Global uniqueness and Lipschitz-stability for the inverse Robin transmission problem

TL;DR

This paper establishes global uniqueness and Lipschitz stability for an inverse Robin transmission problem, providing explicit stability constants and a numerical reconstruction method using PDE solutions and the BFGS algorithm.

Contribution

It introduces a novel approach to quantify Lipschitz stability constants explicitly without relying on analytic estimates, and develops an efficient numerical reconstruction method.

Findings

Proved global uniqueness for the inverse Robin problem.

Derived explicit Lipschitz stability constants from a priori data.

Implemented a BFGS-based algorithm demonstrating effective reconstructions.

Abstract

In this paper, we consider the inverse problem of detecting a corrosion coefficient between two layers of a conducting medium from the Neumann-to-Dirichlet map. This inverse problem is motivated by the description of the index of corrosion in non-destructive testing. We show a monotonicity estimates between the Robin coefficient and the Neumann-to-Dirichlet operator. We prove a global uniqueness result and Lipschitz stability estimate, and show how to quantify the Lipschitz stability constant for a given setting. Our quantification of the Lipschitz constant does not rely on quantitative unique continuation or analytic estimates of special functions. Instead of deriving an analytic estimate, we show that the Lipschitz constant for a given setting can be explicitly calculated from the a priori data by solving finitely many well-posed PDEs. Our arguments rely on standard (non-quantitative) unique continuation, a Runge approximation property, the monotonicity result and the method of localized potentials. To solve the problem numerically, we reformulate the inverse problem into a minimization problem using a least square functional. The reformulation of the minimization problem as a suitable saddle point problem allows us to obtain the optimality conditions by using differentiability properties of the min-sup formulation. The reconstruction is then performed by means of the BFGS algorithm. Finally, numerical results are presented to illustrate the efficiency of the proposed alogorithm.