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AD 2000 - From Simulation to Optimization
June 19-23, 2000
INRIA Sophia Antipolis
Sophia Antipolis, France |
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Organizers
George Corliss, Christele Faure, Andre Galligo, Andreas Griewank, Laurent Hascoet, Uwe Naumann
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Approximate Functions and Gradients in Optimal Control and Optimal Shape Design
by
Olivier Pironneau
Approximate Functions and Gradients in Optimal Control and Optimal Shape Design
Approximate Functions and Gradients in Optimal Control and Optimal Shape Design
Abstract
Control problems involving iterative solutions of the state
equations, PDEs in our case, are difficult for AD because an
adjoint state is attached to each iteration and requires the
storage of the entire history of the iterative procedure.
We propose a method to simplify this problem which takes advantage
of mesh refinement and
- inserts a test in the iteration loop for the state equations to
stop it before convergence
- allows the use of approximate gradients for the discrete
problem.
This results into a family of optimization algorithms for which
the use of approximate gradients is justified and does not prevent
convergence.
The result can be used automatically by AD programs only if the
solver has been written with automatic mesh refinement.
In [Polak97] we find master algorithms for solving finite
dimensional optimization problems with automatic mesh adaption
within the optimization loops when both the cost function value
and its gradient are computed approximately. In this talk we
present a new master algorithm that combines these features and
accounts for the fact that in many instances the cost functions
and gradients need not be computed exactly.
We implement this master algorithm using approximate steepest
descent and Armijo Gradient methods for the solution of 3
problems: a two point boundary value problem where the linear
system corresponding to the ODE is solved approximately only; a
distributed control problem in which the discretized dynamics are
solved using a Domain Decomposition algorithm which can be
implemented on parallelized computers. And an optimal shape design
problem with mesh adaption and iterative solver.
Figure
|
Figure 1 Example of behavior of cost versus iteration number |
for
a gradient method with and without mesh adaption and approximate
gradient due to incomplete iteration cycle in a Gauss-Seidel
solver for the linear system of a two point boundary value problem
arising as the state equation of the optimal control problem.
Footnotes:
1LAN, University of Paris 6.
2EECS, University of California, Berkeley.
Date received: January 31, 2000
Copyright © 2000 by the author(s).
The author(s) of this document and the organizers of the conference
have granted their consent to include this abstract in
Atlas Mathematical Conference Abstracts.
Document # cads-52.