiers.org/biomodels.kisao/KISAO_0000203].

partitioning interval
type: Class
subclass of: partitioning control parameter
subclass of: Restriction
  on property: has type
  some values from: integer
subclass of: Restriction
  on property: is parameter of
  all values from: Class
    intersection of: modeling and simulation algorithm
    intersection of: Restriction
      on property: has characteristic
      some values from: hybridity
definition: This positive integer value specifies after how many steps the internal partitioning of the system should be recalculated.

relative tolerance
type: Class
subclass of: error control parameter
subclass of: Restriction
  on property: has type
  some values from: double
subclass of: Restriction
  on property: is parameter of
  all values from: Restriction
    on property: has characteristic
    some values from: approximate solution
alternative label: RTOL
definition: This parameter is a numeric value specifying the desired relative tolerance the user wants to achieve. A smaller value means that the trajectory is calculated more accurately.

Axiom
comment: EXACT
annotated target: RTOL
annotated source: relative tolerance
annotated property: altLabel

absolute tolerance
type: Class
subclass of: error control parameter
subclass of: Restriction
  on property: is parameter of
  all values from: Restriction
    on property: has characteristic
    some values from: approximate solution
subclass of: Restriction
  on property: has type
  some values from: double
alternative label: ATOL
definition: This parameter is a positive numeric value specifying the desired absolute tolerance the user wants to achieve.

Axiom
annotated target: ATOL
comment: EXACT
annotated source: absolute tolerance
annotated property: altLabel

integrate reduced model
type: Class
subclass of: method switching control parameter
subclass of: Restriction
  on property: is parameter of
  some values from: LSODA
subclass of: Restriction
  on property: has type
  some values from: boolean
definition: This parameter is a boolean value to determine whether the integration shall be performed using the mass conservation laws (true), i.e., reducing the number of system variables or to use the complete model (false).

LSODA maximum non-stiff order
type: Class
subclass of: method switching control parameter
subclass of: Restriction
  on property: has type
  some values from: integer
subclass of: Restriction
  on property: is parameter of
  all values from: LSODA
alternative label: Adams max order
definition: This parameter is a positive integer value specifying the maximal order the non-stiff Adams integration method [http://identifiers.org/biomodels.kisao/KISAO_0000289] shall attempt before switching to the stiff BDF method [http://identifiers.org/biomodels.kisao/KISAO_0000288].

Axiom
annotated target: Adams max order
comment: NARROW
annotated source: LSODA maximum non-stiff order
annotated property: altLabel

LSODA maximum stiff order
type: Class
subclass of: method switching control parameter
subclass of: Restriction
  on property: has type
  some values from: integer
subclass of: Restriction
  on property: is parameter of
  all values from: LSODA
alternative label: BDF max order
definition: This parameter is a positive integer value specifying the maximal order the stiff BDF integration method [http://identifiers.org/biomodels.kisao/KISAO_0000288] shall attempt before switching to smaller internal step sizes.

Axiom
annotated target: BDF max order
comment: NARROW
annotated source: LSODA maximum stiff order
annotated property: altLabel

number of history bins
type: Class
subclass of: granularity control parameter
subclass of: Restriction
  on property: has type
  some values from: integer
subclass of: Restriction
  on property: is parameter of
  some values from: Class
    intersection of: modeling and simulation algorithm
    intersection of: Restriction
      on property: has characteristic
      some values from: delay differential equation problem
definition: The 'number of history bins' is only enabled for models that contain delayed or multistep reactions for specifying the granularity with which the delayed reaction solver should retain the history of species values, for species that participate in delayed reactions.

tau-leaping epsilon
type: Class
subclass of: error control parameter
subclass of: Restriction
  on property: is parameter of
  all values from: tau-leaping method
subclass of: Restriction
  on property: has type
  some values from: double
definition: The leap condition is chosen such that the expected change in the propensity function aj(x) is bounded by Epsilon * a0 where Epsilon is an error control parameter between 0 and 1. This parameter is the basic error control mechanism for the Tau-Leaping algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000039]. As Epsilon decreases the leaps become shorter and the simulation is more accurate.
alternative label: epsilon
alternative label: tolerance

Axiom
comment: EXACT
annotated target: epsilon
annotated source: tau-leaping epsilon
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1063/1.1378322
comment: Gillespie DT. Approximate accelerated stochastic simulation of chemically reacting systems. The Journal of Chemical Physics, Vol. 115 (4), pages 1716-1733 (2001). Section V.
annotated source: tau-leaping epsilon
annotated property: seeAlso

minimum reactions per leap
type: Class
subclass of: method switching control parameter
subclass of: Restriction
  on property: is parameter of
  all values from: Class
    intersection of: Restriction
      on property: is hybrid of
      some values from: Gillespie direct algorithm
    intersection of: Restriction
      on property: is hybrid of
      some values from: tau-leaping method
subclass of: Restriction
  on property: has type
  some values from: integer
definition: 'minimum reactions per leap' parameter is used in hybrid methods, which adaptively switch between the tau-leaping algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000039] to the SSA Direct Method [http://identifiers.org/biomodels.kisao/KISAO_0000029] when the number of reactions in a single tau-leaping leap step is less than the threshold.
alternative label: threshold

Pahle hybrid method
type: Class
subclass of: hybrid method
subclass of: Restriction
  on property: has characteristic
  some values from: deterministic system behaviour
subclass of: Restriction
  on property: has characteristic
  some values from: hybridity
subclass of: Restriction
  on property: has parameter
  on class: particle number lower limit
  qualified cardinality: 1
subclass of: Restriction
  on property: has parameter
  on class: particle number upper limit
  qualified cardinality: 1
subclass of: Restriction
  on property: is hybrid of
  some values from: Gibson-Bruck next reaction algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: stochastic system behaviour
subclass of: Restriction
  on property: is hybrid of
  some values from: Class
    intersection of: modeling and simulation algorithm
    intersection of: Restriction
      on property: has characteristic
      some values from: ordinary differential equation problem
creator: AZ
is implemented in: COPASI
definition: The hybrid method combines the stochastic 'Gibson-Bruck's next reaction method' [http://identifiers.org/biomodels.kisao/KISAO_0000027] with different algorithms for the numerical integration of ODEs [http://identifiers.org/biomodels.kisao/KISAO_0000245 some http://identifiers.org/biomodels.kisao/KISAO_0000374]. The biochemical network is dynamically partitioned into a deterministic and a stochastic subnet depending on the current particle numbers in the system. The user can define limits for when a particle number should be considered low or high. The stochastic subnet contains reactions involving low numbered species as substrate or product. All the other reactions form the deterministic subnet. The two subnets are then simulated in parallel using the stochastic and deterministic solver, respectively. The reaction probabilities in the stochastic subnet are approximated as constant between two stochastic reaction events.

Axiom
annotated target: http://identifiers.org/doi/10.1093/bioinformatics/btl485
comment: COPASI--a COmplex PAthway SImulator. Hoops S, Sahle S, Gauges R, Lee C, Pahle J, Simus N, Singhal M, Xu L, Mendes P, Kummer U. Bioinformatics. 2006 Dec 15;22(24):3067-74. Epub 2006 Oct 10.
annotated source: Pahle hybrid method
annotated property: seeAlso

LSOIBT
type: Class
subclass of: Livermore solver
subclass of: Restriction
  on property: is similar to
  some values from: LSODE
subclass of: Restriction
  on property: has characteristic
  some values from: implicit method type
definition: LSOIBT solves linearly implicit systems in which the matrices involved are all assumed to be block-tridiagonal. Linear systems are solved by the LU method.
alternative label: Livermore solver for ordinary differential equations given in implicit form, with block-tridiagonal Jacobian treatment

Axiom
comment: EXACT
annotated target: Livermore solver for ordinary differential equations given in implicit form, with block-tridiagonal Jacobian treatment
annotated source: LSOIBT
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/isbn/978-0444866073
comment: A. C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman et al. (eds.), North-Holland, Amsterdam, 1983 (vol. 1 of IMACS Transactions on Scientific Computation), pp. 55-64.
annotated source: LSOIBT
annotated property: seeAlso

LSODES
type: Class
subclass of: Livermore solver
subclass of: Restriction
  on property: is similar to
  some values from: LSODE
definition: LSODES solves systems dy/dt = f and in the stiff case treats the Jacobian matrix in general sparse form. It determines the sparsity structure on its own, or optionally accepts this information from the user. It then uses parts of the Yale Sparse Matrix Package (YSMP) to solve the linear systems that arise, by a sparse (direct) LU factorization/backsolve method.
alternative label: Livermore solver for ordinary differential equations with general sparse Jacobian matrix

Axiom
comment: EXACT
annotated target: Livermore solver for ordinary differential equations with general sparse Jacobian matrix
annotated source: LSODES
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/isbn/978-0444866073
comment: A. C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman et al. (eds.), North-Holland, Amsterdam, 1983 (vol. 1 of IMACS Transactions on Scientific Computation), pp. 55-64.
annotated source: LSODES
annotated property: seeAlso

LSODKR
type: Class
subclass of: Livermore solver
subclass of: Restriction
  on property: uses
  some values from: Krylov subspace projection method
subclass of: Restriction
  on property: is similar to
  some values from: LSODPK
subclass of: Restriction
  on property: is similar to
  some values from: LSODE
definition: LSODKR is an initial value ODE solver for stiff and nonstiff systems. It is a variant of the LSODPK [http://identifiers.org/biomodels.kisao/KISAO_0000093] and LSODE [http://identifiers.org/biomodels.kisao/KISAO_0000071] solvers, intended mainly for large stiff systems. The main differences between LSODKR and LSODE [http://identifiers.org/biomodels.kisao/KISAO_0000071] are the following: a) for stiff systems, LSODKR uses a corrector iteration composed of Newton iteration and one of four preconditioned Krylov subspace iteration methods. The user must supply routines for the preconditioning operations, b) within the corrector iteration, LSODKR does automatic switching between functional (fixpoint) iteration and modified Newton iteration, c) LSODKR includes the ability to find roots of given functions of the solution during the integration.
alternative label: Livermore solver for ordinary differential equations, with preconditioned Krylov iteration methods for the Newton correction linear systems, and with root finding.

Axiom
annotated target: http://identifiers.org/isbn/978-0444866073
comment: A. C. Hindmarsh, ODEPACK, A Systematized Collection of ODE Solvers, in Scientific Computing, R. S. Stepleman et al. (eds.), North-Holland, Amsterdam, 1983 (vol. 1 of IMACS Transactions on Scientific Computation), pp. 55-64.
annotated source: LSODKR
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: Livermore solver for ordinary differential equations, with preconditioned Krylov iteration methods for the Newton correction linear systems, and with root finding.
annotated source: LSODKR
annotated property: altLabel

type of solution
type: Class
subclass of: modeling and simulation algorithm characteristic
is organizational: true
definition: A characteristic describing the type of the solution produced by the method, specifically whether it is exact or approximate.
creator: AZ

exact solution
type: Class
subclass of: type of solution
disjoint with: approximate solution
definition: Algorithm, possessing this characteristic, provides an exact solution to the initial problem.

approximate solution
type: Class
subclass of: type of solution
definition: Approximation algorithms are algorithms used to find approximate solutions to optimization problems. Approximation algorithms are often associated with NP-hard problems; since it is unlikely that there can ever be efficient polynomial time exact algorithms solving NP-hard problems, one settles for polynomial time sub-optimal solutions. Unlike heuristics, which usually only find reasonably good solutions reasonably fast, one wants provable solution quality and provable run time bounds. Ideally, the approximation is optimal up to a small constant factor (for instance within 5% of the optimal solution). Approximation algorithms are increasingly being used for problems where exact polynomial-time algorithms are known but are too expensive due to the input size.

type of method
type: Class
subclass of: modeling and simulation algorithm characteristic
is organizational: true
definition: A characteristic describing the way the method finds a solution, specifically whether it solves an equation involving only the current state of the system (explicit) or both the current and the later one (implicit).
creator: AZ

explicit method type
type: Class
subclass of: type of method
disjoint with: implicit method type
creator: AZ
definition: Explicit methods calculate the state of a system at a later time from the state of the system at the current time. Mathematically, if Y(t) is the current system state and Y((t+delta t) is the state at the later time (delta t is a small time step), then, for an explicit method Y(t+delta t) = F(Y(t)), to find Y(t+delta t).

implicit method type
type: Class
subclass of: type of method
creator: AZ
definition: Implicit methods find a solution by solving an equation involving both the current state of the system and the later one. Mathematically, if Y(t) is the current system state and Y((t+delta t) is the state at the later time (delta t is a small time step), then, for an implicit method one solves an equation G(Y(t), Y(t+delta t))=0, to find Y(t+delta t).

Gillespie-like method
type: Class
subclass of: Monte Carlo method
subclass of: Restriction
  on property: has characteristic
  some values from: progression with adaptive time step
subclass of: Restriction
  on property: has characteristic
  some values from: stochastic system behaviour
subclass of: Restriction
  on property: has characteristic
  some values from: discrete variable
is organizational: true
definition: Stochastic simulation algorithm using an approach alike the one described in Gillespie's papers of 1976 and 1977.

error control parameter
type: Class
subclass of: modeling and simulation algorithm parameter
is organizational: true
creator: AZ
definition: Parameter controlling method accuracy.

method switching control parameter
type: Class
subclass of: modeling and simulation algorithm parameter
subclass of: Restriction
  on property: is parameter of
  all values from: Restriction
    on property: uses
    some values from: modeling and simulation algorithm
is organizational: true
creator: AZ
definition: Parameters describing threshold conditions for algorithms that switch between different methods.

granularity control parameter
type: Class
subclass of: modeling and simulation algorithm parameter
is organizational: true
creator: AZ
definition: Parameter controlling granularity.

tau-leaping delta
type: Class
subclass of: partitioning control parameter
subclass of: Restriction
  on property: is parameter of
  all values from: implicit tau-leaping method
subclass of: Restriction
  on property: has type
  some values from: double
definition: Tau-leaping delta specifies how close two symmetric transition rates must be before we classify them as in partial-equilibrium. Only applies to the implicit tau routine [http://identifiers.org/biomodels.kisao/KISAO_0000045].

critical firing threshold
type: Class
subclass of: partitioning control parameter
subclass of: Restriction
  on property: is parameter of
  all values from: nonnegative Poisson tau-leaping method
subclass of: Restriction
  on property: has type
  some values from: integer
definition: The 'nonnegative Poisson tau-leaping method' [http://identifiers.org/biomodels.kisao/KISAO_0000084] is based on the fact that negative populations typically arise from multiple firings of reactions that are only a few firings away from consuming all the molecules of one of their reactants. To focus on those reaction channels, the modified tau-leaping algorithm introduces a second control parameter nc, a positive integer that is usually set somewhere between 5 and 20. Any reaction channel with a positive propensity function that is currently within nc firings of exhausting one of its reactants is then classified as a critical reaction. The modified algorithm chooses tau in such a way that no more than one firing of all the critical reactions can occur during the leap.
alternative label: nonnegative tau-leaping second control parameter

Axiom
comment: EXACT
annotated target: nonnegative tau-leaping second control parameter
annotated source: critical firing threshold
annotated property: altLabel

partitioning control parameter
type: Class
subclass of: modeling and simulation algorithm parameter
is organizational: true
creator: Parameter describing partitioning of the system.

coarse-graining factor
type: Class
subclass of: granularity control parameter
subclass of: Restriction
  on property: is parameter of
  all values from: binomial tau-leaping method
subclass of: Restriction
  on property: has type
  some values from: double
definition: The time in each Monte-Carlo iteration of 'binomial tau-leaping method' [http://identifiers.org/biomodels.kisao/KISAO_0000074] is updated with the time increments tau=f/(a1+a2+...+aM). Here 1/(a1+a2+...+aM) is the averaged microscopic increment of the SSA [http://identifiers.org/biomodels.kisao/KISAO_0000029] and f is a coarse-graining factor, controlling the speed-up.

