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An alternating iterative procedure for the Cauchy problem for the Helmholtz equation
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
2012 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

Let  be a bounded domain in Rn with a Lipschitz boundary Г divided into two parts Г0 and Г1 which do not intersect one another and have a common Lipschitz boundary. We consider the following Cauchy problem for the Helmholtz equation:

$\begin{cases}\Delta u + k^2 u = 0 & \quad \mbox{in} \quad \Omega,\\u = f & \quad \mbox{on} \quad \Gamma_0,\\\partial_{\nu} u = g & \quad \mbox{on} \quad \Gamma_0,\end{cases}$

where k, the wave number, is a positive real constant, аv denotes the outward normal derivative, and f and g are specified Cauchy data on Г0. This problem is ill–posed in the sense that small errors in the Cauchy data f and g may blow up and cause a large error in the solution.

Alternating iterative algorithms for solving this problem are developed and studied. These algorithms are based on the alternating iterative schemes suggested by V.A. Kozlov and V. Maz’ya for solving ill–posed problems. Since these original alternating iterative algorithms diverge for large values of the constant k2 in the Helmholtz equation, we develop a modification of the alterating iterative algorithms that converges for all k2. We also perform numerical experiments that confirm that the proposed modification works.

Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1530
Mathematics
Identifiers
Local ID: LIU-TEK-LIC-2012:15ISBN: 978-91-7519-890-3 (print)OAI: oai:DiVA.org:liu-77300DiVA: diva2:526247
Presentation
2012-05-02, Nobel (BL32), B-huset, ing°ang 23, Campus Valla, Linköpings universitet, Linköping, 15:15 (English)
Supervisors
Available from: 2012-05-11 Created: 2012-05-11 Last updated: 2012-06-04Bibliographically approved
List of papers
1. An alternating iterative procedure for the Cauchy problem for the Helmholtz equation
Open this publication in new window or tab >>An alternating iterative procedure for the Cauchy problem for the Helmholtz equation
2014 (English)In: Inverse Problems in Science and Engineering, ISSN 1741-5977, E-ISSN 1741-5985, Vol. 22, no 1, 45-62 p.Article in journal (Refereed) Published
Abstract [en]

We present a modification of the alternating iterative method, which was introduced by V.A. Kozlov and V. Maz’ya in for solving the Cauchy problem for the Helmholtz equation in a Lipschitz domain. The method is implemented numerically using the finite difference method.

Place, publisher, year, edition, pages
Taylor & Francis, 2014
Mathematics
Identifiers
urn:nbn:se:liu:diva-77298 (URN)10.1080/17415977.2013.827181 (DOI)000328245900005 ()
Conference
6th International Conference "Inverse Problems: Modeling and Simulation", 21-26 May 2012, Antalya, Turkey
Available from: 2012-05-11 Created: 2012-05-11 Last updated: 2017-12-07Bibliographically approved

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Cite
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