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Identification of matrix parameters in elliptic PDEs

Deckelnick K., Hinze M.

Control and Cybernetics

2011

Vol. 40, no 4

957-969

In the present work we treat the inverse problem of identifying the matrix-valued diffusion coefficient of an elliptic PDE from multiple interior measurements with the help of techniques from PDE constrained optimization. We prove existence of solutions using the concept of H-convergence and employ variational discretization for the discrete approximation of solutions. Using a discrete version of H-convergence we are able to establish the strong convergence of the discrete solutions. Finally we present some numerical results.

A note on variational discretization of elliptic Neumann boundary control

Hinze M., Matthes U.

Control and Cybernetics

2009

Vol. 38, no 3

577-591

We consider variational discretization of Neumann-type elliptic optimal control problems with constraints on the control. In this approach the cost functional is approximated by a sequence of functionals, which are obtained by discretizing the state equation with the help of linear finite elements. The control variable is not discretized. Error bounds for control and state are obtained both in two and three space dimensions. Finally, we discuss some implementation issues of a generalized Newton method applied to the numerical solution of the problem class under consideration.