Sampling methods for recovering buried corroded boundaries from partial electrostatic Cauchy data

arXiv (Cornell University)

Abstract

We consider the inverse shape and parameter problem for detecting corrosion from partial boundary measurements. This problem models the non-destructive 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 which is 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

2. Andreas Kleefeld

   Forschungszentrum Jülich · Jülich Supercomputing Centre

   Affiliation as printed

   Forschungszentrum Jülich GmbH , Jülich Supercomputing Centre , Wilhelm-Johnen-Straße , 52425 Jülich , Germany

3. Heejin Lee Aachen Faculty of Medical Engineering and Technomathematics

   FH Aachen · Purdue University West Lafayette

   Affiliation as printed

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

   University of Applied Sciences Aachen , Faculty of Medical Engineering and Technomathematics , Heinrich-Mußmann-Str. 1 , 52428 Jülich , Germany