Adjoint equations
My interest in adjoint equations began with Antony Jameson's research
on optimal aerodynamic design. This technique, using the adjoint
potential, Euler or Navier-Stokes equations, is very efficient in
determining the gradient of an objective function with respect to
many design variables.
Other web pages discuss my interests in the use of adjoint solutions
for optimal design within HYDRA,
and error analysis.
In this page, I discuss my interests in the underlying theory,
formulation, approximation and solution of adjoint equations.
In 1997, Niles Pierce and I considered the proper formulation of
adjoint boundary conditions for the Euler and Navier-Stokes equations,
proving that there is a limited set of objective functions for which
the standard formulation of the problem is wellposed, and determining
some key properties of the solutions to the adjoint quasi-1D and 2D
Euler equations [1]. This led to papers in which we derived a
closed form solution for the adjoint quasi-1D Euler equations, with
and without shocks [2,5].
Working at the interface between mathematics and engineering, I've
always been concerned about explaining key mathematical approaches as
clearly as possible to the engineering community. Paper [3] attempts
to explain the key ideas of optimal design using adjoints, introducing
the ideas at the algebraic level before proceeding to partial differential
equations.
I am a firm advocate of the "fully-discrete" approach to approximating
adjoint equations, in which one starts with the discretisation of the
original nonlinear PDE, and then linearises it and forms its transpose.
This gives numerical results which would be identical to finite difference
sensitivities obtained from the nonlinear calculation with infinitesimal
step size on a computer with infinite machine precision. It is not that
there is anything theoretically wrong with the "continuous" approach of
formulating the adjoint PDE and then approximating it using a discretisation
which might be entirely different to that used for the nonlinear PDE.
Rather, it is pragmatic issues which make me prefer the "fully-discrete"
approach.
One is the fact that the "fully-discrete" method is completely prescriptive;
there is a straightforward process by which one generates the adjoint code,
and this can be significantly aided by the use of
Automatic Differentiation
techniques to automatically generate key pieces of the adjoint code.
Another is that it is possible to solve the discrete adjoint
equations using an adjoint version of the highly-optimised iterative
methods developed for the nonlinear equations. This is the subject
of papers [4,6], with [7] also discussing the complexities that arise
with the imposition of strong boundary conditions.
My latest research interest is in what happens when the underlying nonlinear
solution has a shock, and the conditions under which the computed adjoint
solution will approach the analytic solution as one refines the computational
grid. My first paper on this topic [8] presents numerical results indicating
it is necessary to smear the shock over a few grid points to get a convergent
adjoint discretisation. Subsequent numerical analysis, in joint work with
Stefan Ulbrich, has proved that this is the case.
References
- M.B. Giles and N.A. Pierce
`Adjoint equations in CFD: duality, boundary conditions and solution
behaviour'.
AIAA Paper 97-1850, 1997.
(gzipped postscript file: 192kb)
- M.B. Giles and N.A. Pierce.
`On the properties of solutions of the adjoint Euler equations'.
6th ICFD Conference on Numerical Methods for Fluid Dynamics,
Oxford, UK, 1998.
(gzipped postscript file: 120kb)
- M.B. Giles and N.A. Pierce.
`An introduction to the adjoint approach to design'.
Flow, Turbulence and Combustion, 65(3-4):393-415, 2000.
(PDF file: 184kb)
- M.B. Giles
On the use of Runge-Kutta time-marching and multigrid for the
solution of steady adjoint equations.
Report NA-00/10,
Oxford University Computing Laboratory, 2000.
- M.B. Giles and N.A. Pierce.
`Analytic adjoint solutions for the quasi-one-dimensional Euler equations'.
Journal of Fluid Mechanics, 426:327-345, 2001.
(PDF file: 562kb)
- M.B. Giles
`On the iterative solution of adjoint equations',
in Automatic Differentiation: From Simulation to Optimization,
pages 145-152.
G. Corliss, C. Faure, A. Griewank, L. Hascoet, U. Naumann, editors,
Springer-Verlag, 2001.
(gzipped postscript file: 48kb)
- M.B. Giles, M.C. Duta, J.-D. Muller and N.A. Pierce.
`Algorithm developments for discrete adjoint methods'.
AIAA Journal, 41(2), 2003.
(PDF file: 290kb).
- M.B. Giles.
`Discrete adjoint approximations with shocks'.
Hyperbolic Problems: Theory, Numerics, Applications,
editors T. Hou and E. Tadmor, Springer-Verlag 2003.
(gzipped postscript file: 70kb).
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