Axiom
annotated target: http://identifiers.org/pubmed/15638577
comment: Chatterjee A, Vlachos DG, Katsoulakis MA. Binomial distribution based tau-leap accelerated stochastic simulation. J Chem Phys. 2005;122(2):024112.
annotated source: coarse-graining factor
annotated property: seeAlso

Brownian diffusion accuracy
type: Class
subclass of: error control parameter
subclass of: Restriction
  on property: has type
  some values from: integer
subclass of: Restriction
  on property: is parameter of
  all values from: Brownian diffusion Smoluchowski method
definition: Accuracy code of 'Brownian diffusion Smoluchowski method' [http://identifiers.org/biomodels.kisao/KISAO_0000057], which sets which neighbouring boxes are checked for potential bi-molecular reactions. Consider the reaction A + B -> C and suppose that A and B are within a binding radius of each other. This reaction will always be performed if A and B are in the same virtual box. If accuracy is set to at least 3, then it will also occur if A and B are in nearest-neighbour virtual boxes. If it is at least 7, then the reaction will happen if they are in nearest-neighbour boxes that are separated by periodic boundary conditions. And if it is 9 or 10, then all edge and corner boxes are checked for reactions, which means that no potential reactions are overlooked.
is implemented in: Smoldyn

molecules per virtual box
type: Class
subclass of: virtual box size
subclass of: Restriction
  on property: has type
  some values from: integer
disjoint with: virtual box side length
is implemented in: Smoldyn
definition: Target molecules per virtual box is a parameter of 'Brownian diffusion Smoluchowski method' [http://identifiers.org/biomodels.kisao/KISAO_0000057], which sets the box sizes so that the average number of molecules per box, at simulation initiation, is close to the requested number.

virtual box side length
type: Class
subclass of: virtual box size
subclass of: Restriction
  on property: has type
  some values from: double
is implemented in: Smoldyn
definition: The 'virtual box side length' is a parameter of 'Brownian diffusion Smoluchowski method' [http://identifiers.org/biomodels.kisao/KISAO_0000057]. It requests the length of one side of a box.

surface-bound epsilon
type: Class
subclass of: partitioning control parameter
subclass of: Restriction
  on property: has type
  some values from: double
subclass of: Restriction
  on property: is parameter of
  all values from: Brownian diffusion Smoluchowski method
definition: A parameter of 'Brownian diffusion Smoluchowski method' [http://identifiers.org/biomodels.kisao/KISAO_0000057]. Molecules that are bound to a surface are given locations that are extremely close to that surface. However, this position does not need to be exactly at the surface, and in fact it usually cannot be exactly at the surface due to round-off error. The tolerance for how far a surface-bound molecule is allowed to be away from the surface can be set with the epsilon statement.

neighbour distance
type: Class
subclass of: partitioning control parameter
subclass of: Restriction
  on property: is parameter of
  all values from: Brownian diffusion Smoluchowski method
subclass of: Restriction
  on property: has type
  some values from: double
definition: A parameter of 'Brownian diffusion Smoluchowski method' [http://identifiers.org/biomodels.kisao/KISAO_0000057]. When a surface-bound molecule diffuses off of one surface panel, it can sometimes diffuse onto the neighbouring surface tile. It does so only if the neighbouring panel is declared to be a neighbour and also the neighbour is within a distance that is set with the neighbour distance statement.

virtual box size
type: Class
subclass of: partitioning control parameter
subclass of: Restriction
  on property: is parameter of
  all values from: Brownian diffusion Smoluchowski method
definition: Target size of virtual boxes for 'Brownian diffusion Smoluchowski method' [http://identifiers.org/biomodels.kisao/KISAO_0000057].

Euler method
type: Class
subclass of: one-step method
subclass of: Restriction
  on property: has characteristic
  some values from: progression with fixed time step
subclass of: Restriction
  on property: has characteristic
  some values from: ordinary differential equation problem
subclass of: Restriction
  on property: has characteristic
  some values from: deterministic system behaviour
subclass of: Restriction
  on property: has characteristic
  some values from: continuous variable
subclass of: Restriction
  on property: has characteristic
  some values from: linearity of equation
subclass of: Restriction
  on property: is similar to
  some values from: Class
    intersection of: Runge-Kutta based method
    intersection of: Restriction
      on property: has Runge-Kutta method order
      has value: 1
creator: AZ
is implemented in: ByoDyn
is implemented in: JSim
definition: The Euler method, named after Leonhard Euler, is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value.

Axiom
annotated target: http://identifiers.org/isbn/052143064X
comment: Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes in Fortran 77. Cambridge University Press (2001).
annotated source: Euler method
annotated property: seeAlso

NFSim agent-based simulation method
type: Class
subclass of: multi-state agent-based simulation method
subclass of: Restriction
  on property: is generalization of
  some values from: Gillespie direct algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: exact solution
disjoint with: implicit-state Doob-Gillespie algorithm
created: 2011-04-07
definition: A generalization a rule-based version of 'Gillespie's direct method' (SSA) [http://identifiers.org/biomodels.kisao/KISAO_0000029]. The method is guaranteed to produce the same results as the exact SSA [http://identifiers.org/biomodels.kisao/KISAO_0000029] by cycling over three primary steps. First, NFsim calculates the probability or propensity for each rule to take effect given the current molecular states. Second, it samples the time to the next reaction event and selects the corresponding reaction rule. Finally, NFsim executes the selected reaction by applying the rule and updating the molecular agents accordingly.
creator: AZ
is implemented in: NFSim

Axiom
annotated target: http://identifiers.org/doi/10.1038/nmeth.1546
comment: Sneddon MW, Faeder JR and Emonet T. Efficient modelling, simulation and coarse-graining of biological complexity with NFsim. Nature Methods (2011) 8(2):177-83.
annotated source: NFSim agent-based simulation method
annotated property: seeAlso

cellular automata update method
type: Class
subclass of: rule-based simulation method
subclass of: Restriction
  on property: has characteristic
  some values from: spatial description
subclass of: Restriction
  on property: has characteristic
  some values from: progression with fixed time step
created: 2011-04-07
creator: AZ
alternative label: CA
definition: Cellular automata are mathematical idealizations of physical systems in which space and time are discrete, and physical quantities take on a finite set of discrete values. A cellular automaton consists of a regular uniform lattice (or ''array''), usually infinite in extent, with a discrete variable at each site (''cell''). A cellular automaton evolves in discrete time steps, with the value of the variable at one site being affected by the values of variables at sites in its ''neighbourhood'' on the previous time step. The neighbourhood of a site is typically taken to be the site itself and all immediately adjacent sites. The variables at each site are updated simultaneously (''synchronously''), based on the values of the variables in their neighbourhood at the preceding time step, and according to a definite set of ''local rules''.
alternative label: cellular automata
alternative label: cellular spaces
alternative label: cellular structures
alternative label: homogeneous structures
alternative label: iterative arrays
alternative label: tessellation automata
alternative label: tessellation structures

Axiom
comment: EXACT
annotated target: tessellation structures
annotated source: cellular automata update method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: cellular automata
annotated source: cellular automata update method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: cellular structures
annotated source: cellular automata update method
annotated property: altLabel

Axiom
annotated target: CA
comment: EXACT
annotated source: cellular automata update method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: cellular spaces
annotated source: cellular automata update method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: iterative arrays
annotated source: cellular automata update method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: tessellation automata
annotated source: cellular automata update method
annotated property: altLabel

Axiom
comment: Wolfram, Stephen (1983) Statistical mechanics of cellular automata. Reviews of Modern Physics 55 (3): 601-644.
annotated target: http://identifiers.org/doi/10.1103/RevModPhys.55.601
annotated source: cellular automata update method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: homogeneous structures
annotated source: cellular automata update method
annotated property: altLabel

hard-particle molecular dynamics
type: Class
subclass of: modeling and simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: progression with adaptive time step
subclass of: Restriction
  on property: has characteristic
  some values from: spatial description
created: 2011-05-05
definition: A collision-driven molecular dynamics algorithm for a system of non-spherical particles.
creator: AZ

Axiom
annotated target: http://identifiers.org/doi/10.1016/j.jcp.2004.08.014
comment: Aleksandar Donev, Salvatore Torquato, Frank H. Stillinger, Neighbor list collision-driven molecular dynamics simulation for nonspherical hard particles. I. Algorithmic details, Journal of Computational Physics, v.202 n.2, p.737-764, 20 January 2005
annotated source: hard-particle molecular dynamics
annotated property: seeAlso

first-passage Monte Carlo algorithm
type: Class
subclass of: Monte Carlo method
subclass of: Restriction
  on property: has characteristic
  some values from: spatial description
subclass of: Restriction
  on property: has characteristic
  some values from: stochastic system behaviour
subclass of: Restriction
  on property: has characteristic
  some values from: exact solution
subclass of: Restriction
  on property: has characteristic
  some values from: progression with adaptive time step
created: 2011-05-05
alternative label: AED DKMC
alternative label: AED diffusion kinetic Monte Carlo method
creator: AZ
definition: We present a novel Monte Carlo algorithm for N diffusing finite particles that react on collisions. Using the theory of first-passage processes and time dependent Green's functions, we break the difficult N-body problem into independent single- and two-body propagations circumventing numerous diffusion hops used in standard Monte Carlo simulations. The new algorithm is exact, extremely efficient, and applicable to many important physical situations in arbitrary integer dimensions.
alternative label: asynchronous event-driven diffusion Monte Carlo

Axiom
annotated target: AED diffusion kinetic Monte Carlo method
comment: EXACT
annotated source: first-passage Monte Carlo algorithm
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1103/PhysRevLett.97.230602
comment: T. Oppelstrup, V. V. Bulatov, G. H. Gilmer, M. H. Kalos, and B. Sadigh. First-passage Monte- Carlo algorithm: Diffusion without all the hops. Physical Review Letters, 97(23):230602, 2006.
annotated source: first-passage Monte Carlo algorithm
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: asynchronous event-driven diffusion Monte Carlo
annotated source: first-passage Monte Carlo algorithm
annotated property: altLabel

Axiom
annotated target: AED DKMC
comment: EXACT
annotated source: first-passage Monte Carlo algorithm
annotated property: altLabel

Gill method
type: Class
subclass of: Runge-Kutta based method
subclass of: Restriction
  on property: has characteristic
  some values from: progression with fixed time step
subclass of: Restriction
  on property: has Runge-Kutta method order
  has value: 4
subclass of: Restriction
  on property: has characteristic
  some values from: explicit method type
created: 2011-05-09
creator: AZ
definition: Gill's fourth order method is a Runge-Kutta method for approximating the solution of the initial value problem y'(x) = f(x,y); y(x0) = y0 which evaluates the integrand,f(x,y), four times per step. This method is a fourth order procedure for which Richardson extrapolation can be used.
alternative label: Gill's method
alternative label: Runge-Kutta-Gill method

Axiom
annotated target: http://identifiers.org/doi/10.1017/S0305004100026414
comment: Gill, S. 1951. A process for the step-by-step integration of differential equations in an automatic digital computing machine. Proc. Cambridge Philos. Soc., 47, pp 96-108.
annotated source: Gill method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: Gill's method
annotated source: Gill method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: Runge-Kutta-Gill method
annotated source: Gill method
annotated property: altLabel

Metropolis Monte Carlo algorithm
type: Class
subclass of: Monte Carlo method
subclass of: Restriction
  on property: has characteristic
  some values from: approximate solution
subclass of: Restriction
  on property: has characteristic
  some values from: stochastic system behaviour
subclass of: Restriction
  on property: has characteristic
  some values from: spatial description
subclass of: Restriction
  on property: has characteristic
  some values from: progression with fixed time step
created: 2011-05-09
definition: A general method, suitable for fast computing machines, for investigating such properties as equations of state for substances consisting of interacting individual molecules is described. The method consists of a modified Monte Carlo integration [http://identifiers.org/biomodels.kisao/KISAO_0000051] over configuration space.
creator: AZ
is implemented in: CompuCell3D
alternative label: Metropolis algorithm
alternative label: Metropolis-Hastings algorithm

Axiom
comment: EXACT
annotated target: Metropolis-Hastings algorithm
annotated source: Metropolis Monte Carlo algorithm
annotated property: altLabel

Axiom
comment: EXACT
annotated target: Metropolis algorithm
annotated source: Metropolis Monte Carlo algorithm
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1063/1.1699114
comment: Metropolis, N.; Rosenbluth, A.W.; Rosenbluth, M.N.; Teller, A.H.; Teller, E. (1953). Equations of State Calculations by Fast Computing Machines. Journal of Chemical Physics 21 (6): 1087-1092.
annotated source: Metropolis Monte Carlo algorithm
annotated property: seeAlso

Adams-Bashforth method
type: Class
subclass of: Adams method
subclass of: Restriction
  on property: has characteristic
  some values from: explicit method type
created: 2011-05-09
creator: AZ
definition: Given an initial value problem: y' = f(x,y), y(x0) = y0 together with additional starting values y1 = y(x0 + h), . . . , yk-1 = y(x0 + (k-1) h) the k-step Adams-Bashforth method is an explicit linear multistep method that approximates the solution, y(x) at x = x0+kh, of the initial value problem by yk = yk - 1 + h * ( a0 f(xk - 1,yk - 1) + a1 f(xk - 2,yk - 2) + . . . + ak - 1 f(x0,y0) ), where a0, a1, . . . , ak - 1 are constants.
alternative label: explicit Adams method

Axiom
annotated target: http://identifiers.org/isbn/978-3-540-56670-0
comment: Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems (2nd ed.), Berlin: Springer Verlag
annotated source: Adams-Bashforth method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: explicit Adams method
annotated source: Adams-Bashforth method
annotated property: altLabel

Adams-Moulton method
type: Class
subclass of: Adams method
subclass of: Restriction
  on property: has characteristic
  some values from: implicit method type
created: 2011-05-09
creator: AZ
definition: The (k-1)-step Adams-Moulton method is an implicit linear multistep method that iteratively approximates the solution, y(x) at x = x0+kh, of the initial value problem by yk = yk - 1 + h * ( b0 f(xk,yk) + b1 f(xk - 1,yk - 1) + . . . + bk - 1 f(x1,y1) ), where b1, . . . , bk - 1 are constants.
is implemented in: VCell
alternative label: implicit Adams method

Axiom
comment: EXACT
annotated target: implicit Adams method
annotated source: Adams-Moulton method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/isbn/978-3-540-56670-0
comment: Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems (2nd ed.), Berlin: Springer Verlag
annotated source: Adams-Moulton method
annotated property: seeAlso

multistep method
type: Class
subclass of: modeling and simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: continuous variable
subclass of: Class
  complement of: Restriction
    on property: has characteristic
    some values from: spatial description
subclass of: Restriction
  on property: has characteristic
  some values from: approximate solution
subclass of: Restriction
  on property: has characteristic
  some values from: progression with fixed time step
created: 2011-05-09
is organizational: true
definition: A numerical method for differential equations which is based on several values of the solution.
creator: AZ
alternative label: multi-value method

Axiom
annotated target: http://identifiers.org/isbn/978-3-540-56670-0
comment: Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems (2nd ed.), Berlin: Springer Verlag
annotated source: multistep method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: multi-value method
annotated source: multistep method
annotated property: altLabel

KINSOL
type: Class
subclass of: Newton-type method
subclass of: Class
  complement of: Restriction
    on property: has characteristic
    some values from: linearity of equation
subclass of: Restriction
  on property: uses
  some values from: Krylov subspace projection method
subclass of: Restriction
  on property: has characteristic
  some values from: differential-algebraic equation problem
created: 2011-05-09
creator: AZ
alternative label: FKINSOL
definition: KINSOL solves algebraic systems in real N-space, written as F(u)=0, F:RN->RN, given an initial guess u0. The basic method is either a modified or an inexact Newton iteration [http://identifiers.org/biomodels.kisao/KISAO_0000408]. The linear systems that arise are solved with either a direct (dense or banded) solver (serial version only), or one of the Krylov iterative solvers [http://identifiers.org/biomodels.kisao/KISAO_0000354]. In the Krylov case, the user can (optionally) supply a right preconditioner.
alternative label: NKSOL
alternative label: Newton-Krylov solver for nonlinear algebraic systems
is implemented in: SUNDIALS

Axiom
annotated target: Newton-Krylov solver for nonlinear algebraic systems
comment: RELATED
annotated source: KINSOL
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.4249/scholarpedia.2860
comment: Alan C. Hindmarsh and Radu Serban (2007) Sundails equation solvers, Scholarpedia, 2(3):2860.
annotated source: KINSOL
annotated property: seeAlso

Axiom
annotated target: http://identifiers.org/doi/10.1145/1089014.1089020
comment: Hindmarsh, A. C., et al., SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers, ACM Trans. Math. Softw., 31:363-396, 2005.
annotated source: KINSOL
annotated property: seeAlso

Axiom
annotated target: NKSOL
comment: RELATED
annotated source: KINSOL
annotated property: altLabel

Axiom
annotated target: FKINSOL
comment: RELATED
annotated source: KINSOL
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1137/0911026
comment: BROWN, P. N. AND SAAD, Y. 1990. Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Stat. Comput. 11, 450-481.
annotated source: KINSOL
annotated property: seeAlso

IDA
type: Class
subclass of: IDA-like method
subclass of: Class
  complement of: Restriction
    on property: has characteristic
    some values from: linearity of equation
subclass of: Restriction
  on property: has characteristic
  some values from: differential-algebraic equation problem
subclass of: Restriction
  on property: is similar to
  some values from: DASPK
subclass of: Restriction
  on property: uses
  some values from: Krylov subspace projection method
subclass of: Restriction
  on property: uses
  some values from: backward differentiation formula
subclass of: Restriction
  on property: uses
  some values from: scaled preconditioned generalized minimal residual method
created: 2011-05-09
creator: AZ
definition: IDA solves real differential-algebraic systems in N-space, in the general form F(t,y,y')=0, y(t0)=y0, y'(t0)=y'0. At each step, a Newton iteration [http://identifiers.org/biomodels.kisao/KISAO_0000408] leads to linear systems Jx=b, which are solved by one of five methods - two direct (dense or band; serial version only) and three Krylov [http://identifiers.org/biomodels.kisao/KISAO_0000354] (GMRES [http://identifiers.org/biomodels.kisao/KISAO_0000353], BiCGStab [http://identifiers.org/biomodels.kisao/KISAO_0000392], or TFQMR [http://identifiers.org/biomodels.kisao/KISAO_0000396]). IDA is written in C, but derived from the package DASPK [http://identifiers.org/biomodels.kisao/KISAO_0000355] which is written in Fortran.
is implemented in: SUNDIALS
is implemented in: VCell
alternative label: implicit differential-algebraic solver
alternative label: solver for differential-algebraic equation systems

