Finite element approximation of source term identification with TV-regularization

Abstract

In this paper we investigate the problem of recovering the source f in the elliptic system from an observation z of the state u on a part of the boundary , where the functions and j are given. For particular interest in reconstructing probably discontinuous sources, we use the standard least squares method with the total variation regularization, i.e. we consider a minimizer of the minimization problem as a reconstruction. Here denotes the unique weak solution of the above elliptic system which depends on the source term f , is the total variation of f , is the regularization parameter, the admissible set where , are given constants, and is the Banach space of all bounded total variation functions. We approximate the problem with piecewise linear and continuous finite elements, where denotes the corresponding finite element approximation of u. This leads to the minimization problem where . In theorems and we provide the numerical analysis for the discrete solutions f h of , and also propose an algorithm to stably solve this discrete minimization problem, where we are led by the algorithmic developments of Bartels (2012 SIAM J. Numer. Anal. 50 1162–80); Tian and Yuan (2016 Inverse Problems 32 115011). In particular we prove that the iteration sequence generated by this algorithm converges to a minimizer of , and that a convergence measure of the kind is satisfied. Finally, a numerical experiment is presented to illustrate our theoretical findings.