18th GAMM-Seminar Leipzig on
Multigrid and related methods for optimization problems

Max-Planck-Institute for Mathematics in the Sciences
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  18th GAMM-Seminar
January, 24th-26th, 2002
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  Abstract Angela Kunoth, Fri, 14.30-14.55 Previous Contents Next  
  Iterative Solution of Linear-Quadratic Elliptic Control Problems in Wavelet Discretization
Angela Kunoth (Universität Bonn)

For the numerical solution of a control problem governed by an elliptic boundary value problem with boundary control, wavelet techniques are employed. A quadratic cost functional involving natural norms of the state and the control is to be minimized. Firstly the constraint is formulated as a saddle point problem, allowing to handle varying boundary conditions explicitly.

Deviating from standard approaches, then biorthogonal wavelets are used to derive an equivalent infinite discretized control problem which involves only l_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. This together with the l_2-setting circumvents the preconditioning problem since all operators have uniformly bounded condition numbers independent of the discretization.

In order to numerically solve the resulting (finite-dimensional) linear system of the weakly coupled saddle point problems, fully iterative methods are presented and their convergence is shown. The one class of methods can be viewed as inexact gradient or conjugate gradient schemes, consisting of an outer iteration which alternatingly picks the two saddle point problems, and an inner iteration to solve each of the saddle point problems, exemplified in terms of the Uzawa algorithm. The other class is an All-In-One Solver applied to the normal equations.

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Concept, Design and Realisation
[O->]Jens Burmeister (Uni Kiel), Kai Helms (MPI Leipzig)
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