Axiom
annotated target: http://identifiers.org/doi/10.4249/scholarpedia.2860
comment: Alan C. Hindmarsh and Radu Serban (2007) Sundails equation solvers, Scholarpedia, 2(3):2860.
annotated source: IDA
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: implicit differential-algebraic solver
annotated source: IDA
annotated property: altLabel

Axiom
comment: RELATED
annotated target: solver for differential-algebraic equation systems
annotated source: IDA
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1137/0915088
comment: BROWN, P. N., HINDMARSH, A. C., AND PETZOLD, L. R. 1994. Using Krylov methods in the solution of large-scale differential-algebraic systems. SIAM J. Sci. Comput. 15, 1467-1488.
annotated source: IDA
annotated property: seeAlso

Axiom
annotated target: http://identifiers.org/doi/10.1145/1089014.1089020
comment: Hindmarsh, A. C., et al., SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers, ACM Trans. Math. Softw., 31:363-396, 2005.
annotated source: IDA
annotated property: seeAlso

Axiom
comment: Alan C. Hindmarsh and Radu Serban (2007) Sundails equation solvers, Scholarpedia, 2(3):2860.
annotated target: IDA
annotated source: IDA
annotated property: label

finite volume method
type: Class
subclass of: partial differential equation discretization method
subclass of: Restriction
  on property: has characteristic
  some values from: mesh-based
created: 2011-05-09
creator: AZ
alternative label: FVM
definition: The finite volume method is a method for representing and evaluating partial differential equations in the form of algebraic equations, which attempts to emulate continuous conservation laws of physics.
is implemented in: VCell

Axiom
comment: EXACT
annotated target: FVM
annotated source: finite volume method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/isbn/0898715342
comment: Y. Saad. 2003. Iterative Methods for Sparse Linear Systems (2nd ed.). Soc. for Industrial and Applied Math., Philadelphia, PA, USA.
annotated source: finite volume method
annotated property: seeAlso

Euler-Maruyama method
type: Class
subclass of: one-step method
subclass of: Restriction
  on property: has characteristic
  some values from: continuous variable
subclass of: Restriction
  on property: is generalization of
  some values from: Euler method
subclass of: Restriction
  on property: has characteristic
  some values from: implicit method type
subclass of: Restriction
  on property: has characteristic
  some values from: stochastic differential equation problem
subclass of: Restriction
  on property: has characteristic
  some values from: stochastic system behaviour
subclass of: Restriction
  on property: has characteristic
  some values from: progression with fixed time step
created: 2011-05-09
creator: AZ
definition: The Euler-Maruyama method is a method for the approximate numerical solution of a stochastic differential equation, which truncates the Ito and Stratonovich Taylor series of the exact solution after the first order stochastic terms. This converges to the Ito solution with strong global order accuracy 1/2 or weak global order accuracy 1. It is a simple generalization of the Euler method [http://identifiers.org/biomodels.kisao/KISAO_0000261] for ordinary differential equations to stochastic differential equations.
alternative label: stochastic Euler scheme

Axiom
comment: EXACT
annotated target: stochastic Euler scheme
annotated source: Euler-Maruyama method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1098/rspa.2003.1247
comment: Burrage, K., Burrage, P.M. and Tian, T. (2004) Numerical methods for strong solutions of stochastic differential equations: an overview. Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences, 460 (2041). pp. 373-402.
annotated source: Euler-Maruyama method
annotated property: seeAlso

Milstein method
type: Class
subclass of: multistep method
subclass of: Restriction
  on property: has characteristic
  some values from: stochastic differential equation problem
subclass of: Restriction
  on property: has characteristic
  some values from: stochastic system behaviour
subclass of: Restriction
  on property: has characteristic
  some values from: implicit method type
created: 2011-05-10
creator: AZ
definition: The Milstein method is a technique for the approximate numerical solution of a stochastic differential equation.

Axiom
annotated target: http://identifiers.org/isbn/079233213X
comment: Milstein M. G. Numerical Integration of Stochastic Differential Equations (Mathematics and Its Applications) Springer, 2004.
annotated source: Milstein method
annotated property: seeAlso

backward differentiation formula
type: Class
subclass of: multistep method
subclass of: Restriction
  on property: has characteristic
  some values from: implicit method type
subclass of: Restriction
  on property: has characteristic
  some values from: linearity of equation
subclass of: Restriction
  on property: has characteristic
  some values from: ordinary differential equation problem
subclass of: Restriction
  on property: has characteristic
  some values from: deterministic system behaviour
disjoint with: Adams method
created: 2011-05-10
creator: AZ
alternative label: BDF
is implemented in: ByoDyn
is implemented in: GSL
alternative label: Gear method
alternative label: Gear's method
definition: The backward differentiation formulas (BDF) are implicit multistep methods based on the numerical differentiation of a given function and are wildly used for integration of stiff differential equations.
is implemented in: iBioSim

Axiom
comment: EXACT
annotated target: Gear's method
annotated source: backward differentiation formula
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/isbn/0136266061
comment: C. William Gear. 1971. Numerical Initial Value Problems in Ordinary Differential Equations. Prentice Hall PTR, Upper Saddle River, NJ, USA.
annotated source: backward differentiation formula
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: Gear method
annotated source: backward differentiation formula
annotated property: altLabel

Axiom
annotated target: BDF
comment: EXACT
annotated source: backward differentiation formula
annotated property: altLabel

Adams method
type: Class
subclass of: multistep method
subclass of: Restriction
  on property: has characteristic
  some values from: deterministic system behaviour
subclass of: Restriction
  on property: has characteristic
  some values from: linearity of equation
subclass of: Restriction
  on property: has characteristic
  some values from: ordinary differential equation problem
created: 2011-05-10
creator: AZ
definition: Adams' methods are multi-step methods used for the numerical integration of initial value problems in Ordinary Differential Equations (ODE's). Adams' algorithm consists of two parts: firstly, a starting procedure which provides y1, ... , yk-1 ( approximations to the exact solution at the points x0 + h, ... , x0 + (k - 1)h ) and, secondly, a multistep formula to obtain an approximation to the exact solution y(x0 + kh). This is then applied recursively, based on the numerical approximation of k successive steps, to compute y(x0 + (k + 1)h).
is implemented in: ByoDyn

Axiom
annotated target: http://identifiers.org/isbn/978-3-540-56670-0
comment: Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems (2nd ed.), Berlin: Springer Verlag
annotated source: Adams method
annotated property: seeAlso

Merson method
type: Class
subclass of: embedded Runge-Kutta method
subclass of: Restriction
  on property: has Runge-Kutta method order
  has value: 4
created: 2011-05-10
definition: A five-stage Runge-Kutta method with fourth-order accuracy.
creator: AZ
is implemented in: JSim
alternative label: KM
alternative label: Kutta-Merson method
alternative label: Merson's method
alternative label: Runge-Kutta-Merson method

Axiom
comment: EXACT
annotated target: KM
annotated source: Merson method
annotated property: altLabel

Axiom
annotated target: http://nla.gov.au/nla.cat-vn870866
comment: R.H. Merson, An operational method for the study of integration processes, in: Proceedings of the Symposium on Data Processing, Weapons Research Establishment, Salisbury, Australia, 1957, pp. 110-1-110-25.
annotated source: Merson method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: Merson's method
annotated source: Merson method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: Runge-Kutta-Merson method
annotated source: Merson method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: Kutta-Merson method
annotated source: Merson method
annotated property: altLabel

Hammer-Hollingsworth method
type: Class
subclass of: Runge-Kutta based method
subclass of: Restriction
  on property: has characteristic
  some values from: progression with fixed time step
subclass of: Restriction
  on property: has characteristic
  some values from: implicit method type
created: 2011-05-10
creator: AZ
definition: The numerical integration of ordinary differential equations by the use of Gaussian quadrature methods.

Axiom
annotated target: urn:issn:0891-6837
comment: P. C. Hammer & J. W. Hollingsworth, Trapezoidal methods of approximating solutions of differential equations, MTAC, v. 9, 1955, pp. 92-96. MR 17, 302.
annotated source: Hammer-Hollingsworth method
annotated property: seeAlso

Lobatto method
type: Class
subclass of: Runge-Kutta based method
subclass of: Class
  complement of: Restriction
    on property: has characteristic
    some values from: symplecticness
subclass of: Restriction
  on property: has characteristic
  some values from: progression with fixed time step
subclass of: Restriction
  on property: has characteristic
  some values from: implicit method type
created: 2011-05-10
creator: AZ
definition: There are three families of Lobatto methods, called IIIA, IIIB and IIIC. These are named after Rehuel Lobatto. All are implicit Runge-Kutta methods, have order 2s − 2 and they all have c1 = 0 and cs = 1.
alternative label: implicit Runge-Kutta method based on Lobatto quadrature

Axiom
annotated target: urn:issn:1088-6842(e)
comment: J. C. Butcher, Integration processes based on Radau quadrature formulas, Math. Comp., v. 18, 1964, pp. 233-244.
annotated source: Lobatto method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: implicit Runge-Kutta method based on Lobatto quadrature
annotated source: Lobatto method
annotated property: altLabel

Butcher-Kuntzmann method
type: Class
subclass of: Runge-Kutta based method
subclass of: Restriction
  on property: has characteristic
  some values from: symplecticness
subclass of: Restriction
  on property: has characteristic
  some values from: progression with fixed time step
subclass of: Restriction
  on property: has characteristic
  some values from: implicit method type
created: 2011-05-10
creator: AZ
definition: From a theoretical point of view, the Butcher-Kuntzmann Runge-Kutta methods belong to the best step-by-step methods for nonstiff problems. These methods integrate first-order initial-value problems by means of formulas based on Gauss-Legendre quadrature, and combine excellent stability features with the property of superconvergence at the step points.
alternative label: Gauss method

Axiom
comment: EXACT
annotated target: Gauss method
annotated source: Butcher-Kuntzmann method
annotated property: altLabel

Axiom
annotated target: urn:issn:1088-6842(e)
comment: Butcher, J. C. (1964) Implicit Runge-Kutta processes. Math. Comput. 18, 50-64.
annotated source: Butcher-Kuntzmann method
annotated property: seeAlso

Axiom
annotated target: http://identifiers.org/doi/10.1002/zamm.19660460519
comment: F. Ceschin, J. Kuntzmann, Problèmes différentiels de conditions initiales. Paris 1963. Dunod. @fr
annotated source: Butcher-Kuntzmann method
annotated property: seeAlso

Heun method
type: Class
subclass of: Runge-Kutta based method
subclass of: Restriction
  on property: has characteristic
  some values from: progression with fixed time step
subclass of: Restriction
  on property: has Runge-Kutta method order
  has value: 2
subclass of: Restriction
  on property: has characteristic
  some values from: explicit method type
created: 2011-05-10
creator: AZ
alternative label: Heun's method
definition: The method is named after Karl L. W. M. Heun and is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It can be seen as extension of the Euler method [http://identifiers.org/biomodels.kisao/KISAO_0000261] into two-stage second-order Runge-Kutta method.

Axiom
annotated target: http://identifiers.org/isbn/978-3-540-56670-0
comment: Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems (2nd ed.), Berlin: Springer Verlag
annotated source: Heun method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: Heun's method
annotated source: Heun method
annotated property: altLabel

embedded Runge-Kutta method
type: Class
subclass of: Runge-Kutta based method
subclass of: Restriction
  on property: has characteristic
  some values from: explicit method type
subclass of: Restriction
  on property: has characteristic
  some values from: progression with adaptive time step
created: 2011-05-10
is organizational: true
creator: AZ
definition: An embedded Runge-Kutta method is a method in which two Runge-Kutta estimates are obtained using the same auxiliary functions ki but with a different linear combination of these functions so that one estimate has an order one greater than the other.
alternative label: embedded RK

Axiom
annotated target: http://identifiers.org/isbn/978-3-540-56670-0
comment: Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems (2nd ed.), Berlin: Springer Verlag
annotated source: embedded Runge-Kutta method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: embedded RK
annotated source: embedded Runge-Kutta method
annotated property: altLabel

Zonneveld method
type: Class
subclass of: embedded Runge-Kutta method
created: 2011-05-10
creator: AZ
definition: An embedded Runge-Kutta method, proposed by J.A. Zonneveld in 1964.

Axiom
annotated target: http://trove.nla.gov.au/work/21424455
comment: Zonneveld, J. A (1970). Automatic numerical integration Mathematisch Centrum, Amsterdam
annotated source: Zonneveld method
annotated property: seeAlso

Radau method
type: Class
subclass of: Runge-Kutta based method
subclass of: Restriction
  on property: has characteristic
  some values from: progression with fixed time step
subclass of: Restriction
  on property: has characteristic
  some values from: implicit method type
created: 2011-05-11
creator: AZ
definition: Implicit Runge-Kutta methods based on Radau quadrature.
is implemented in: JSim
alternative label: implicit Runge-Kutta method based on Radau quadrature

Axiom
comment: EXACT
annotated target: implicit Runge-Kutta method based on Radau quadrature
annotated source: Radau method
annotated property: altLabel

Axiom
annotated target: urn:issn:1088-6842(e)
comment: J. C. Butcher, Integration processes based on Radau quadrature formulas, Math. Comput., 18(1964), 233-244.
annotated source: Radau method
annotated property: seeAlso

Verner method
type: Class
subclass of: embedded Runge-Kutta method
created: 2011-05-10
creator: AZ
definition: The first high order embedded Runge-Kutta formulas that avoid the drawback of giving identically zero error estimates for quadrature problems y' = f(x) were constructed by Verner in 1978.
alternative label: Verner's method

Axiom
annotated target: http://identifiers.org/doi/10.1137/0728027
comment: J.H. Verner, Some Runge-Kutta formula pairs, SIAM J. Numer. Anal 28 (1991) 496-511
annotated source: Verner method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: Verner's method
annotated source: Verner method
annotated property: altLabel

Lagrangian sliding fluid element algorithm
type: Class
subclass of: modeling and simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: progression with fixed time step
subclass of: Restriction
  on property: has characteristic
  some values from: partial differential equation problem
subclass of: Restriction
  on property: has characteristic
  some values from: deterministic system behaviour
subclass of: Restriction
  on property: has characteristic
  some values from: approximate solution
subclass of: Restriction
  on property: has characteristic
  some values from: spatial description
subclass of: Restriction
  on property: has characteristic
  some values from: continuous variable
created: 2011-05-11
creator: AZ
alternative label: BTEX
definition: Because the analytic solutions to the partial differential equations require convolution integration, solutions are obtained relatively efficiently by a fast numerical method. Our approach centers on the use of a sliding fluid element algorithm for capillary convection, with the time step set equal to the length step divided by the fluid velocity. Radial fluxes by permeation between plasma, interstitial fluid, and cells and axial diffusion exchanges within each time step are calculated analytically. The method enforces mass conservation unless there is regional consumption.
is implemented in: JSim
alternative label: LSFEA
alternative label: blood-tissue exchange method

Axiom
annotated target: BTEX
comment: EXACT
annotated source: Lagrangian sliding fluid element algorithm
annotated property: altLabel

Axiom
comment: EXACT
annotated target: LSFEA
annotated source: Lagrangian sliding fluid element algorithm
annotated property: altLabel

Axiom
comment: EXACT
annotated target: blood-tissue exchange method
annotated source: Lagrangian sliding fluid element algorithm
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/pubmed/1449234
comment: Bassingthwaighte JB, Chan IS, and Wang CY. Computationally efficient algorithms for capillary convection-permeation-diffusion models for blood-tissue exchange. Ann Biomed Eng 20: 687-725, 1992.
annotated source: Lagrangian sliding fluid element algorithm
annotated property: seeAlso

finite difference method
type: Class
subclass of: partial differential equation discretization method
subclass of: Restriction
  on property: has characteristic
  some values from: approximate solution
subclass of: Restriction
  on property: has characteristic
  some values from: progression with fixed time step
subclass of: Restriction
  on property: has characteristic
  some values from: mesh-based
subclass of: Restriction
  on property: has characteristic
  some values from: deterministic system behaviour
created: 2011-05-11
creator: AZ
alternative label: FDM
definition: The finite difference method is based on local approximations of the partial derivatives in a Partial Differential Equation, which are derived by low order Taylor series expansions.

