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An accelerated alternating procedure for the Cauchy problem for the Helmholtz equation
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
2014 (English)In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 68, no 1-2, 44-60 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we study the Cauchy problem for the Helmholtz equation. This problem appears in various applications and is severely ill–posed. The modified alternating procedure has been proposed by the authors for solving this problem but the convergence has been rather slow. We demonstrate how to instead use conjugate gradient methods for accelerating the convergence. The main idea is to introduce an artificial boundary in the interior of the domain. This addition of the interior boundary allows us to derive an inner product that is natural for the application and that gives us a proper framework for implementing the steps of the conjugate gradient methods. The numerical results performed using the finite difference method show that the conjugate gradient based methods converge considerably faster than the modified alternating iterative procedure studied previously.

Place, publisher, year, edition, pages
Elsevier, 2014. Vol. 68, no 1-2, 44-60 p.
Keyword [en]
Cauchy problem; alternating iterative method; conjugate gradient methods; inverse problem; ill–posed problem
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-105877DOI: 10.1016/j.camwa.2014.05.002ISI: 000338816300004OAI: oai:DiVA.org:liu-105877DiVA: diva2:711804
Available from: 2014-04-11 Created: 2014-04-11 Last updated: 2017-12-05Bibliographically approved
In thesis
1. Iterative Methods for Solving the Cauchy Problem for the Helmholtz Equation
Open this publication in new window or tab >>Iterative Methods for Solving the Cauchy Problem for the Helmholtz Equation
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The inverse problem of reconstructing the acoustic, or electromagnetic, field from inexact measurements on a part of the boundary of a domain is important in applications, for instance for detecting the source of acoustic noise. The governing equation for the applications we consider is the Helmholtz equation. More precisely, in this thesis we study the case where Cauchy data is available on a part of the boundary and we seek to recover the solution in the whole domain. The problem is ill-posed in the sense that small errors in the Cauchy data may lead to large errors in the recovered solution. Thus special regularization methods that restore the stability with respect to measurements errors are used.

In the thesis, we focus on iterative methods for solving the Cauchy problem. The methods are based on solving a sequence of well-posed boundary value problems. The specific choices for the boundary conditions used are selected in such a way that the sequence of solutions converges to the solution for the original Cauchy problem. For the iterative methods to converge, it is important that a certain bilinear form, associated with the boundary value problem, is positive definite. This is sometimes not the case for problems with a high wave number.

The main focus of our research is to study certain modifications to the problem that restore positive definiteness to the associated bilinear form. First we add an artificial interior boundary inside the domain together with a jump condition that includes a parameter μ. We have shown by selecting an appropriate interior boundary and sufficiently large value for μ, we get a convergent iterative regularization method. We have proved the convergence of this method. This method converges slowly. We have therefore developed two conjugate gradient type methods and achieved much faster convergence. Finally, we have attempted to reduce the size of the computational domain by solving well–posed problems only in a strip between the outer and inner boundaries. We demonstrate that by alternating between Robin and Dirichlet conditions on the interior boundary, we can get a convergent iterative regularization method. Numerical experiments are used to illustrate the performance of the  methods suggested.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2014. 12 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1593
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-105879 (URN)10.3384/diss.diva-105879 (DOI)978-91-7519-350-2 (ISBN)
Public defence
2014-05-09, ACAS, A–huset, Campus Valla, Linköpings universitet, Linköping, 13:15 (English)
Opponent
Supervisors
Note

An invalid ISRN (LIU-TEK-LIC-2012:15) is stated on page 2. The ISRN belongs to the Licentiate thesis, published in 2012.

Available from: 2014-04-11 Created: 2014-04-11 Last updated: 2014-04-11Bibliographically approved

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Berntsson, FredrikKozlov, VladimirMpinganzima, LydieTuresson, Bengt-Ove

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