IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

Wavelet Methods for Elliptic Boundary Value Problems in Optimal Control

  • Sept. 28, 2001
  • 4:30 p.m.
  • LeConte 312

Abstract

In the talk wavelets will be employed for the fast numerical solution of a control problem governed by an elliptic boundary value problem with boundary control. A quadratic cost functional involving natural norms of the state and the control is to be minimized. Firstly the constraint, the elliptic boundary value problem, is formulated as a saddle point problem, allowing to handle varying boundary conditions explicitly. This is combined with a fictitious domain approach in order to cover also more complicated boundaries.

Deviating from standard approaches, then (biorthogonal) wavelets are used to derive an equivalent infinite discretized control problem which involves only $\ell _ 2$--norms and --operators.

Classical methods from optimization yield the corresponding optimality conditions in terms of two weakly coupled (still infinite) saddle point problems for which a unique solution exists. For deriving finite--dimensional systems which are uniformly invertible, stability of the discretizations has to be ensured.

In order to numerically solve the resulting (finite--dimensional) linear system of the weakly coupled saddlepoint problems, a fully iterative method is proposed which can be viewed as an inexact gradient scheme. It consists of a gradient algorithm as an outer iteration which alternatingly picks the two saddle point problems, and an inner iteration to solve each of the saddle point problems like e.g. the Uzawa algorithm. It is proved here that this strategy converges, provided that the inner systems are solved sufficiently well. Moreover, since the system matrix is well--conditioned, it is shown that in combination with a nested iteration strategy this iteration is asymptotically optimal in the sense that it provides the solution on discretization level J with an overall amount of arithmetic operations that is proportional to the number of unknowns N_J on that level.

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