Axiom
comment: EXACT
annotated target: FDM
annotated source: finite difference method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/isbn/0898715342
comment: Y. Saad. 2003. Iterative Methods for Sparse Linear Systems (2nd ed.). Soc. for Industrial and Applied Math., Philadelphia, PA, USA.
annotated source: finite difference method
annotated property: seeAlso

MacCormack method
type: Class
subclass of: finite difference method
disjoint with: Crank-Nicolson method
created: 2011-05-11
creator: AZ
definition: In computational fluid dynamics, the MacCormack method is a widely used discretization scheme for the numerical solution of hyperbolic partial differential equations. This second-order finite difference method [http://identifiers.org/biomodels.kisao/KISAO_0000307] is introduced by R. W. MacCormack in 1969.
is implemented in: JSim

Axiom
annotated target: http://identifiers.org/doi/10.2514/2.6901
comment: MacCormack, R. W., The Effect of viscosity in hypervelocity impact cratering, AIAA Paper, 69-354 (1969).
annotated source: MacCormack method
annotated property: seeAlso

Crank-Nicolson method
type: Class
subclass of: finite difference method
subclass of: Restriction
  on property: has characteristic
  some values from: implicit method type
created: 2011-05-11
creator: AZ
definition: In numerical analysis, the Crank-Nicolson method is a finite difference method [http://identifiers.org/biomodels.kisao/KISAO_0000307] used for numerically solving the heat equation and similar partial differential equations. It is a second-order method in time, implicit in time, and is numerically stable. The method was developed by John Crank and Phyllis Nicolson in the mid 20th century.

Axiom
annotated target: http://identifiers.org/doi/10.1007/BF02127704
comment: Crank, J.; Nicolson, P. (1947). A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type. Proc. Camb. Phil. Soc. 43 (1): 50-67.
annotated source: Crank-Nicolson method
annotated property: seeAlso

method of lines
type: Class
subclass of: partial differential equation discretization method
subclass of: Restriction
  on property: has characteristic
  some values from: continuous variable
subclass of: Restriction
  on property: has characteristic
  some values from: mesh-based
subclass of: Restriction
  on property: has characteristic
  some values from: progression with adaptive time step
created: 2011-05-11
creator: AZ
alternative label: MOL
alternative label: NMOL
alternative label: NUMOL
definition: The method of lines is a general technique for solving partial differential equations (PDEs) by typically using finite difference relationships for the spatial derivatives and ordinary differential equations for the time derivative.

Axiom
comment: EXACT
annotated target: NUMOL
annotated source: method of lines
annotated property: altLabel

Axiom
comment: EXACT
annotated target: MOL
annotated source: method of lines
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/isbn/0126241309
comment: Schiesser, W. E. (1991). The Numerical Method of Lines. Academic Press.
annotated source: method of lines
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: NMOL
annotated source: method of lines
annotated property: altLabel

Axiom
comment: EXACT
annotated target: method of lines
annotated source: method of lines
annotated property: label

type of domain geometry handling
type: Class
subclass of: modeling and simulation algorithm characteristic
is organizational: true
creator: AZ

S-System power-law canonical differential equations solver
type: Class
subclass of: modeling and simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: deterministic system behaviour
subclass of: Restriction
  on property: has characteristic
  some values from: progression with adaptive time step
subclass of: Restriction
  on property: has characteristic
  some values from: approximate solution
subclass of: Restriction
  on property: has characteristic
  some values from: ordinary differential equation problem
created: 2011-05-20
creator: AZ
is implemented in: E-Cell
alternative label: ESSYNS GMA
definition: Ordinary differential equations can be recast into a nonlinear canonical form called an S-system. Evidence for the generality of this class comes from extensive empirical examples that have been recast and from the discovery that sets of differential equations and functions, recognized as among the most general, are special cases of S-systems. Identification of this nonlinear canonical form suggests a radically different approach to numerical solution of ordinary differential equations. By capitalizing on the regular structure of S-systems, efficient formulas for a variable-order, variable-step Taylor-series method are developed.

Axiom
annotated target: http://identifiers.org/doi/10.1137/0727042
comment: Irvine, D.H. and Savageau, M.A., Efficient solution of nonlinear ordinary differential equations expressed in S-System canonical form, Siam. J. Numer. Anal., 27:704-735, 1990.
annotated source: S-System power-law canonical differential equations solver
annotated property: seeAlso

Axiom
annotated target: ESSYNS GMA
comment: EXACT
annotated source: S-System power-law canonical differential equations solver
annotated property: altLabel

lattice gas automata
type: Class
subclass of: deterministic cellular automata update algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: exact solution
disjoint with: multiparticle lattice gas automata
created: 2011-05-23
creator: AZ
alternative label: LGA
alternative label: LGCA
definition: Lattice gas automata methods are a series of cellular automata methods used to simulate fluid flows. From the LGCA, it is possible to derive the macroscopic Navier-Stokes equations.
alternative label: lattice gas cellular automata

Axiom
comment: EXACT
annotated target: LGA
annotated source: lattice gas automata
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/isbn/3-540-66973-6
comment: Dieter A. Wolf-Gladrow (2000). Lattice-Gas Cellular Automata and Lattice Boltzmann Models. Springer.
annotated source: lattice gas automata
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: lattice gas cellular automata
annotated source: lattice gas automata
annotated property: altLabel

Axiom
comment: EXACT
annotated target: LGCA
annotated source: lattice gas automata
annotated property: altLabel

enhanced Greens function reaction dynamics
type: Class
subclass of: Smoluchowski equation based method
subclass of: Restriction
  on property: has characteristic
  some values from: discrete variable
subclass of: Restriction
  on property: is similar to
  some values from: Greens function reaction dynamics
subclass of: Restriction
  on property: has characteristic
  some values from: progression with adaptive time step
subclass of: Restriction
  on property: has characteristic
  some values from: exact solution
subclass of: Restriction
  on property: has characteristic
  some values from: spatial description
created: 2011-05-23
creator: AZ
definition: GFRD [http://identifiers.org/biomodels.kisao/KISAO_0000058] decomposes the multi­body reaction diffusion problem to a set of single and two body problems. Analytical solutions for two body reaction diffusion are available via Smoluchowski equation. eGFRD allows to solve each sub­problem asynchronously by introducing the concept of first passage processes.
alternative label: eGFRD

Axiom
comment: EXACT
annotated target: eGFRD
annotated source: enhanced Greens function reaction dynamics
annotated property: altLabel

Axiom
comment: EXACT
annotated target: enhanced Greens function reaction dynamics
annotated source: enhanced Greens function reaction dynamics
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1007/978-3-540-88562-7_3
comment: K. Takahashi, An exact Brownian dynamics method for cell simulation, in Computational Methods in Systems Biology, vol. 5307/2008.
annotated source: enhanced Greens function reaction dynamics
annotated property: seeAlso

E-Cell multi-algorithm simulation method
type: Class
subclass of: hybrid method
subclass of: Restriction
  on property: has characteristic
  some values from: hybridity
created: 2011-05-23
definition: A modular meta-algorithm with a discrete event scheduler that can incorporate any type of time-driven simulation algorithm. It was shown that this meta-algorithm can efficiently drive simulation models with different simulation algorithms with little intrusive modification to the algorithms themselves. Only a few additional methods to handle communications between computational modules are required.
creator: AZ
is implemented in: E-Cell

Axiom
annotated target: http://identifiers.org/pubmed/14990450
comment: Takahashi, K., Kaizu, K., Hu, B., and Tomita, M., A multi-algorithm, multi-timescale method for cell simulation. Bioinformatics, in press.
annotated source: E-Cell multi-algorithm simulation method
annotated property: seeAlso

Gauss-Legendre Runge-Kutta method
type: Class
subclass of: Runge-Kutta based method
subclass of: Restriction
  on property: has characteristic
  some values from: continuous variable
subclass of: Restriction
  on property: has characteristic
  some values from: progression with fixed time step
subclass of: Restriction
  on property: has characteristic
  some values from: implicit method type
created: 2011-05-26
creator: AZ
alternative label: Open Formula
definition: So called 'Open Formula', two points formula, three points formula, four points formula, five points formula and six points formula of the Runge-Kutta method to solve the initial value problem of the ordinary differential equation. These formulas use the points and weights from the Gauss-Legendre Quadrature formulas for finding the value of the definite integral.
is implemented in: iBioSim

Axiom
comment: EXACT
annotated target: Open Formula
annotated source: Gauss-Legendre Runge-Kutta method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/isbn/960-8457-54-8
comment: Maitree Podisuk, Sirirat Khuntidilokwongsa, and Witchaya Rattanametawee. 2006. Gauss-Legendre Quadrature formula in Runge-Kutta method with modified model of Newton cooling law. In Proceedings of the 8th WSEAS international conference on Mathematical methods and computational techniques in electrical engineering (MMACTEE'06). World Scientific and Engineering Academy and Society (WSEAS), Stevens Point, Wisconsin, USA, 312-317.
annotated source: Gauss-Legendre Runge-Kutta method
annotated property: seeAlso

Monte Carlo method
type: Class
subclass of: modeling and simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: stochastic differential equation problem
subclass of: Restriction
  on property: has characteristic
  some values from: stochastic system behaviour
subclass of: Restriction
  on property: has characteristic
  some values from: discrete variable
created: 2011-05-26
is organizational: true
creator: AZ
alternative label: MC
definition: Monte Carlo methods (or Monte Carlo experiments) are a class of computational algorithms that rely on repeated random sampling to compute their results.

Axiom
annotated target: http://identifiers.org/doi/10.2307/2280232
comment: Metropolis, N.; Ulam, S. (1949). "The Monte Carlo Method". Journal of the American Statistical Association (American Statistical Association) 44 (247): 335-341.
annotated source: Monte Carlo method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: MC
annotated source: Monte Carlo method
annotated property: altLabel

BioRica hybrid method
type: Class
subclass of: hybrid method
subclass of: Restriction
  on property: has characteristic
  some values from: progression with adaptive time step
subclass of: Restriction
  on property: is hybrid of
  some values from: modeling and simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: hybridity
created: 2011-05-26
creator: AZ
is implemented in: BioRica
definition: The simulation schema for a given BioRica node is given by a hybrid algorithm that deals with continuous time and allows for discrete events that roll back the time according to these discrete interruptions.

Axiom
annotated target: http://identifiers.org/pubmed/19425152
comment: Cvijovic M, Soueidan H, Sherman DJ, Klipp E, Nikolski M. Exploratory simulation of cell ageing using hierarchical models. Genome Inform. 2008;21:114-25.
annotated source: BioRica hybrid method
annotated property: seeAlso

Cash-Karp method
type: Class
subclass of: embedded Runge-Kutta method
subclass of: Restriction
  on property: uses
  some values from: Class
    intersection of: Runge-Kutta based method
    intersection of: Restriction
      on property: has Runge-Kutta method order
      has value: 4
subclass of: Restriction
  on property: uses
  some values from: Class
    intersection of: Runge-Kutta based method
    intersection of: Restriction
      on property: has Runge-Kutta method order
      has value: 5
created: 2011-05-26
creator: AZ
definition: An family of explicit Runge-Kutta formulas, which are very efficient for problems with smooth solution as well as problems having rapidly varying solutions. Each member of this family consists of a fifty-order formula that contains embedded formulas of all orders 1 through 4. By computing solutions at several different orders, it is possible to detect sharp fronts or discontinuities before all the function evaluations defining the full Runge-Kutta step have been computed.
is implemented in: Cain
is implemented in: GSL

Axiom
annotated target: http://identifiers.org/doi/10.1145/79505.79507
comment: J. R. Cash, A. H. Karp. "A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides", ACM Transactions on Mathematical Software 16: 201-222, 1990.
annotated source: Cash-Karp method
annotated property: seeAlso

hybridity
type: Class
subclass of: modeling and simulation algorithm characteristic
created: 2011-05-27
creator: AZ
definition: The basic idea of hybrid simulation methods is to combine the advantages of complementary simulation approaches: the whole system is subdivided into appropriate parts and different simulation methods operate on these parts at the same time.

equation-free probabilistic steady-state approximation
type: Class
subclass of: accelerated stochastic simulation algorithm
subclass of: Restriction
  on property: has parameter
  some values from: minimum fast/discrete reaction occurrences number
subclass of: Restriction
  on property: has parameter
  some values from: maximum discrete number
subclass of: Restriction
  on property: has parameter
  some values from: minimum fast rate
subclass of: Restriction
  on property: has parameter
  some values from: number of samples
created: 2011-06-02
creator: AZ
definition: We present a probabilistic steady-state approximation that separates the time scales of an arbitrary reaction network, detects the convergence of a marginal distribution to a quasi-steady-state, directly samples the underlying distribution, and uses those samples to accurately predict the state of the system, including the effects of the slow dynamics, at future times. The numerical method produces an accurate solution of both the fast and slow reaction dynamics while, for stiff systems, reducing the computational time by orders of magnitude. The developed theory makes no approximations on the shape or form of the underlying steady-state distribution and only assumes that it is ergodic. <...> The developed theory may be applied to any type of kinetic Monte Carlo simulation to more efficiently simulate dynamically stiff systems, including existing exact, approximate, or hybrid stochastic simulation techniques.

Axiom
annotated target: http://identifiers.org/doi/10.1063/1.2131050
comment: Salis H, Kaznessis YN. An equation-free probabilistic steady-state approximation: dynamic application to the stochastic simulation of biochemical reaction networks. J Chem Phys. 2005 Dec 1;123(21):214106.
annotated source: equation-free probabilistic steady-state approximation
annotated property: seeAlso

nested stochastic simulation algorithm
type: Class
subclass of: accelerated stochastic simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: approximate solution
subclass of: Restriction
  on property: uses
  some values from: Gillespie direct algorithm
created: 2011-06-02
creator: AZ
definition: This multiscale method is a small modification of the Gillespie's direct method [http://identifiers.org/biomodels.kisao/KISAO_0000029], in the form of a nested SSA, with inner loops for the fast reactions, and outer loop for the slow reactions. The number of groups can be more than two, and the grouping into fast and slow variables can be done dynamically in an adaptive version of the scheme.
alternative label: nested SSA

Axiom
annotated target: http://identifiers.org/pubmed/16321076
comment: E W., Liu D. and Vanden-Eijnden E. Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J Chem Phys. 2005 Nov 15;123(19):194107.
annotated source: nested stochastic simulation algorithm
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: nested SSA
annotated source: nested stochastic simulation algorithm
annotated property: altLabel

minimum fast/discrete reaction occurrences number
type: Class
subclass of: error control parameter
subclass of: Restriction
  on property: is parameter of
  all values from: equation-free probabilistic steady-state approximation
subclass of: Restriction
  on property: has type
  some values from: integer
created: 2011-06-02
creator: AZ
definition: Parameter of 'equation-free probabilistic steady-state approximation' method [http://identifiers.org/biomodels.kisao/KISAO_0000323], which describes the minimum number of fast/discrete reaction occurrences before their effects cause convergence to a quasi-steady-state distribution.

Axiom
annotated target: http://identifiers.org/doi/10.1063/1.2131050
comment: Salis H, Kaznessis YN. An equation-free probabilistic steady-state approximation: dynamic application to the stochastic simulation of biochemical reaction networks. J Chem Phys. 2005 Dec 1;123(21):214106.
annotated source: minimum fast/discrete reaction occurrences number
annotated property: seeAlso

number of samples
type: Class
subclass of: error control parameter
subclass of: Restriction
  on property: is parameter of
  all values from: equation-free probabilistic steady-state approximation
subclass of: Restriction
  on property: has type
  some values from: integer
created: 2011-06-02
creator: AZ
definition: Parameter of 'equation-free probabilistic steady-state approximation' method [http://identifiers.org/biomodels.kisao/KISAO_0000323], which determines the number of samples taken from the distribution.

Axiom
annotated target: http://identifiers.org/doi/10.1063/1.2131050
comment: Salis H, Kaznessis YN. An equation-free probabilistic steady-state approximation: dynamic application to the stochastic simulation of biochemical reaction networks. J Chem Phys. 2005 Dec 1;123(21):214106.
annotated source: number of samples
annotated property: seeAlso

maximum discrete number
type: Class
subclass of: partitioning control parameter
subclass of: Restriction
  on property: is parameter of
  all values from: equation-free probabilistic steady-state approximation
subclass of: Restriction
  on property: has type
  some values from: integer
created: 2011-06-02
creator: AZ
definition: Parameter of 'equation-free probabilistic steady-state approximation' method [http://identifiers.org/biomodels.kisao/KISAO_0000323], which controls the maximum number of molecules of some reactant species in order for the reaction to be considered discrete.

Axiom
annotated target: http://identifiers.org/doi/10.1063/1.2131050
comment: Salis H, Kaznessis YN. An equation-free probabilistic steady-state approximation: dynamic application to the stochastic simulation of biochemical reaction networks. J Chem Phys. 2005 Dec 1;123(21):214106.
annotated source: maximum discrete number
annotated property: seeAlso

minimum fast rate
type: Class
subclass of: partitioning control parameter
subclass of: Restriction
  on property: has type
  some values from: double
subclass of: Restriction
  on property: is parameter of
  all values from: equation-free probabilistic steady-state approximation
created: 2011-06-02
creator: AZ
definition: Parameter of 'equation-free probabilistic steady-state approximation' method [http://identifiers.org/biomodels.kisao/KISAO_0000323], which controls the minimum rate of the reaction in order for it to be considered fast.

