Estimation of discontinuous parameters in linear elliptic equations by a regularized inverse problem

Abstract

To calculate the parameters k in a stationary model Aku=f from measurements yˆ of the solution u, i.e. to solve the inverse problem, usually a regularized minimization problem is solved and its solution kˆ is considered as estimate of the true parameters. We focus on a situation which occurs e.g. for the coefficient functions k of a linear elliptic operator Aku=−divk∇u on a domain Ω⊂Rd and the total variation as regularization term. As functions k∈BV(Ω) of bounded variation can be discontinuous, existence of strong solutions u of Aku=f cannot be guaranteed and existence of minimizers cannot be obtained by standard methods. In this article, we prove solvability and stability for a general regularized minimization problem under weak assumptions, which particularly hold in case of BV(Ω) as parameter space due to a higher integrability result.