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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

6 Issues per year


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A second order Newton method for sound soft inverse obstacle scattering

R. Kress1 / N. Tezel2 / F. Yaman3

1Institut für Numerische Angewandte Mathematik, Universität Göttingen, 37083 Göttingen, Germany. Email:

2Istanbul Technical University, Electrical and Electronics Engineering Faculty, 34469 Maslak, Istanbul, Turkey. Email:

3Istanbul Technical University, Electrical and Electronics Engineering Faculty, 34469 Maslak, Istanbul, Turkey. Email:

Citation Information: Journal of Inverse and Ill-posed Problems. Volume 17, Issue 2, Pages 173–185, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: 10.1515/JIIP.2009.015, March 2009

Publication History

Received:
2008-08-28
Published Online:
2009-03-30

Abstract

A new second order Newton method for reconstructing the shape of a sound soft scatterer from the measured far-field pattern for scattering of time harmonic plane waves is presented. This method extends a hybrid between regularized Newton iterations and decomposition methods that has been suggested and analyzed in a number of papers by Kress and Serranho [Inverse Problems 19: 91–104, 2003, Inverse Problems 21: 773–784, 2005, J. Comput. Appl. Math. 204: 418–427, 2007, Inverse Problems 22: 663–680, 2006, Inverse Problems and Imaging 1: 691–712, 2007] and has some features in common with the second degree method for ill-posed nonlinear problems as considered by Hettlich and Rundell [SIAM J.Numer. Anal. 37: 587–620, 2000]. The main idea of our iterative method is to use Huygen's principle, i.e., represent the scattered field as a single-layer potential. Given an approximation for the boundary of the scatterer, this leads to an ill-posed integral equation of the first kind that is solved via Tikhonov regularization. Then, in a second order Taylor expansion, the sound soft boundary condition is employed to update the boundary approximation. In an iterative procedure, these two steps are alternated until some stopping criterium is satisfied. We describe the method in detail and illustrate its feasibility through examples with exact and noisy data.

Key words.: Inverse obstacle scattering; ill-posed problem; Newton iteration

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