Axiom
annotated target: http://identifiers.org/doi/10.1063/1.2131050
comment: Salis H, Kaznessis YN. An equation-free probabilistic steady-state approximation: dynamic application to the stochastic simulation of biochemical reaction networks. J Chem Phys. 2005 Dec 1;123(21):214106.
annotated source: minimum fast rate
annotated property: seeAlso

constant-time kinetic Monte Carlo algorithm
type: Class
subclass of: accelerated stochastic simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: exact solution
subclass of: Class
  complement of: Restriction
    on property: has characteristic
    some values from: spatial description
created: 2011-06-03
creator: AZ
alternative label: SSA-CR
definition: The computational cost of the original SSA [http://identifiers.org/biomodels.kisao/KISAO_0000029] scaled linearly with the number of reactions in the network. Gibson and Bruck developed a logarithmic scaling version of the SSA which uses a priority queue or binary tree for more efficient reaction selection [http://identifiers.org/biomodels.kisao/KISAO_0000027]. More generally, this problem is one of dynamic discrete random variate generation which finds many uses in kinetic Monte Carlo and discrete event simulation. We present here a constant-time algorithm, whose cost is independent of the number of reactions, enabled by a slightly more complex underlying data structure.

Axiom
comment: EXACT
annotated target: SSA-CR
annotated source: constant-time kinetic Monte Carlo algorithm
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/pubmed/18513044
comment: Slepoy A, Thompson AP, Plimpton SJ. A constant-time kinetic Monte Carlo algorithm for simulation of large biochemical reaction networks. J Chem Phys. 2008 May 28;128(20):205101.
annotated source: constant-time kinetic Monte Carlo algorithm
annotated property: seeAlso

R-leaping algorithm
type: Class
subclass of: accelerated stochastic simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: approximate solution
created: 2011-06-03
definition: A novel algorithm is proposed for the acceleration of the exact stochastic simulation algorithm by a predefined number of reaction firings (R-leaping) that may occur across several reaction channels. In the present approach, the numbers of reaction firings are correlated binomial distributions and the sampling procedure is independent of any permutation of the reaction channels. This enables the algorithm to efficiently handle large systems with disparate rates, providing substantial computational savings in certain cases.
creator: AZ
alternative label: R-leap method

Axiom
annotated target: http://identifiers.org/pubmed/16964997
comment: Auger A, Chatelain P, Koumoutsakos P. R-leaping: accelerating the stochastic simulation algorithm by reaction leaps. J Chem Phys. 2006 Aug 28;125(8):084103.
annotated source: R-leaping algorithm
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: R-leap method
annotated source: R-leaping algorithm
annotated property: altLabel

exact R-leaping algorithm
type: Class
subclass of: accelerated stochastic simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: exact solution
subclass of: Restriction
  on property: is similar to
  some values from: R-leaping algorithm
created: 2011-06-03
creator: AZ
alternative label: ER-leap method
definition: We present a SSA which, similar to R-leap [http://identifiers.org/biomodels.kisao/KISAO_0000330], accelerates SSA [http://identifiers.org/biomodels.kisao/KISAO_0000029] by executing multiple reactions per algorithmic step, but which samples the reactant trajectories from the same probability distribution as the SSA. This 'exact R-leap' or 'ER-leap' algorithm is a modification of the R-leap algorithm which is both exact and capable of substantial speed-up over SSA.
alternative label: exact R-leap method
alternative label: exact accelerated stochastic simulation algorithm

Axiom
annotated target: ER-leap method
comment: EXACT
annotated source: exact R-leaping algorithm
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/pubmed/2852436
comment: Eric Mjolsness, David Orendorff, Philippe Chatelain, and Petros Koumoutsakos. An exact accelerated stochastic simulation algorithm. J Chem Phys. 2009 April 14; 130(14): 144110.
annotated source: exact R-leaping algorithm
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: exact R-leap method
annotated source: exact R-leaping algorithm
annotated property: altLabel

Axiom
comment: EXACT
annotated target: exact accelerated stochastic simulation algorithm
annotated source: exact R-leaping algorithm
annotated property: altLabel

ER-leap initial leap
type: Class
subclass of: granularity control parameter
subclass of: Restriction
  on property: is parameter of
  all values from: exact R-leaping algorithm
subclass of: Restriction
  on property: has type
  some values from: integer
created: 2011-06-03
creator: AZ
alternative label: L
definition: L (initial step) is a parameter of 'exact R-leaping method' [http://identifiers.org/biomodels.kisao/KISAO_0000331]. ''We will assume that the reaction event to be bounded occurs within a run of L events in the SSA algorithm[http://identifiers.org/biomodels.kisao/KISAO_0000029], in order to execute L reactions at once in the manner of the R-leap algorithm[http://identifiers.org/biomodels.kisao/KISAO_0000230]''.

Axiom
annotated target: http://identifiers.org/pubmed/2852436
comment: Eric Mjolsness, David Orendorff, Philippe Chatelain, and Petros Koumoutsakos. An exact accelerated stochastic simulation algorithm. J Chem Phys. 2009 April 14; 130(14): 144110.
annotated source: ER-leap initial leap
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: L
annotated source: ER-leap initial leap
annotated property: altLabel

accelerated stochastic simulation algorithm
type: Class
subclass of: Gillespie-like method
subclass of: Restriction
  on property: is similar to
  some values from: Gillespie direct algorithm
created: 2011-06-03
is organizational: true
creator: AZ
definition: An algorithm, which accelerates SSA [http://identifiers.org/biomodels.kisao/KISAO_0000029] either at the expense of its accuracy or exact.
alternative label: accelerated SSA

Axiom
comment: EXACT
annotated target: accelerated SSA
annotated source: accelerated stochastic simulation algorithm
annotated property: altLabel

multiparticle lattice gas automata
type: Class
subclass of: deterministic cellular automata update algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: exact solution
subclass of: Restriction
  on property: is similar to
  some values from: lattice gas automata
created: 2011-06-03
creator: AZ
definition: An algorithm which allows for an arbitrary number of particles, while keeping the benefits of the cellular automata approach [http://identifiers.org/biomodels.kisao/KISAO_0000315].
alternative label: multiparticle lattice gas cellular automata

Axiom
comment: EXACT
annotated target: multiparticle lattice gas cellular automata
annotated source: multiparticle lattice gas automata
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1142/S0129183194000052
comment: Chopard B., et al.Multiparticle lattice gas automata for reaction diffusion systems. Int. J. Mod. Phys. C 1994;5:47-63.
annotated source: multiparticle lattice gas automata
annotated property: seeAlso

generalized stochastic simulation algorithm
type: Class
subclass of: Gillespie-like method
subclass of: Restriction
  on property: is generalization of
  some values from: Gillespie direct algorithm
created: 2011-06-03
is organizational: true
creator: AZ
definition: Gillespie direct method [http://identifiers.org/biomodels.kisao/KISAO_0000029] follows unit-by-unit changes in the total numbers of each reactant species, it is especially well suited to the study of systems in which reactant densities are low and the application of methods based on continuum approximations, such as the traditional ordinary differential equations of chemical kinetics, is questionable. The 'generalized stochastic simulation algorithm' branch presents methods, which extend Gillespie direct method [http://identifiers.org/biomodels.kisao/KISAO_0000029] to suit to systems with other characteristics.

D-leaping method
type: Class
subclass of: generalized stochastic simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: approximate solution
created: 2011-06-03
creator: AZ
definition: We propose a novel, accelerated algorithm for the approximate stochastic simulation of biochemical systems with delays. The present work extends existing accelerated algorithms by distributing, in a time adaptive fashion, the delayed reactions so as to minimize the computational effort while preserving their accuracy.

Axiom
annotated target: http://identifiers.org/doi/10.1016/j.jcp.2009.05.004
comment: Bayati B, Chatelain P, Koumoutsakos. D-leaping: Accelerating stochastic simulation algorithms for reactions with delays. Journal of Computational Physics. 2009 Sept;228(16):5909-5918.
annotated source: D-leaping method
annotated property: seeAlso

finite element method
type: Class
subclass of: partial differential equation discretization method
subclass of: Restriction
  on property: has characteristic
  some values from: deterministic system behaviour
subclass of: Restriction
  on property: has characteristic
  some values from: approximate solution
subclass of: Restriction
  on property: has characteristic
  some values from: mesh-based
created: 2011-06-07
definition: A numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler method [http://identifiers.org/biomodels.kisao/KISAO_0000261], Runge-Kutta [http://identifiers.org/biomodels.kisao/KISAO_0000064], etc.
creator: AZ
alternative label: FEA
alternative label: FEM
alternative label: finite element analysis

Axiom
comment: EXACT
annotated target: FEA
annotated source: finite element method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: FEM
annotated source: finite element method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: finite element analysis
annotated source: finite element method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1109/MAP.2007.376627
comment: Pelosi G. The finite-element method, Part I: R. L. Courant: Historical Corner. IEEE Antennas and Propagation Magazine, Vol. 49, No. 2. (April 2007), pp. 180-182.
annotated source: finite element method
annotated property: seeAlso

h-version of the finite element method
type: Class
subclass of: finite element method
created: 2011-06-07
creator: AZ
definition: Classical form of the 'finite element method' [http://identifiers.org/biomodels.kisao/KISAO_0000337], in which polynomials of fixed degree p are used and the mesh is refined to increase accuracy. Can be considered as a special case of the h-p version [http://identifiers.org/biomodels.kisao/KISAO_0000340].
alternative label: h-FEM
alternative label: h-method

Axiom
comment: EXACT
annotated target: h-method
annotated source: h-version of the finite element method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1016/0168-874X(94)90003-5
comment: Babuska, I., T. Strouboulis, A. Mathur, C. S. Upadhyay, Pollution error in the h-version of the finite element method and the local quality of a-posteriori error estimators, Finite Elements in Analysis and Design, 17: 273-321, 1994.
annotated source: h-version of the finite element method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: h-FEM
annotated source: h-version of the finite element method
annotated property: altLabel

p-version of the finite element method
type: Class
subclass of: finite element method
created: 2011-06-07
creator: AZ
definition: The p version of 'finite element method' [http://identifiers.org/biomodels.kisao/KISAO_0000337] uses a fixed mesh but increases the polynomial degree p to increase accuracy. Can be considered as a special case of the h-p version [http://identifiers.org/biomodels.kisao/KISAO_0000340].
alternative label: p-FEM
alternative label: p-method

Axiom
annotated target: http://identifiers.org/doi/10.1137/0718033
comment: Babuska, I., B.A. Szabo and I.N. Katz, The p-version of the finite element method, SIAM J. Numer. Anal., 18: 515-545, 1981.
annotated source: p-version of the finite element method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: p-method
annotated source: p-version of the finite element method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: p-FEM
annotated source: p-version of the finite element method
annotated property: altLabel

h-p version of the finite element method
type: Class
subclass of: finite element method
subclass of: Restriction
  on property: is generalization of
  some values from: p-version of the finite element method
subclass of: Restriction
  on property: is generalization of
  some values from: h-version of the finite element method
disjoint with: mixed finite element method
created: 2011-06-07
creator: AZ
definition: In h-p version of 'finite difference method' [http://identifiers.org/biomodels.kisao/KISAO_0000337] the two approaches of mesh refinement and degree enchacement are combined.
alternative label: hp-FEM
alternative label: hp-method

Axiom
comment: EXACT
annotated target: hp-FEM
annotated source: h-p version of the finite element method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1007/BF00272624
comment: Babuska, I., and B. Guo, The h-p version of the finite element method, Comput. Mech., 1: 203-220, 1986.
annotated source: h-p version of the finite element method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: hp-method
annotated source: h-p version of the finite element method
annotated property: altLabel

mixed finite element method
type: Class
subclass of: finite element method
created: 2011-06-07
definition: A 'finite element method' [http://identifiers.org/biomodels.kisao/KISAO_0000337] in which both stress and displacement fields are approximated as primary variables.
creator: AZ

Axiom
annotated target: http://identifiers.org/doi/10.1016/0045-7825(90)90168-L
comment: D. N. Arnold. 1990. Mixed finite element methods for elliptic problems. Comput. Methods Appl. Mech. Eng. 82, 1-3 (September 1990), 281-300.
annotated source: mixed finite element method
annotated property: seeAlso

level set method
type: Class
subclass of: partial differential equation discretization method
subclass of: Class
  complement of: Restriction
    on property: has characteristic
    some values from: linearity of equation
created: 2011-06-07
creator: AZ
definition: An algorithm for moving surfaces under their curvature. This algorithm rely on numerically solving Hamilton-Jacobi equations with viscous terms, using approximation techniques from hyperbolic conservation laws.
alternative label: LSM
alternative label: level-set method

Axiom
comment: EXACT
annotated target: LSM
annotated source: level set method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: level-set method
annotated source: level set method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1016/0021-9991(88)90002-2
comment: Osher, S.; Sethian, J. A. (1988), "Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations", J. Comput. Phys. 79: 12-49.
annotated source: level set method
annotated property: seeAlso

generalized finite element method
type: Class
subclass of: partial differential equation discretization method
subclass of: Restriction
  on property: is generalization of
  some values from: finite element method
subclass of: Restriction
  on property: has characteristic
  some values from: approximate solution
subclass of: Restriction
  on property: has characteristic
  some values from: deterministic system behaviour
subclass of: Restriction
  on property: has characteristic
  some values from: meshless
subclass of: Restriction
  on property: has characteristic
  some values from: continuous variable
created: 2011-06-07
creator: AZ
alternative label: GFEM
alternative label: PUM
definition: The GFEM is a generalization of the classical 'finite element method' [http://identifiers.org/biomodels.kisao/KISAO_0000337] — in its h [http://identifiers.org/biomodels.kisao/KISAO_0000338], p [http://identifiers.org/biomodels.kisao/KISAO_0000339], and h-p versions [http://identifiers.org/biomodels.kisao/KISAO_0000340]— as well as of the various forms of meshless methods used in engineering.
alternative label: partition of unity method

Axiom
comment: EXACT
annotated target: partition of unity method
annotated source: generalized finite element method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: GFEM
annotated source: generalized finite element method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1142/S0219876204000083
comment: Babuska, Ivo; Uday Banerjee, John E. Osborn (June 2004). Generalized Finite Element Methods: Main Ideas, Results, and Perspective. International Journal of Computational Methods 1 (1): 67-103.
annotated source: generalized finite element method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: PUM
annotated source: generalized finite element method
annotated property: altLabel

h-p cloud method
type: Class
subclass of: partial differential equation discretization method
subclass of: Restriction
  on property: has characteristic
  some values from: continuous variable
subclass of: Restriction
  on property: has characteristic
  some values from: deterministic system behaviour
subclass of: Restriction
  on property: has characteristic
  some values from: meshless
subclass of: Restriction
  on property: has characteristic
  some values from: approximate solution
created: 2011-06-09
definition: A meshless method, which uses a partition of unity to construct the family of h-p cloud functions.
creator: AZ
alternative label: h-p clouds
alternative label: method of clouds

Axiom
comment: EXACT
annotated target: h-p clouds
annotated source: h-p cloud method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: method of clouds
annotated source: h-p cloud method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1002/(SICI)1098-2426(199611)12:6<673::AID-NUM3>3.0.CO
comment: Duarte, C. A. and Oden, J. T. (1996), H-p clouds—an h-p meshless method. Numerical Methods for Partial Differential Equations, 12: 673-705.
annotated source: h-p cloud method
annotated property: seeAlso

mesh-based
type: Class
subclass of: type of domain geometry handling
disjoint with: meshless
creator: AZ
definition: In most large-scale numerical simulations of physical phenomena, a large percentage of the overall computational effort is expended on technical details connected with meshing. These details include, in particular, grid generation, mesh adaptation to domain geometry, element or cell connectivity, grid motion and separation to model fracture, fragmentation, free surfaces, etc.

meshless
type: Class
subclass of: type of domain geometry handling
creator: AZ
definition: Most meshless methods require a scattered set of nodal points in the domain of interest. In these methods, there may be no fixed connectivities between the nodes, unlike the finite element or finite difference methods. This feature has significant implications in modeling some physical phenomena that are characterized by a continuous change in the geometry of the domain under analysis.

extended finite element method
type: Class
subclass of: partial differential equation discretization method
subclass of: Restriction
  on property: has characteristic
  some values from: meshless
subclass of: Restriction
  on property: is generalization of
  some values from: finite element method
subclass of: Restriction
  on property: has characteristic
  some values from: approximate solution
subclass of: Restriction
  on property: has characteristic
  some values from: deterministic system behaviour
created: 2011-06-09
definition: A numerical method to model arbitrary discontinuities in continuous bodies that does not require the mesh to conform to the discontinuities nor significant mesh refinement near singularities. In X-FEM the standard finite element approximation [http://identifiers.org/biomodels.kisao/KISAO_0000337] is enriched and the approximation space is extended by an additional family of functions.
creator: AZ
alternative label: X-FEM
alternative label: XFEM

Axiom
comment: EXACT
annotated target: XFEM
annotated source: extended finite element method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1007/s00466-002-0391-2
comment: Stazi, F. L.; Budyn, E.; Chessa, J.; Belytschko, T. An extended finite element method with higher-order elements for curved cracks. Computational Mechanics, Volume 31, Issue 1-2, pp. 38-48 (2003).
annotated source: extended finite element method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: X-FEM
annotated source: extended finite element method
annotated property: altLabel

method of finite spheres
type: Class
subclass of: partial differential equation discretization method
subclass of: Restriction
  on property: has characteristic
  some values from: meshless
subclass of: Restriction
  on property: has characteristic
  some values from: approximate solution
subclass of: Restriction
  on property: has characteristic
  some values from: deterministic system behaviour
created: 2011-06-09
creator: AZ
alternative label: MFS
definition: Method of finite spheres is truly meshless in the sense that the nodes are placed and the numerical integration is performed without a mesh. Some of the novel features of the method of finite spheres are the numerical integration scheme and the way in which the Dirichlet boundary conditions are incorporated.

