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Inverse scattering for planar cracks via nonlinear integral equations
Author(s) -
Ivanyshyn O.,
Kress R.
Publication year - 2007
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.970
Subject(s) - mathematics , linearization , integral equation , inverse scattering problem , mathematical analysis , inverse problem , scattering , nonlinear system , boundary value problem , laplace transform , plane (geometry) , laplace's equation , geometry , physics , optics , quantum mechanics
Abstract We present a Newton‐type method for reconstructing planar sound‐soft or perfectly conducting cracks from far‐field measurements for one time‐harmonic scattering with plane wave incidence. Our approach arises from a method suggested by Kress and Rundell ( Inv. Probl. 2005; 21 (4):1207–1223) for an inverse boundary value problem for the Laplace equation. It was extended to inverse scattering problems for sound‐soft obstacles ( Mathematical Methods in Scattering Theory and Biomedical Engineering . World Scientific: Singapore, 2006; 39–50) and for sound‐hard cracks ( Inv. Probl. 2006; 22 (6)). In both cases it was shown that the method gives accurate reconstructions with reasonable stability against noisy data. The approach is based on a pair of nonlinear and ill‐posed integral equations for the unknown boundary. The integral equations are solved by linearization, i.e. by regularized Newton iterations. Numerical reconstructions illustrate the feasibility of the method. Copyright © 2007 John Wiley & Sons, Ltd.