Sampling Methods for Recovering Buried Corroded Boundaries from Partial Electrostatic Cauchy Data

SIAM Journal on Applied Mathematics, vol. 85, pp. 2215–2241

Abstract

Abstract. We consider the inverse shape and parameter problem for detecting corrosion from partial boundary measurements. This problem models the nondestructive testing for a partially buried object from electrostatic measurements on the accessible part of the boundary. The main novelty is the extension of the linear sampling and factorization methods to an electrostatic problem with partial measurements. These methods so far have only mainly applied to recovering interior defects, which is a simpler problem. Another important aspect of this paper is in our numerics, where we derive a system of boundary integral equations to recover the mixed Green’s function needed for our inversion. With this, we are able to analytically and numerically solve the inverse shape problem. For the inverse parameter problem, we prove uniqueness and Lipschitz-stability (in a finite dimensional function space) assuming that one has the associated Neumann-to-Dirichlet operator on the accessible part of the boundary.

Authors 3

1. Isaac Harris

   Purdue University West Lafayette

   Affiliation as printed

   Department of Mathematics, Purdue University, West Lafayette, IN 47907 USA

2. Andreas Kleefeld Aachen Faculty of Medical Engineering and Technomathematics

   FH Aachen · Forschungszentrum Jülich · Jülich Supercomputing Centre

   Affiliation as printed

   Forschungszentrum Jülich GmbH, Jülich Supercomputing Centre, Wilhelm-Johnen-Straße, 52425 Jülich, Germany and University of Applied Sciences Aachen, Faculty of Medical Engineering and Technomathematics, Heinrich-Mußmann-Str. 1, 52428, Jülich, Germany

3. Heejin Lee

   Duke Kunshan University

   Affiliation as printed

   Zu Chongzhi Center for Mathematics and Computational Sciences, Duke Kunshan University, Kunshan, 215316 Jiangsu Province, People’s Republic of China

   Zu Chongzhi Center for Mathematics and Computational Sciences, Duke Kunshan University, Kunshan, 215316 Jiangsu Province, People's Republic of China