Axiom
comment: EXACT
annotated target: MFS
annotated source: method of finite spheres
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1007/s004660050481
comment: S. De and K. J. Bathe. The method of finite spheres. Computational Mechanics, Volume 25, Number 4, 329-345.
annotated source: method of finite spheres
annotated property: seeAlso

probability-weighted dynamic Monte Carlo method
type: Class
subclass of: accelerated stochastic simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: approximate solution
created: 2011-06-09
creator: AZ
alternative label: PW-DMC
definition: We have developed a probability-weighted DMC method by incorporating the weighted sampling algorithm of equilibrium molecular simulations. This new algorithm samples the slow reactions very efficiently and makes it possible to simulate in a computationally efficient manner the reaction kinetics of physical systems in which the rates of reactions vary by several orders of magnitude.
alternative label: probability-weighted DMC

Axiom
annotated target: http://identifiers.org/doi/10.1021/jp011404w
comment: Resat H, Wiley HS, Dixon DA. Probability-weighted dynamic Monte Carlo method for reaction kinetics simulations. J Phys Chem B. 2001;105:11026-11034.
annotated source: probability-weighted dynamic Monte Carlo method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: probability-weighted DMC
annotated source: probability-weighted dynamic Monte Carlo method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: PW-DMC
annotated source: probability-weighted dynamic Monte Carlo method
annotated property: altLabel

multinomial tau-leaping method
type: Class
subclass of: tau-leaping method
subclass of: Restriction
  on property: is generalization of
  some values from: binomial tau-leaping method
created: 2011-06-09
creator: AZ
alternative label: MtauL
definition: The multinomial tau-leaping method is an extension of the binomial tau-leaping method [http://identifiers.org/biomodels.kisao/KISAO_0000074] to networks with arbitrary multiple-channel reactant dependencies. Improvements were achieved by a combination of three factors: First, tau-leaping steps are determined simply and efficiently using a-priori information and Poisson distribution based estimates of expectation values for reaction numbers. Second, networks are partitioned into closed groups of reactions and corresponding reactants in which no group reactant set is found in any other group. Third, product formation is factored into upper bound estimation of the number of times a particular reaction occurs.

Axiom
annotated target: http://identifiers.org/pubmed/17343434
comment: Pettigrew MF, Resat H. Multinomial tau-leaping method for stochastic kinetic simulations. J Chem Phys. 2007 Feb 28;126(8):084101.
annotated source: multinomial tau-leaping method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: MtauL
annotated source: multinomial tau-leaping method
annotated property: altLabel

hybrid method
type: Class
equivalent class: Class
  intersection of: modeling and simulation algorithm
  intersection of: Restriction
    on property: has characteristic
    some values from: hybridity
subclass of: modeling and simulation algorithm
created: 2011-06-09
is organizational: true
definition: A simulation methods which combines the advantages of complementary simulation approaches: the whole system is subdivided into appropriate parts and different simulation methods operate on these parts at the same time.
creator: AZ

generalized minimal residual algorithm
type: Class
subclass of: Krylov subspace projection method
subclass of: Restriction
  on property: is generalization of
  some values from: minimal residual method
subclass of: Class
  complement of: Restriction
    on property: has characteristic
    some values from: spatial description
created: 2011-06-10
creator: AZ
definition: An iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace. The generalized minimal residual method extends the minimal residual method (MINRES) [http://identifiers.org/biomodels.kisao/KISAO_0000388], which is only applicable to symmetric systems, to non-symmetric systems.
alternative label: GMRES

Axiom
annotated target: http://identifiers.org/doi/10.1137/0907058
comment: Youcef Saad and Martin H. Schultz, GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems. SIAM J. Sci. and Stat. Comput. 7, 856 (1986).
annotated source: generalized minimal residual algorithm
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: GMRES
annotated source: generalized minimal residual algorithm
annotated property: altLabel

Krylov subspace projection method
type: Class
subclass of: iterative method for linear system
subclass of: Restriction
  on property: has characteristic
  some values from: deterministic system behaviour
subclass of: Restriction
  on property: has characteristic
  some values from: continuous variable
created: 2011-06-10
creator: AZ
alternative label: Krylov subspace method
definition: Krylov subspace method is an iterative linear equation method, which builds up Krylov subspaces and look for good approximations to eigenvectors and invariant subspaces within the Krylov spaces.

Axiom
annotated target: http://identifiers.org/isbn/0898715342
comment: Y. Saad. 2003. Iterative Methods for Sparse Linear Systems (2nd ed.). Soc. for Industrial and Applied Math., Philadelphia, PA, USA.
annotated source: Krylov subspace projection method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: Krylov subspace method
annotated source: Krylov subspace projection method
annotated property: altLabel

DASPK
type: Class
subclass of: IDA-like method
subclass of: Restriction
  on property: uses
  some values from: scaled preconditioned generalized minimal residual method
subclass of: Restriction
  on property: uses
  some values from: DASSL
subclass of: Restriction
  on property: is similar to
  some values from: IDA
subclass of: Restriction
  on property: has characteristic
  some values from: differential-algebraic equation problem
subclass of: Restriction
  on property: uses
  some values from: generalized minimal residual algorithm
created: 2011-06-10
creator: AZ
alternative label: DDASPK
definition: In DASPK, we have combined the time-stepping methods of DASSL [http://identifiers.org/biomodels.kisao/KISAO_0000255] with preconditioned iterative method GMRES [http://identifiers.org/biomodels.kisao/KISAO_0000386], for solving large-scale systems of DAEs of the form F(t, y, y') = 0, where F, y, y' are N-dimensional vectors, and a consistent set of initial conditions y(t0) = y0, y'(t0) = y'0 is given. DASPK is written in Fortran.
alternative label: SDASPK
alternative label: differential algebraic system solver with Krylov preconditioning

Axiom
annotated target: DDASPK
comment: NARROW
annotated source: DASPK
annotated property: altLabel

Axiom
comment: EXACT
annotated target: differential algebraic system solver with Krylov preconditioning
annotated source: DASPK
annotated property: altLabel

Axiom
comment: NARROW
annotated target: SDASPK
annotated source: DASPK
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1137/0915088
comment: Peter N. Brown, Alan C. Hindmarsh, and Linda R. Petzold. 1994. Using Krylov methods in the solution of large-scale differential-algebraic systems. SIAM J. Sci. Comput. 15, 6 (November 1994), 1467-1488.
annotated source: DASPK
annotated property: seeAlso

DASSL
type: Class
subclass of: IDA-like method
subclass of: Restriction
  on property: has characteristic
  some values from: differential-algebraic equation problem
created: 2011-06-10
creator: AZ
definition: DASSL is designed for the numerical solution of implicit systems of differential/algebraic equations written in the form F(t,y,y')=0, where F, y, and y' are vectors, and initial values for y and y' are given.
alternative label: DDASSL
alternative label: SDASSL
alternative label: differential algebraic system solver

Axiom
comment: NARROW
annotated target: SDASSL
annotated source: DASSL
annotated property: altLabel

Axiom
annotated target: http://www.nea.fr/abs/html/nesc9918.html
comment: Linda R. Petzold: A Description of DASSL: A Differential/Algebraic System Solver, SAND82-8637 (September 1982).
annotated source: DASSL
annotated property: seeAlso

Axiom
annotated target: DDASSL
comment: NARROW
annotated source: DASSL
annotated property: altLabel

Axiom
comment: EXACT
annotated target: differential algebraic system solver
annotated source: DASSL
annotated property: altLabel

conjugate gradient method
type: Class
subclass of: Krylov subspace projection method
subclass of: Restriction
  on property: has characteristic
  some values from: symmetricity of matrix
created: 2011-06-10
creator: AZ
alternative label: CG
definition: Conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positive-definite. The conjugate gradient method is an iterative method, so it can be applied to sparse systems that are too large to be handled by direct methods. Such systems often arise when numerically solving partial differential equations.

Axiom
annotated target: CG
comment: EXACT
annotated source: conjugate gradient method
annotated property: altLabel

Axiom
annotated target: urn:issn:0091-0635
comment: M. R. Hestenes and E. L. Stiefel, Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 49 (1952), p. 409.
annotated source: conjugate gradient method
annotated property: seeAlso

biconjugate gradient method
type: Class
subclass of: Krylov subspace projection method
subclass of: Restriction
  on property: is generalization of
  some values from: conjugate gradient method
created: 2011-06-10
creator: AZ
alternative label: BCG
alternative label: Bi-CG
alternative label: BiCG
definition: The biconjugate gradient method provides a generalization of conjugate gradient method [http://identifiers.org/biomodels.kisao/KISAO_0000357] to non-symmetric matrices.

Axiom
annotated target: BiCG
comment: EXACT
annotated source: biconjugate gradient method
annotated property: altLabel

Axiom
annotated target: BCG
comment: EXACT
annotated source: biconjugate gradient method
annotated property: altLabel

Axiom
annotated target: Bi-CG
comment: EXACT
annotated source: biconjugate gradient method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1007/BFb0080109
comment: Fletcher, R. (1976). "Conjugate gradient methods for indefinite systems". Numerical Analysis. Lecture Notes in Mathematics (Springer Berlin / Heidelberg) 506: 73-89.
annotated source: biconjugate gradient method
annotated property: seeAlso

implicit-state Doob-Gillespie algorithm
type: Class
subclass of: multi-state agent-based simulation method
subclass of: Restriction
  on property: is similar to
  some values from: StochSim nearest-neighbour algorithm
subclass of: Restriction
  on property: is generalization of
  some values from: Gillespie direct algorithm
created: 2011-06-13
creator: AZ
definition: The algorithm uses a representation of the system together with a super-approximation of its ‘event horizon’ (all events that may happen next), and a specific correction scheme to obtain exact timings. Being completely local and not based on any kind of enumeration, this algorithm has a per event time cost which is independent of (i) the size of the set of generable species (which can even be infinite), and (ii) independent of the size of the system (ie, the number of agent instances). The algorithm can be refined, using concepts derived from the classical notion of causality, so that in addition to the above one also has that the even cost is depending (iii) only logarithmically on the size of the model (ie, the number of rules).

Axiom
annotated target: http://identifiers.org/isbn/3-540-76636-7 978-3-540-76636-0
comment: V. Danos, J. Feret, W. Fontana, and J. Krivine. 2007. Scalable simulation of cellular signalling networks. In Proceedings of the 5th Asian conference on Programming languages and systems (APLAS'07), Zhong Shao (Ed.). Springer-Verlag, Berlin, Heidelberg, 139-157.
annotated source: implicit-state Doob-Gillespie algorithm
annotated property: seeAlso

rule-based simulation method
type: Class
subclass of: modeling and simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: discrete variable
created: 2011-06-13
is organizational: true
creator: AZ
definition: Rule-based models provide a powerful alternative to approaches that require explicit enumeration of all possible molecular species of a system. Such models consist of formal rules governing interactive behaviour. Rule-based simulation methods are methods, used to simulated such models.

Adams predictor-corrector method
type: Class
subclass of: Adams method
subclass of: Restriction
  on property: uses
  some values from: Adams-Bashforth method
subclass of: Restriction
  on property: uses
  some values from: Adams-Moulton method
created: 2011-06-16
creator: AZ
is implemented in: GSL
definition: The combination of evaluating a single explicit integration method ('Adams-Bashforth method' [http://identifiers.org/biomodels.kisao/KISAO_0000279]) (the predictor step) in order to provide a good initial guess for the successive evaluation of an implicit method ('Adams-Moulton method' [http://identifiers.org/biomodels.kisao/KISAO_0000280]) (the corrector step) using iteration.

Axiom
annotated target: urn:asin:B0000EFQ8B
comment: Moulton, Forest R. (1926), New methods in exterior ballistics, University of Chicago Press.
annotated source: Adams predictor-corrector method
annotated property: seeAlso

NDSolve method
type: Class
subclass of: hybrid method
subclass of: Restriction
  on property: is hybrid of
  some values from: method of lines
subclass of: Restriction
  on property: is hybrid of
  some values from: embedded Runge-Kutta method
subclass of: Restriction
  on property: is hybrid of
  some values from: Class
    intersection of: Runge-Kutta based method
    intersection of: Restriction
      on property: has characteristic
      some values from: implicit method type
subclass of: Restriction
  on property: is hybrid of
  some values from: Adams predictor-corrector method
subclass of: Restriction
  on property: is hybrid of
  some values from: partitioned Runge-Kutta method
subclass of: Restriction
  on property: is hybrid of
  some values from: backward differentiation formula
created: 2011-06-16
creator: AZ
is implemented in: Mathematica
definition: The Mathematica computation system function NDSolve is a general numerical differential equation solver. It can handle a wide range of ordinary differential equations as well as some partial differential equations. NDSolve can also solve some differential-algebraic equations, which are typically a mix of differential and algebraic equations.

Axiom
annotated target: http://identifiers.org/isbn/978-1-57955-058-5
comment: Advanced Numerical Differential Equation Solving in Mathematica, Wolfram Mathematica Tutorial Collection, 2008.
annotated source: NDSolve method
annotated property: seeAlso

symplecticness
type: Class
subclass of: modeling and simulation algorithm characteristic
created: 2011-06-16
creator: AZ
definition: Roughly speaking, ‘symplecticness’ is a characteristic property possessed by the solutions of Hamiltonian problems. A numerical method is called symplectic if, when applied to Hamiltonian problems, it generates numerical solutions which inherit the property of symplecticness (phase volume preservation).

Axiom
annotated target: http://identifiers.org/doi/10.1017/S0962492900002282
comment: J. M. Sanz-Serna (1992). Symplectic integrators for Hamiltonian problems: an overview. Acta Numerica, 1, pp 243-286
annotated source: symplecticness
annotated property: seeAlso

partitioned Runge-Kutta method
type: Class
subclass of: Runge-Kutta based method
subclass of: Restriction
  on property: has characteristic
  some values from: symplecticness
subclass of: Restriction
  on property: uses
  on class: Runge-Kutta based method
  min qualified cardinality: 2
created: 2011-06-16
creator: AZ
definition: If a Hamiltonian system possesses a natural partitioning, it is possible to integrate its certain components using one Runge-Kutta method and other components using a different Runge-Kutta method. The overall s-stage scheme is called a partitioned Runge-Kutta method.
alternative label: PRK
alternative label: SPRK
alternative label: symplectic partitioned Runge-Kutta method

Axiom
comment: EXACT
annotated target: SPRK
annotated source: partitioned Runge-Kutta method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: symplectic partitioned Runge-Kutta method
annotated source: partitioned Runge-Kutta method
annotated property: altLabel

Axiom
comment: P. Rentrop (1985): Partitioned Runge-Kutta methods with stiffness detection and stepsize control. Numer. Math., Vol. 47, p.545-564.
annotated target: http://identifiers.org/doi/10.1007/BF01389456
annotated source: partitioned Runge-Kutta method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: PRK
annotated source: partitioned Runge-Kutta method
annotated property: altLabel

partial differential equation discretization method
type: Class
subclass of: modeling and simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: spatial description
subclass of: Restriction
  on property: has characteristic
  some values from: partial differential equation problem
created: 2011-06-27
is organizational: true
definition: A method which solves partial differential equations by discretizing them, i.e. approximating them by equations that involve a finite number of unknowns.
creator: AZ

type of problem
type: Class
subclass of: modeling and simulation algorithm characteristic
created: 2011-06-29
is organizational: true
definition: A characteristic describing the type of the problems which can be solved by the algorithm.
creator: AZ

stochastic differential equation problem
type: Class
subclass of: type of differential equation
creator: AZ
alternative label: SDE problem

Axiom
comment: EXACT
annotated target: SDE problem
annotated source: stochastic differential equation problem
annotated property: altLabel

partial differential equation problem
type: Class
subclass of: type of differential equation
creator: AZ
alternative label: PDE problem

Axiom
comment: EXACT
annotated target: PDE problem
annotated source: partial differential equation problem
annotated property: altLabel

differential-algebraic equation problem
type: Class
subclass of: type of differential equation
creator: AZ
alternative label: DAE

Axiom
annotated target: DAE
comment: EXACT
annotated source: differential-algebraic equation problem
annotated property: altLabel

ordinary differential equation problem
type: Class
subclass of: type of differential equation
creator: AZ
alternative label: ODE problem

Axiom
comment: EXACT
annotated target: ODE problem
annotated source: ordinary differential equation problem
annotated property: altLabel

delay differential equation problem
type: Class
subclass of: type of differential equation
creator: AZ
alternative label: DDE problem

linearity of equation
type: Class
subclass of: type of problem
created: 2011-07-19
creator: AZ
definition: Linear differential equations are of the form Ly = f, where the differential operator L is a linear operator, y is the unknown function, and the right hand side f is a given function of the same nature as y.

one-step method
type: Class
subclass of: modeling and simulation algorithm
subclass of: Class
  complement of: Restriction
    on property: has characteristic
    some values from: spatial description
subclass of: Restriction
  on property: has characteristic
  some values from: approximate solution
created: 2011-06-30
is organizational: true
definition: A numerical method for differential equations which uses one starting value at each step.
creator: AZ

Axiom
annotated target: http://identifiers.org/isbn/978-3-540-56670-0
comment: Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems (2nd ed.), Berlin: Springer Verlag
annotated source: one-step method
annotated property: seeAlso

implicit midpoint rule
type: Class
subclass of: Runge-Kutta based method
subclass of: Restriction
  on property: has Runge-Kutta method order
  has value: 2
subclass of: Restriction
  on property: has characteristic
  some values from: implicit method type
created: 2011-06-30
creator: AZ
is implemented in: GSL
definition: The implicit midpoint rule is a second-order case of the more general implicit s-stage Runge-Kutta methods [http://identifiers.org/biomodels.kisao/KISAO_0000064 and (http://identifiers.org/biomodels.kisao/KISAO_0000245 some http://identifiers.org/biomodels.kisao/KISAO_0000240)].
alternative label: implicit Gaussian second order Runge-Kutta method

Axiom
comment: EXACT
annotated target: implicit Gaussian second order Runge-Kutta method
annotated source: implicit midpoint rule
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/isbn/978-3-540-56670-0
comment: Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems (2nd ed.), Berlin: Springer Verlag.
annotated source: implicit midpoint rule
annotated property: seeAlso

Bulirsch-Stoer algorithm
type: Class
subclass of: Richardson extrapolation based method
subclass of: Restriction
  on property: uses
  some values from: modified midpoint method
subclass of: Restriction
  on property: has characteristic
  some values from: deterministic system behaviour
subclass of: Restriction
  on property: has characteristic
  some values from: ordinary differential equation problem
created: 2011-07-01
creator: AZ
alternative label: GBS
alternative label: Gragg-Bulirsch-Stoer algorithm
definition: The Bulirsch-Stoer method is an adaptive method which uses Gragg's modified midpoint method [http://identifiers.org/biomodels.kisao/KISAO_0000382] to estimate the solution of an initial value problem for various step sizes. The estimates are fit to a "diagonal" rational function or a polynomial as a function of the step size and the limit as the step size tends to zero is taken as the final estimate.

