On parameter identification problems for elliptic boundary value problems in divergence form, Part I: An abstract framework

TL;DR

This paper develops a general abstract functional analytic framework for parameter identification problems in elliptic PDEs, ensuring well-posedness and properties like the tangential cone condition to support inverse problem solutions.

Contribution

It introduces a broad, unified framework for elliptic boundary value inverse problems, including conditions for the parameter-to-state map and the tangential cone condition.

Findings

Framework applies to diverse elliptic boundary value problems.

Parameter-to-state operators satisfy the tangential cone condition.

Covers inverse medium and terahertz tomography problems.

Abstract

Parameter identification problems for partial differential equations are an important subclass of inverse problems. The parameter-to-state map, which maps the parameter of interest to the respective solution of the PDE or state of the system, plays the central role in the (usually nonlinear) forward operator. Consequently, one is interested in well-definedness and further analytic properties such as continuity and differentiability of this operator w.r.t. the parameter in order to make sure that techniques from inverse problems theory may be successfully applied to solve the inverse problem. In this work, we present a general functional analytic framework suited for the study of a huge class of parameter identification problems including a variety of elliptic boundary value problems (in divergence form) with Dirichlet, Neumann, Robin or mixed boundary conditions. In particular, we show that the corresponding parameter-to-state operators fulfil, under suitable conditions, the tangential cone condition, which is often postulated for numerical solution techniques. This framework particularly covers the inverse medium problem and an inverse problem that arises in terahertz tomography.