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
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