Axiom
comment: EXACT
annotated target: Gragg-Bulirsch-Stoer algorithm
annotated source: Bulirsch-Stoer algorithm
annotated property: altLabel

Axiom
comment: EXACT
annotated target: GBS
annotated source: Bulirsch-Stoer algorithm
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1007/BF02165234
comment: Bulirsch R. and Stoer J. (1966) Numerical treatment of ordinary differential equations by extrapolation methods, Numerische Mathematik, Volume 8, Number 1, 1-13.
annotated source: Bulirsch-Stoer algorithm
annotated property: seeAlso

Richardson extrapolation based method
type: Class
subclass of: one-step method
created: 2011-07-01
is organizational: true
definition: A method based on ideas of Richardson extrapolation, which is a process for obtaining increased accuracy in a discretized approximation by extrapolating results from coarse discretizations to an arbitrarily fine one.
creator: AZ

Axiom
annotated target: http://identifiers.org/doi/10.1098/rsta.1911.0009
comment: Richardson, L. F. (1911). The approximate arithmetical solution by finite differences of physical problems including differential equations, with an application to the stresses in a masonry dam. Philosophical Transactions of the Royal Society of London, Series A 210 (459-470): 307-357.
annotated source: Richardson extrapolation based method
annotated property: seeAlso

midpoint method
type: Class
subclass of: Runge-Kutta based method
subclass of: Restriction
  on property: has Runge-Kutta method order
  has value: 2
subclass of: Restriction
  on property: has characteristic
  some values from: explicit method type
created: 2011-07-01
creator: AZ
definition: The midpoint method is an explicit method for approximating the solution of the initial value problem y' = f(x,y); y(x0) = y0 at x for a given step size h. For the midpoint method the derivative of y(x) is approximated by the symmetric difference y'(x) = ( y(x+h) - y(x-h) ) / 2h + O(h2).

Axiom
annotated target: http://identifiers.org/isbn/978-3-540-56670-0
comment: Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems (2nd ed.), Berlin: Springer Verlag.
annotated source: midpoint method
annotated property: seeAlso

modified midpoint method
type: Class
subclass of: Runge-Kutta based method
subclass of: Restriction
  on property: has characteristic
  some values from: explicit method type
created: 2011-07-01
creator: AZ
alternative label: Gragg's method
alternative label: Gragg's modified midpoint method
definition: The modified midpoint method is globally a second order method for approximating the solution of the initial value problem y' = f(x, y), y(x0) = y0, which advances a vector of dependent variables y(x) from a point x to a point x + H by a sequence of n substeps each of size h, h = H/n.

Axiom
comment: EXACT
annotated target: Gragg's method
annotated source: modified midpoint method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/isbn/052143064X
comment: Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes in Fortran 77. Cambridge University Press (2001).
annotated source: modified midpoint method
annotated property: seeAlso

Bader-Deuflhard method
type: Class
subclass of: Richardson extrapolation based method
subclass of: Restriction
  on property: is generalization of
  some values from: Bulirsch-Stoer algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: ordinary differential equation problem
subclass of: Restriction
  on property: uses
  some values from: semi-implicit midpoint rule
created: 2011-07-01
creator: AZ
is implemented in: GSL
definition: The Bader-Deuflhard method is an extrapolation method based on a semi-implicit discretization [http://identifiers.org/biomodels.kisao/KISAO_0000387]. It is a generalization of the Bulirsch-Stoer algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000379] for solving ordinary differential equations.

Axiom
annotated target: http://identifiers.org/doi/10.1007/BF01418331
comment: Bader, G. and Deuflhard, P. "A Semi-Implicit Mid-Point Rule for Stiff Systems of Ordinary Differential Equations." Numer. Math. 41, 373-398, 1983.
annotated source: Bader-Deuflhard method
annotated property: seeAlso

semi-implicit midpoint rule
type: Class
subclass of: Runge-Kutta based method
subclass of: Restriction
  on property: has characteristic
  some values from: implicit method type
created: 2011-07-01
definition: A semi-implicit version of the midpoint method that has an even error series [http://identifiers.org/biomodels.kisao/KISAO_0000381].
creator: AZ

Axiom
annotated target: http://identifiers.org/doi/10.1007/BF01418331
comment: Bader, G. and Deuflhard, P. "A Semi-Implicit Mid-Point Rule for Stiff Systems of Ordinary Differential Equations." Numer. Math. 41, 373-398, 1983.
annotated source: semi-implicit midpoint rule
annotated property: seeAlso

scaled preconditioned generalized minimal residual method
type: Class
subclass of: generalized minimal residual algorithm
subclass of: Restriction
  on property: uses
  some values from: preconditioning technique
created: 2011-07-18
definition: A scaled preconditioned version of 'generalized minimal residual algorithm' [http://identifiers.org/biomodels.kisao/KISAO_0000353]. For linear system Ax = b a preconditioner matrix P that approximates A is sought, for which linear system Px = b can be solved easily. Preconditioning is applied on the left only. Scaling is done using diagonal matrix D whose diagonal elements are weights w^i = rtol|y^i| +atol^i, where rtol is 'relative tolerance' [http://identifiers.org/biomodels.kisao/KISAO_0000209] and atol is 'absolute tolerance' [http://identifiers.org/biomodels.kisao/KISAO_0000211].
creator: AZ
alternative label: SPGMR

Axiom
comment: EXACT
annotated target: SPGMR
annotated source: scaled preconditioned generalized minimal residual method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1137/0915088
comment: Peter N. Brown, Alan C. Hindmarsh, and Linda R. Petzold. 1994. Using Krylov methods in the solution of large-scale differential-algebraic systems. SIAM J. Sci. Comput. 15, 6 (November 1994), 1467-1488.
annotated source: scaled preconditioned generalized minimal residual method
annotated property: seeAlso

minimal residual method
type: Class
subclass of: Krylov subspace projection method
subclass of: Restriction
  on property: has characteristic
  some values from: symmetricity of matrix
created: 2011-07-18
creator: AZ
alternative label: MINRES
definition: The 'minimal residual method' is an algorithm for the numerical solution of indefinite symmertic systems of linear equations.

Axiom
annotated target: http://identifiers.org/doi/10.1137/0712047
comment: Paige C.C., Saunders M.A. Solution of sparse indefinite systems of linear equations, SIAM Journal on Numerical Analysis (1975) Volume: 12, Issue: 4, Publisher: SIAM, Pages: 617-629.
annotated source: minimal residual method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: MINRES
annotated source: minimal residual method
annotated property: altLabel

quasi-minimal residual method
type: Class
subclass of: ingenious conjugate gradients-squared method
created: 2011-07-18
creator: AZ
alternative label: QMR
definition: The QMR algorithm is a robust iterative solver for general nonsingular non-Hermitian linear systems. The method uses a robust implementation of the look-ahead Lanczos algorithm to generate basis vectors for the Krylov subspaces Kn(r0, A). The QMR iterates are characterized by a quasi-minimal residual property over Kn(r0, A).

Axiom
comment: EXACT
annotated target: QMR
annotated source: quasi-minimal residual method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1007/BF01385726
comment: Roland W. Freund and Noël M. Nachtigal. QMR: a quasi-minimal residual method for non-Hermitian linear systems. NUMERISCHE MATHEMATIK Volume 60, Number 1, 315-339.
annotated source: quasi-minimal residual method
annotated property: seeAlso

biconjugate gradient stabilized method
type: Class
subclass of: improved biconjugate gradient method
disjoint with: ingenious conjugate gradients-squared method
definition: An iterative method for the numerical solution of nonsymmetric linear systems. It is a variant of the biconjugate gradient method (BiCG) [http://identifiers.org/biomodels.kisao/KISAO_0000358] and has faster and smoother convergence than the original BiCG.
alternative label: Bi-CGSTAB
alternative label: BiCGSTAB

Axiom
annotated target: BiCGSTAB
comment: EXACT
annotated source: biconjugate gradient stabilized method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1137/0913035
comment: Van der Vorst, H. A. (1992). Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems. SIAM Journal on Scientific and Statistical Computing 13: 631-644.
annotated source: biconjugate gradient stabilized method
annotated property: seeAlso

Axiom
annotated target: Bi-CGSTAB
comment: EXACT
annotated source: biconjugate gradient stabilized method
annotated property: altLabel

ingenious conjugate gradients-squared method
type: Class
subclass of: improved biconjugate gradient method
created: 2011-07-18
definition: A Lanczos-type method for nonsymmetric sparse linear systems. The method is based on a polynomial variant of the conjugate gradients algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000357]. Although related to the so-called bi-conjugate gradients (Bi-CG) algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000358], it does not involve adjoint matrix-vector multiplications, and the expected convergence rate is about twice that of the Bi-CG algorithm.
creator: AZ
alternative label: CGS

Axiom
annotated target: http://identifiers.org/doi/10.1137/0910004
comment: Peter Sonneveld. 1989. CGS, a fast Lanczos-type solver for nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 10, 1 (January 1989), 36-52.
annotated source: ingenious conjugate gradients-squared method
annotated property: seeAlso

Axiom
annotated target: CGS
comment: EXACT
annotated source: ingenious conjugate gradients-squared method
annotated property: altLabel

quasi-minimal residual variant of biconjugate gradient stabilized method
type: Class
subclass of: biconjugate gradient stabilized method
created: 2011-07-19
creator: AZ
alternative label: QMRCGSTAB
definition: QMRCGSTAB is a quasi-minimal residual (QMR) variant of the Bi-CGSTAB algorithm [http://identifiers.org/biomodels.kisao/KISAO_0000394] of van der Vorst for solving nonsymmetric linear systems. The motivation for the QMR variant is to obtain smoother convergence behavior of the underlying method.

Axiom
comment: EXACT
annotated target: QMRCGSTAB
annotated source: quasi-minimal residual variant of biconjugate gradient stabilized method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1137/0915023
comment: T. F. Chan, E. Gallopoulos, V. Simoncini, T. Szeto, and C. H. Tong. 1994. A quasi-minimal residual variant of the Bi-CGSTAB algorithm for nonsymmetric systems. SIAM J. Sci. Comput. 15, 2 (March 1994), 338-347.
annotated source: quasi-minimal residual variant of biconjugate gradient stabilized method
annotated property: seeAlso

improved biconjugate gradient method
type: Class
subclass of: Krylov subspace projection method
subclass of: Restriction
  on property: is similar to
  some values from: biconjugate gradient method
created: 2011-07-19
is organizational: true
creator: AZ
definition: An 'improved biconjugate gradient method' branch contains algorithms which can be viewed as improvements over some of drawbacks of BCG [http://identifiers.org/biomodels.kisao/KISAO_0000358], such as (1) the need for matrix-vector multiplications with A^T (which can be inconvenient as well as doubling the number of matrix-vector multiplications compared to CG [http://identifiers.org/biomodels.kisao/KISAO_0000357] for each increase in the degree of the underlying Krylov subspace), (2) the possibility of breakdowns and (3) erratic convergence behavior.

transpose-free quasi-minimal residual algorithm
type: Class
subclass of: quasi-minimal residual method
created: 2011-07-19
definition: A version of CGS [http://identifiers.org/biomodels.kisao/KISAO_0000393] which 'quasi-minimizes' the residual in the space spanned by the vectors generated by the CGS iteration.
creator: AZ
alternative label: TFQMR

Axiom
comment: EXACT
annotated target: TFQMR
annotated source: transpose-free quasi-minimal residual algorithm
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1137/0914029
comment: Roland W. Freund. 1993. A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems. SIAM J. Sci. Comput. 14, 2 (March 1993), 470-482.
annotated source: transpose-free quasi-minimal residual algorithm
annotated property: seeAlso

preconditioning technique
type: Class
subclass of: modeling and simulation algorithm
subclass of: Restriction
  on property: is used by
  some values from: iterative method for linear system
created: 2011-07-19
creator: AZ
definition: Preconditioning is simply a means of transforming the original linear system into one which has the same solution, but which is likely to be easier to solve with an iterative solver.

Axiom
annotated target: http://identifiers.org/isbn/0898715342
comment: Saad, Yousef (2003). Iterative methods for sparse linear systems (2nd ed.). SIAM.
annotated source: preconditioning technique
annotated property: seeAlso

iterative method for linear system
type: Class
subclass of: modeling and simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: differential-algebraic equation problem
subclass of: Restriction
  on property: has characteristic
  some values from: approximate solution
subclass of: Restriction
  on property: has characteristic
  some values from: linearity of equation
created: 2011-07-19
is organizational: true
creator: AZ

Axiom
annotated target: http://identifiers.org/isbn/0898715342
comment: Saad, Yousef (2003). Iterative methods for sparse linear systems (2nd ed.). SIAM.
annotated source: iterative method for linear system
annotated property: seeAlso

homogeneousness of equation
type: Class
subclass of: linearity of equation
created: 2011-07-19
creator: AZ
definition: Homogeneous equations are of the form Ly = 0, where the differential operator L is a linear operator and y is the unknown function.

symmetricity of matrix
type: Class
subclass of: type of problem
created: 2011-07-19
creator: AZ
definition: In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose.

type of differential equation
type: Class
subclass of: type of problem
created: 2011-07-19
is organizational: true
creator: AZ

steady state method
type: Class
subclass of: modeling and simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: continuous variable
created: 2012-01-17
is organizational: true
definition: A method looking for a steady state of a dynamic system.
creator: AZ
comment: Requested by Frank T. Bergmann on Sunday, November 27, 2011 4:45:30 PM.

Newton-type method
type: Class
subclass of: steady state method
subclass of: Restriction
  on property: has parameter
  some values from: absolute tolerance
subclass of: Restriction
  on property: has parameter
  some values from: relative tolerance
subclass of: Restriction
  on property: has parameter
  some values from: maximum number of steps
subclass of: Restriction
  on property: has characteristic
  some values from: continuous variable
created: 2012-01-18
definition: A method which attacks the solution of a nonlinear problem F(x) = 0 by solving a sequence of liner problems of the same kind. The solution of the system F(x)=0 can be interpreted as a steady state of a dynamic system x'(t)=F(x(t)). The Newton approach will only work if the fixed point [http://identifiers.org/biomodels.teddy/TEDDY_0000086] of the dinamic system is attractive [http://identifiers.org/biomodels.teddy/TEDDY_0000094].
creator: AZ
comment: Requested by Frank T. Bergmann on Sunday, November 27, 2011 4:45:30 PM.

