On rational approximation methods for inverse source problems

Martin Hanke; William Rundell

doi:10.3934/ipi.2011.5.185

Article Contents

2011, Volume 5, Issue 1: 185-202. Doi: 10.3934/ipi.2011.5.185

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On rational approximation methods for inverse source problems

Martin Hanke^1, and
William Rundell^2,

1.
Institute of Mathematics, Johannes Gutenberg-Universität, 55099 Mainz
2.
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368

Received: February 28, 2010

Revised: June 30, 2010

Published: January 2011

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Abstract

The basis of most imaging methods is to detect hidden obstacles or inclusions within a body when one can only make measurements on an exterior surface. Such is the ubiquity of these problems, the underlying model can lead to a partial differential equation of any of the major types, but here we focus on the case of steady-state electrostatic or thermal imaging and consider boundary value problems for Laplace's equation. Our inclusions are interior forces with compact support and our data consists of a single measurement of (say) voltage/current or temperature/heat flux on the external boundary. We propose an algorithm that under certain assumptions allows for the determination of the support set of these forces by solving a simpler "equivalent point source" problem, and which uses a Newton scheme to improve the corresponding initial approximation.

Keywords:

Inverse source problem,
rational approximation.

Mathematics Subject Classification: Primary: 35R20.

Citation:

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