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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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Volume 16, Issue 8


Iterative methods for planar crack reconstruction in semi-infinite domains

R. Kress
  • Institut für Numerische und Angewandte Mathematik, Universität Göttingen, 37083 Göttingen, Germany. Email:
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/ N. Vintonyak
  • Department of Applied Mathematics and Computer Science, Ivan Franko National University of Lviv, 79000 Lviv, Ukraine. Email:
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Published Online: 2008-01-24 | DOI: https://doi.org/10.1515/JIIP.2008.047


We consider the problem of determining the shape and location of cracks from Cauchy data on the boundary of semi-infinite domains modeling the reconstruction of cracks within a heat conducting medium from temperature and heat flux measurements. Our reconstructions are based on a pair of nonlinear integral equations for the unknown crack and the unknown flux jump on the crack that are linear with respect to the flux and nonlinear with respect to the crack. We propose two different iteration methods employing the following idea: Given an approximate reconstruction for the crack we first solve one of the equations for the flux and subsequently linearize the other equation for updating the crack. The foundations for this approach for solving the inverse problem in semi-infinite domains are provided and numerical experiments exhibit the feasibility of both methods and their stability with respect to noisy data.

Key words.: Laplace equation; inverse boundary value problem; Green's function; nonlinear integral equation; iteration methods

About the article

Received: 2008-04-23

Published Online: 2008-01-24

Published in Print: 2008-12-01

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 16, Issue 8, Pages 743–761, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/JIIP.2008.047.

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