Analysis of the Hessian for inverse scattering problems. Part  III: Inverse medium scattering of electromagnetic waves in three  dimensions

Tan Bui-Thanh; Omar Ghattas

doi:10.3934/ipi.2013.7.1139

Article Contents

2013, Volume 7, Issue 4: 1139-1155. Doi: 10.3934/ipi.2013.7.1139

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Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions

Tan Bui-Thanh^1, and
Omar Ghattas^2,

1.
Department of Aerospace Engineering and Engineering Mechanics, Institute for Computational Engineering & Sciences, The University of Texas at Austin, Austin, TX 78712
2.
Institute for Computational Engineering & Sciences, Jackson School of Geosciences, and Department of Mechanical Engineering, The University of Texas at Austin, Austin, TX 78712

Received: October 31, 2012

Revised: August 31, 2013

Published: October 2013

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Abstract

Continuing our previous work [6, Inverse Problems, 2012, 28, 055002] and [5, Inverse Problems, 2012, 28, 055001], we address the ill-posedness of the inverse scattering problem of electromagnetic waves due to an inhomogeneous medium by studying the Hessian of the data misfit. We derive and analyze the Hessian in both Hölder and Sobolev spaces. Using an integral equation approach based on Newton potential theory and compact embeddings in Hölder and Sobolev spaces, we show that the Hessian can be decomposed into three components, all of which are shown to be compact operators. The implication of the compactness of the Hessian is that for small data noise and model error, the discrete Hessian can be approximated by a low-rank matrix. This in turn enables fast solution of an appropriately regularized inverse problem, as well as Gaussian-based quantification of uncertainty in the estimated inhomogeneity.

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Mathematics Subject Classification: Primary: 49N45, 49K20, 49K40, 49K30; Secondary: 49J50, 49N60.

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