Integral equations for inverse problems in corrosion detection from partial Cauchy data

Fioralba Cakoni,Rainer Kress

doi:10.3934/ipi.2007.1.229

Abstract

We consider the inverse problem to recover a part $\Gamma_c$ of the boundary of a simply connected planar domain $D$ from a pair of Cauchy data of a harmonic function $u$ in $D$ on the remaining part $\partial D\setminus \Gamma_c$ when $u$ satisfies a homogeneous impedance boundary condition on $\Gamma_c$. Our approach extends a method that has been suggested by Kress and Rundell [17] for recovering the interior boundary curve of a doubly connected planar domain from a pair of Cauchy data on the exterior boundary curve and is based on a system of nonlinear integral equations. As a byproduct, these integral equations can also be used for the problem to extend incomplete Cauchy data and to solve the inverse problem to recover an impedance profile on a known boundary curve. We present the mathematical foundation of the method and illustrate its feasibility by numerical examples.

Highlights

We consider a connected bounded domain D ⊂ R2 with piece-wise smooth boundary ∂D
One can think of u as representing the electrostatic potential in a conducting body D of which only the portion Γm of the boundary is accessible to measurements. In this application, the above inverse problem can be interpreted as to determine the shape of the inaccessible portion Γc of the boundary from a knowledge of the imposed voltage u|Γm and the measured resulting current ∂u/∂ν|Γm on Γm
To derive nonlinear integral equations that are equivalent to the inverse problem we represent the solution u of (1.1)–(1.3) as surface superposition of point sources given by the fundamental solution

Summary

Introduction

We consider a connected bounded domain D ⊂ R2 with piece-wise smooth boundary ∂D. Key words and phrases: Inverse boundary value problem, integral equations, partial boundary measurements, impedance boundary condition. In this application, the above inverse problem can be interpreted as to determine the shape of the inaccessible portion Γc of the boundary from a knowledge of the imposed voltage u|Γm and the measured resulting current ∂u/∂ν|Γm on Γm Various applications of this problem (or slightly modified versions) are discussed in [1, 2, 5] (see the references therein) where, in general, the authors consider only the reconstruction of the boundary impedance λ as a function of space on the inaccessible portion of the boundary.

Uniqueness for the inverse problem

Nonlinear integral equations

Completion of Cauchy data

The iteration scheme

Numerical examples