A TRUST REGION METHOD FOR PARABOLIC BOUNDARY CONTROL PROBLEMS

A TRUST REGION METHOD FOR PARABOLIC BOUNDARY CONTROL PROBLEMS

C.T.KELLEY AND E.W.SACHS

Abstract.In this paper we develop a trust region algorithm for constrained parabolic boundary control problems. The method is a projected form of the Steihaug trust-region-CG method with a smoothing step added at each iteration to improve performance in the global phase and provide mesh-independent sup-norm convergence in the terminal phase.

Key words.Trust region methods,inexact Newton methods,optimal control

AMS subject classifications.49K20,65F10,49M15,65J15,65K10

1.Introduction.In this paper we show how methods that combine CG iteration and trust re-gion globalization for optimization problems subject to simple bounds can be applied in a infinite dimensional setting to parabolic optimal control problems.This paper addresses the global conver-gence questions left open by our previous work[11],[13],[14],on fast multilevel algorithms for local convergence by showing how trust region-CG algorithms can solve the coarse mesh problems needed to initialize the multilevel method in an efficient and mesh-independent way.The algorithm uses the postsmoothing step from[14]and[11]to improve the performance of the iteration.

For unconstrained problems,our approach differs from[18]only in that after a step and trust region radius have been accepted,a smoothing iteration like those in[11]and[14]is attempted. Unlike our previous work,however,an Armijo[1]line search is added to the smoothing step to ensure decrease in the objective function.This new form of the smoothing step is a scaled steepest descent algorithm.The local theory from[11]and[14]implies that full smoothing steps are taken near the solution.

The effect of this in the infinite dimensional case is to allow one to make sup-norm error estimates in the terminal phase of the iteration[11],[14].In the constrained case,we differ from the algorithm in[8]in more ways.We use trust region and solve unconstrained trust region problems,using the reduced Hessian at the current point to build the quadratic model.The reason for this is to make the trust region problem as easy to solve as possible and to eliminate the need to explicitly compute a generalized Cauchy point.We update the active set after the trust region step has been computed with a scaled projected gradient step(similar to[8]).The scaling serves the purpose of becoming an inexact implementation of the algorithm in[11]and[14]when full steps are taken.We then obtain fast local convergence in the norm.Our local convergence theory does not depend on identification of the active set infinitely many iterations but rather applies the measure-theoretic ideas in[14].Hence,our trust region algorithm becomes an inexact projected Newton method in the terminal phase of the iteration with local convergence properties covered by

Kelley@ncsu.edu).The research of this author was supported by National Science Foundation grant#DMS-9321938and North Atlantic Treaty Organization grant#CRG920067. Computing was partially supported by an allocation of time from the North Carolina Supercomputing Center.

Universit¨a t Trier,FB IV–Mathematik and Graduiertenkolleg Mathematische Optimierung,54296Trier,Germany (sachs@uni-trier.de).The research of this author was supported by North Atlantic Treaty Organization grant #CRG920067.

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