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The Factorization Method for Inverse Problems$
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Andreas Kirsch and Natalia Grinberg

Print publication date: 2007

Print ISBN-13: 9780199213535

Published to Oxford Scholarship Online: September 2008

DOI: 10.1093/acprof:oso/9780199213535.001.0001

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The factorization method for Maxwell's equations

The factorization method for Maxwell's equations

(p.109) 5 The factorization method for Maxwell's equations
The Factorization Method for Inverse Problems

Andreas Kirsch

Natalia Grinberg

Oxford University Press

This chapter examines the time-harmonic Maxwell's equations for isotropic and non-magnetic but possibly inhomogeneous media. It begins by formulating the direct scattering problem for arbitrary measurable and essentially bounded permittivities. It derives an integro-differential equation of Lippmann-Schwinger type, proving equivalence with the weak formulation and the Fredholm property. As in the previous chapters, the chapter introduces the far field patterns and the far field operator, derives some of the most important properties, and formulates the inverse scattering problem. The general results of Chapters 1 and 2 are then applied, and a characterization of the contrast in terms of a Picard series is derived involving only known quantities. For the scalar case of Chapter 4, certain critical values of the frequency have to be excluded, which are eigenvalues of an interior transmission eigenvalue problem for Maxwell's equations. It is shown that there exists a quantifiable number of these values.

Keywords:   weak formulation, far field operator, interior transmission eigenvalues, isotropic, non-magnetic, inhomogeneous, direct scattering problem, permittivities, Lippmann-Schwinger, Fredholm property

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