Convergence and error analysis of a numerical method for the identification of matrix parameters in elliptic PDEs

We analyze a numerical method for solving the inverse problem of identifying the diffusion matrix in an elliptic PDE from distributed noisy measurements. We use a regularized least-squares approach in which the state equations are given by a finite element discretization of the elliptic PDE. The unknown matrix parameters act as control variables and are handled with the help of variational discretization as introduced in (Hinze M 2005 Comput. Optim. Appl. 30 45-61). For a suitable coupling of Tikhonov regularization parameter, finite element grid size and noise level we are able to prove L-2-convergence of the discrete solutions to the unique norm-minimal solution of the identification problem; corresponding convergence rates can be obtained provided that a suitable projected source condition is fulfilled. Finally, we present a numerical experiment which supports our theoretical findings.