Axiom
annotated target: http://identifiers.org/isbn/9783540210993
comment: Deuflhard, P. Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Springer Series in Computational Mathematics, Vol. 35, 1st ed. 2004.
annotated source: Newton-type method
annotated property: seeAlso

ordinary Newton method
type: Class
subclass of: Newton-type method
created: 2012-01-18
definition: A 'Newton-type method' [http://identifiers.org/biomodels.kisao/KISAO_0000408] which solves the general nonlinear problem F(x)=0 by applying successive linearization F'(x[k])deltax[k]=-F(x[k]), x[k+1]=x[k]+deltax[k], k=0,1,...
creator: AZ

Axiom
annotated target: http://identifiers.org/isbn/9783540210993
comment: Deuflhard, P. Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Springer Series in Computational Mathematics, Vol. 35, 1st ed. 2004.
annotated source: ordinary Newton method
annotated property: seeAlso

simlified Newton method
type: Class
subclass of: Newton-type method
created: 2012-01-18
definition: A 'Newton-type method' [http://identifiers.org/biomodels.kisao/KISAO_0000408] which is characterized by keeping the initial derivative throughout the whole iteration: F'(x[0])deltax[k]=-F(x[k]), x[k+1]=x[k]+deltax[k], k=0,1,...
creator: AZ

Axiom
annotated target: http://identifiers.org/isbn/9783540210993
comment: Deuflhard, P. Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Springer Series in Computational Mathematics, Vol. 35, 1st ed. 2004.
annotated source: simlified Newton method
annotated property: seeAlso

Newton-like method
type: Class
subclass of: Newton-type method
created: 2012-01-18
definition: A 'Newton-type method' [http://identifiers.org/biomodels.kisao/KISAO_0000408] which is characterized by the fact that, in finite dimension, the Jacodian matrices are either replaced by some fixed 'close by' Jacobian F'(z) with z not equal to the initial guess x[0], or by some approximation so that: M'(x[0])deltax[k]=-F(x[k]), x[k+1]=x[k]+deltax[k], k=0,1,...
creator: AZ

Axiom
annotated target: http://identifiers.org/isbn/9783540210993
comment: Deuflhard, P. Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Springer Series in Computational Mathematics, Vol. 35, 1st ed. 2004.
annotated source: Newton-like method
annotated property: seeAlso

inexact Newton method
type: Class
subclass of: Newton-type method
subclass of: Restriction
  on property: uses
  some values from: Class
    intersection of: modeling and simulation algorithm
    intersection of: Restriction
      on property: has characteristic
      some values from: approximate solution
created: 2012-01-18
creator: AZ
definition: For extremely large scale nonlinear problems the arising linear systems for the Newton corrections can no longer be solved directly ('exactly'), but must be solved iterativly ('inexactly) - which gives the name inexact Newton methods. The whole scheme then consists of an inner iteration (at Newton step k): F'(x[k])deltaxi[k]=-F(x[k])+ri[k], k=0,1,... xi[k+1]=x[k]+deltaxi[k], i=0,1,..,imax[k] in terms of residuals ri[k] and an outer iteration where, given x[0], the iterates are defined as x[k+1]=xi[k+1] for i=imax[k], k=0,1,...
alternative label: iterative Newton method
alternative label: truncated Newton method

Axiom
comment: EXACT
annotated target: iterative Newton method
annotated source: inexact Newton method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: truncated Newton method
annotated source: inexact Newton method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/isbn/9783540210993
comment: Deuflhard, P. Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Springer Series in Computational Mathematics, Vol. 35, 1st ed. 2004.
annotated source: inexact Newton method
annotated property: seeAlso

exact Newton method
type: Class
subclass of: Newton-type method
subclass of: Restriction
  on property: uses
  some values from: Class
    intersection of: modeling and simulation algorithm
    intersection of: Restriction
      on property: has characteristic
      some values from: exact solution
created: 2012-01-18
creator: AZ
definition: Any of the finite dimensional Newton-type methods [http://identifiers.org/biomodels.kisao/KISAO_0000408] requires the numerical solution of the linear equations F'(x[k])deltax[k]=-F(x[k]). Whenever direct elimination methods are applicable, we speak of exact Newton methods.
alternative label: direct Newton method

Axiom
annotated target: http://identifiers.org/isbn/9783540210993
comment: Deuflhard, P. Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Springer Series in Computational Mathematics, Vol. 35, 1st ed. 2004.
annotated source: exact Newton method
annotated property: seeAlso

maximum number of steps
type: Class
subclass of: granularity control parameter
subclass of: Restriction
  on property: has type
  some values from: integer
subclass of: Restriction
  on property: is parameter of
  all values from: Newton-type method
created: 2012-01-18
creator: AZ
definition: Maximum number of steps the 'Newton-type method' [http://identifiers.org/biomodels.kisao/KISAO_0000408] algorithm is allowed to take.

partial least squares regression method
type: Class
subclass of: partial least squares regression-like method
subclass of: Restriction
  on property: has parameter
  on class: type of validation
  qualified cardinality: 1
subclass of: Restriction
  on property: has parameter
  on class: number of partial least squares components
  qualified cardinality: 1
created: 2012-01-18
creator: AZ
definition: Multivariate regression method based on estimated latent variables. Related to Principal Component Analysis (PCA) and Principal Component Regression (PCR).
alternative label: PLSR
alternative label: PLSR method
comment: Requested by Kristin Tøndel on Thursday, October 13, 2011 10:46:04 AM.

Axiom
comment: EXACT
annotated target: PLSR
annotated source: partial least squares regression method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/biomodels.kisao/10.1007/BFb0062108
comment: Wold S, Martens H, Wold H (1983) The multivariate calibration method in chemistry solved by the PLS method. In: Lecture notes in Mathematics, Matrix Pencils. Heidelberg: Springer-Verlag. pp. 286-293.
annotated source: partial least squares regression method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: PLSR method
annotated source: partial least squares regression method
annotated property: altLabel

hierarchical cluster-based partial least squares regression method
type: Class
subclass of: partial least squares regression-like method
subclass of: Restriction
  on property: has parameter
  some values from: number of partial least squares components
subclass of: Restriction
  on property: has parameter
  some values from: standardising of variables
subclass of: Restriction
  on property: has parameter
  on class: matrix for clusterization
  qualified cardinality: 1
subclass of: Restriction
  on property: uses
  some values from: partial least squares regression method
subclass of: Restriction
  on property: has parameter
  some values from: mean-centring of variables
subclass of: Restriction
  on property: has characteristic
  some values from: hybridity
created: 2012-01-18
creator: AZ
alternative label: HC-PLSR
alternative label: Multivariate regression method based on separating the observations into clusters and generating Partial Least Squares Regression (PLSR) [http://identifiers.org/biomodels.kisao/KISAO_0000416] models within each cluster. This local regression analysis is suitable for very non-linear systems. PLSR is a regression method based on estimated latent variables, related to Principal Component Analysis (PCA) and Principal Component Regression (PCR). Hierarchical cluster-based partial least squares regression method uses fuzzy C-means clustering, PLSR and Linear Discriminant Analysis (LDA), Quadratic Discriminant Analysis (QDA) or Naive Bayes for classification of new observations to be predicted.
comment: Requested by Kristin Tøndel on Thursday, October 13, 2011 11:13:17 AM.

Axiom
annotated target: http://identifiers.org/doi/10.1186/1752-0509-5-90
comment: Tøndel K, Indahl UG, Gjuvsland AB, Vik JO, Hunter P, et al. (2011) Hierarchical Cluster-based Partial Least Squares Regression is an efficient tool for metamodelling of nonlinear dynamic models. BMC Systems Biology 5: 90.
annotated source: hierarchical cluster-based partial least squares regression method
annotated property: seeAlso

N-way partial least squares regression method
type: Class
subclass of: partial least squares regression-like method
subclass of: Restriction
  on property: has parameter
  on class: number of N-way partial least squares regression factors
  qualified cardinality: 1
created: 2012-01-18
creator: AZ
definition: Multivariate regression method that can be used on N-way data. Based on Partial Least Squares Regression (PLSR) [http://identifiers.org/biomodels.kisao/KISAO_0000416], which is a regression method based on estimated latent variables. PLSR is related to Principal Component Analysis (PCA) and Principal Component Regression (PCR).
alternative label: N-PLS
alternative label: N-way PLSR
alternative label: N-way partial least squares method
comment: Requested by Kristin Tøndel on Thursday, October 13, 2011 11:33:06 AM.

Axiom
comment: EXACT
annotated target: N-way partial least squares method
annotated source: N-way partial least squares regression method
annotated property: altLabel

Axiom
comment: EXACT
annotated target: N-way PLSR
annotated source: N-way partial least squares regression method
annotated property: altLabel

Axiom
annotated target: http://identifiers.org/doi/10.1002/(SICI)1099-128X(199601)10:1<47::AID-CEM400>3.0.CO
comment: Bro R (1996) Multiway calibration. Multilinear PLS. J. Chemometrics 10: 47-6.
annotated source: N-way partial least squares regression method
annotated property: seeAlso

Axiom
annotated target: http://www.models.life.ku.dk/nwaytoolbox
comment: N-way toolbox website
annotated source: N-way partial least squares regression method
annotated property: seeAlso

Axiom
comment: EXACT
annotated target: N-PLS
annotated source: N-way partial least squares regression method
annotated property: altLabel

metamodelling method
type: Class
subclass of: modeling and simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: deterministic system behaviour
created: 2012-01-18
is organizational: true
creator: AZ
definition: Deterministic dynamic models of complex biological systems contain a large number of parameters and state variables, related through nonlinear differential equations with various types of feedback. A metamodel of such a dynamic model is a statistical approximation model that maps variation in parameters and initial conditions (inputs) to variation in features of the trajectories of the state variables (outputs) throughout the entire biologically relevant input space.

Axiom
annotated target: http://identifiers.org/doi/10.1186/1752-0509-5-90
comment: Tøndel K, Indahl UG, Gjuvsland AB, Vik JO, Hunter P, et al. (2011) Hierarchical Cluster-based Partial Least Squares Regression is an efficient tool for metamodelling of nonlinear dynamic models. BMC Systems Biology 5: 90.
annotated source: metamodelling method
annotated property: seeAlso

number of partial least squares components
type: Class
subclass of: modeling and simulation algorithm parameter
subclass of: Restriction
  on property: has type
  some values from: integer
subclass of: Restriction
  on property: is parameter of
  all values from: partial least squares regression-like method
created: 2012-01-18
creator: AZ
definition: Parameter used by 'partial least squares regression method' [http://identifiers.org/biomodels.kisao/KISAO_0000416] describing number of PLS components to include in the regression analysis.

Axiom
annotated target: http://identifiers.org/biomodels.kisao/10.1007/BFb0062108
comment: Wold S, Martens H, Wold H (1983) The multivariate calibration method in chemistry solved by the PLS method. In: Lecture notes in Mathematics, Matrix Pencils. Heidelberg: Springer-Verlag. pp. 286-293.
annotated source: number of partial least squares components
annotated property: seeAlso

type of validation
type: Class
subclass of: modeling and simulation algorithm parameter
subclass of: Restriction
  on property: is parameter of
  all values from: partial least squares regression-like method
created: 2012-01-18
creator: AZ
definition: Parameter of 'partial least squares regression method' [http://identifiers.org/biomodels.kisao/KISAO_0000416] describing how validation is performed. Possible values include cross-validation and test set validation.

number of N-way partial least squares regression factors
type: Class
subclass of: modeling and simulation algorithm parameter
subclass of: Restriction
  on property: is parameter of
  all values from: N-way partial least squares regression method
subclass of: Restriction
  on property: has type
  some values from: integer
created: 2012-01-18
creator: AZ
definition: Parameter of 'N-way partial least squares regression method' [http://identifiers.org/biomodels.kisao/KISAO_0000369] describing the number of factors to compute.
alternative label: number of factors

Axiom
comment: EXACT
annotated target: number of factors
annotated source: number of N-way partial least squares regression factors
annotated property: altLabel

partial least squares regression-like method
type: Class
subclass of: metamodelling method
subclass of: Restriction
  on property: has parameter
  on class: type of validation
  qualified cardinality: 1
created: 2012-01-18
is organizational: true
creator: AZ
definition: Method for building regression models between independent and dependent variables.

mean-centring of variables
type: Class
subclass of: variables preprocessing parameter
subclass of: Restriction
  on property: is parameter of
  all values from: partial least squares regression-like method
subclass of: Restriction
  on property: has type
  some values from: boolean
disjoint with: standardising of variables
created: 2012-01-18
definition: A boolean parameter of the 'hierarchical cluster-based partial least squares regression method' [http://identifiers.org/biomodels.kisao/KISAO_0000417] specifying whether the variables were mean-centred prior to the regression analysis.
creator: AZ

Axiom
annotated target: http://identifiers.org/doi/10.1186/1752-0509-5-90
comment: Tøndel K, Indahl UG, Gjuvsland AB, Vik JO, Hunter P, et al. (2011) Hierarchical Cluster-based Partial Least Squares Regression is an efficient tool for metamodelling of nonlinear dynamic models. BMC Systems Biology 5: 90.
annotated source: mean-centring of variables
annotated property: seeAlso

standardising of variables
type: Class
subclass of: variables preprocessing parameter
subclass of: Restriction
  on property: has type
  some values from: boolean
subclass of: Restriction
  on property: is parameter of
  all values from: partial least squares regression-like method
created: 2012-01-18
definition: A boolean parameter of the 'hierarchical cluster-based partial least squares regression method' [http://identifiers.org/biomodels.kisao/KISAO_0000417] specifying whether the variables were standardised (divided by their standard deviations) prior to the regression analysis.
creator: AZ

Axiom
annotated target: http://identifiers.org/doi/10.1186/1752-0509-5-90
comment: Tøndel K, Indahl UG, Gjuvsland AB, Vik JO, Hunter P, et al. (2011) Hierarchical Cluster-based Partial Least Squares Regression is an efficient tool for metamodelling of nonlinear dynamic models. BMC Systems Biology 5: 90.
annotated source: standardising of variables
annotated property: seeAlso

number of clusters
type: Class
subclass of: clusterization parameter
subclass of: Restriction
  on property: has type
  some values from: integer
disjoint with: matrix for clusterization
created: 2012-01-18
creator: AZ
definition: Parameter specifying the number of clusters used by C-means algorithm.

Axiom
annotated target: http://identifiers.org/doi/10.1186/1752-0509-5-90
comment: Tøndel K, Indahl UG, Gjuvsland AB, Vik JO, Hunter P, et al. (2011) Hierarchical Cluster-based Partial Least Squares Regression is an efficient tool for metamodelling of nonlinear dynamic models. BMC Systems Biology 5: 90.
annotated source: number of clusters
annotated property: seeAlso

matrix for clusterization
type: Class
subclass of: clusterization parameter
subclass of: Restriction
  on property: is parameter of
  all values from: hierarchical cluster-based partial least squares regression method
created: 2012-01-18
definition: A matrix to do the clustering in 'hierarchical cluster-based partial least squares regression method' [http://identifiers.org/biomodels.kisao/KISAO_0000417].
creator: AZ

Axiom
annotated target: http://identifiers.org/doi/10.1186/1752-0509-5-90
comment: Tøndel K, Indahl UG, Gjuvsland AB, Vik JO, Hunter P, et al. (2011) Hierarchical Cluster-based Partial Least Squares Regression is an efficient tool for metamodelling of nonlinear dynamic models. BMC Systems Biology 5: 90.
annotated source: matrix for clusterization
annotated property: seeAlso

clusterization parameter
type: Class
subclass of: modeling and simulation algorithm parameter
is organizational: true
creator: AZ
definition: Parameter used by algorithms performing clusterization.

variables preprocessing parameter
type: Class
subclass of: modeling and simulation algorithm parameter
is organizational: true
creator: AZ

IDA-like method
type: Class
subclass of: Newton-type method
created: 2012-05-24
is organizational: True
creator: AZ
definition: Solves real differential-algebraic systems in N-space, in the general form F(t,y,y')=0, y(t0)=y0, y'(t0)=y'0. At each step, a Newton iteration [http://identifiers.org/biomodels.kisao/KISAO_0000408] leads to linear systems Jx=b, which are solved by one of five methods - two direct (dense or band; serial version only) and three Krylov [http://identifiers.org/biomodels.kisao/KISAO_0000354] (GMRES [http://identifiers.org/biomodels.kisao/KISAO_0000353], BiCGStab [http://identifiers.org/biomodels.kisao/KISAO_0000392], or TFQMR [http://identifiers.org/biomodels.kisao/KISAO_0000396]).

Axiom
annotated target: http://identifiers.org/doi/10.1145/1089014.1089020
comment: Hindmarsh, A. C., et al., SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers, ACM Trans. Math. Softw., 31:363-396, 2005.
annotated source: IDA-like method
annotated property: seeAlso

CVODE-like method
type: Class
subclass of: modeling and simulation algorithm
subclass of: Restriction
  on property: has characteristic
  some values from: ordinary differential equation problem
subclass of: Class
  union of: Restriction
    on property: uses
    some values from: Adams-Moulton method
  union of: Restriction
    on property: uses
    some values from: backward differentiation formula
created: 2012-05-24
is organizational: True
creator: AZ
definition: Solves ODE initial value problems, in real N-space, written as y'=f(t,y), y(t0)=y0. It is capable for stiff and non-stiff systems and uses two different linear multi-step methods, namely the Adam-Moulton [http://identifiers.org/biomodels.kisao/KISAO_0000280] method and the backward differentiation formula [http://identifiers.org/biomodels.kisao/KISAO_0000288].

Axiom
annotated target: http://identifiers.org/doi/10.1145/1089014.1089020
comment: Hindmarsh, A. C., et al., SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers, ACM Trans. Math. Softw., 31:363-396, 2005.
annotated source: CVODE-like method
annotated property: seeAlso

Thing
is organizational: true