Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem

Bastian Gebauer; Nuutti Hyvönen

doi:10.3934/ipi.2008.2.355

Article Contents

2008, Volume 2, Issue 3: 355-372. Doi: 10.3934/ipi.2008.2.355

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Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem

Bastian Gebauer^1, and
Nuutti Hyvönen^2,

1.
Institut für Mathematik, Johannes Gutenberg-Universität Maint, 55099 Mainz
2.
Institute of Mathematics, Helsinki University of Technology, FI-02015 HUT

Received: November 30, 2007

Revised: May 31, 2008

Published: July 2008

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Abstract

In various imaging problems the task is to use the Cauchy data of the solutions to an elliptic boundary value problem to reconstruct the coefficients of the corresponding partial differential equation. Often the examined object has known background properties but is contaminated by inhomogeneities that cause perturbations of the coefficient functions. The factorization method of Kirsch provides a tool for locating such inclusions. In this paper, the factorization technique is studied in the framework of coercive elliptic partial differential equations of the divergence type: Earlier it has been demonstrated that the factorization algorithm can reconstruct the support of a strictly positive (or negative) definite perturbation of the leading order coefficient, or if that remains unperturbed, the support of a strictly positive (or negative) perturbation of the zeroth order coefficient. In this work we show that these two types of inhomogeneities can, in fact, be located simultaneously. Unlike in the earlier articles on the factorization method, our inclusions may have disconnected complements and we also weaken some other a priori assumptions of the method. Our theoretical findings are complemented by two-dimensional numerical experiments that are presented in the framework of the diffusion approximation of optical tomography.

Keywords:

Factorization method,
inverse elliptic boundary value problems,
inclusions.

Mathematics Subject Classification: Primary: 35R30; Secondary: 65N21.

Citation:

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