Why linear sampling really seems to work

Martin Hanke

doi:10.3934/ipi.2008.2.373

Article Contents

2008, Volume 2, Issue 3: 373-395. Doi: 10.3934/ipi.2008.2.373

This issue Previous Article Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem Next Article On the regularization of the inverse conductivity problem with discontinuous conductivities

Why linear sampling really seems to work

Martin Hanke^1,

1.
Institute of Mathematics, Johannes Gutenberg-Universität, 55099 Mainz

Received: May 31, 2008

Revised: May 31, 2008

Published: July 2008

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Abstract

We reconsider the Linear Sampling Method by Colton and Kirsch, and provide an analysis which may serve as a justification of the method for problems where the Factorization Method is known to work. As a by-product, however, we obtain convincing arguments that one popular implementation of the Linear Sampling Method may not be as robust as is commonly believed. Our approach stems from the theory of regularization methods for linear ill-posed operator equations. More precisely, we derive a novel asymptotic analysis of the Tikhonov method if the exact right-hand side is inconsistent, i.e., does not belong to the (dense) range of the corresponding operator. It appears possible that our results can be a starting point to derive a calibration of standard implementations of the Linear Sampling Method, in order to obtain reconstructions of the scattering obstacles that go beyond an approximate localization of their respective positions.

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Mathematics Subject Classification: Primary: 35R30, 65N21.

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