Parameter identification for an elliptic partial differential equation with distributed noisy data

Abstract

This paper is devoted to the identification of a parameter in an elliptic partial differential equation from noisy distributed data. It can be divided into two parts. In the first part we only consider smooth parameters and we propose a well-posed formulation, which takes measurement errors into account. After stating an existence result, optimality conditions are derived from the Frechet derivative of the Lagrange functional. Finite-element discretization is then introduced and numerical experiments as well as error estimates are presented. The second part deals with the identification of discontinuous parameters. We adapt the previously presented method by using total variation regularization. Finite-dimensional approximations of bounded variation functions are then studied. We point out the importance of the decomposition of the domain when using approximating subspaces consisting of piecewise constant functions. Moreover, the non-differentiability of total variation is circumvented; numerical experiments are given.