A multigrid approach to the optimization of systems governed by differential equations

Abstract

We consider the optimization of systems governed by differential equations. In such problems, one has a set of design variables along with a set of state variables, the two sets of variables being related through a set of differential equation constraints. The overall computational cost of optimization is determined by the level of discretizatio n used to numerically solve the governing differential equations. If a fine discretization is used, one expects a greater degree of physical and mathematical fidelity to the problem under consideration, but the large number of state variables can make the cost of optimization prohibitive. We present here a multigrid algorithm that uses solutions to optimization problems based on coarser discretizations, which are less expensive to compute, in a systematic manner to help us obtain the solution of the optimization problem based on a finer discretization. Of interest is the fact that the approach is applicable in situations where multigrid applied only to the solution of the differential equation might not be applicable or effective. We give evidence (both theoretical and numerical) that a multigrid approach can often be successful in the more general setting of optimization.