liu.seSearch for publications in DiVA
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Iterative Methods for Solving the Cauchy Problem for the Helmholtz Equation
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
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: urn:nbn:se:liu:diva-105879DOI: 10.3384/diss.diva-105879ISBN: 978-91-7519-350-2 (print)OAI: oai:DiVA.org:liu-105879DiVA: diva2:711818
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
List of papers
1. Numerical Solution of the Cauchy Problem for the Helmholtz Equation
Open this publication in new window or tab >>Numerical Solution of the Cauchy Problem for the Helmholtz Equation
2014 (English)Report (Other academic)
Abstract [en]

The Cauchy problem for the Helmholtz equation appears in applications related to acoustic or electromagnetic wave phenomena. The problem is ill–posed in the sense that the solution does not depend on the data in a stable way. In this paper we give a detailed study of the problem. Specifically we investigate how the ill–posedness depends on the shape of the computational domain and also on the wave number. Furthermore, we give an overview over standard techniques for dealing with ill–posed problems and apply them to the problem.

Place, publisher, year, edition, pages
Linköping University Electronic Press, 2014. 16 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2014:04
Keyword
Helmholtz equation, Cauchy Problem, Ill-Posed, Regularization, Numerical Methods.
National Category
Computational Mathematics Mathematics
Identifiers
urn:nbn:se:liu:diva-105707 (URN)LiTH-MAT-R--2014/04--SE (ISRN)
Available from: 2014-04-03 Created: 2014-04-03 Last updated: 2014-04-11Bibliographically approved
2. 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
National Category
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
3. An accelerated alternating procedure for the Cauchy problem for the Helmholtz equation
Open this publication in new window or tab >>An accelerated alternating procedure for the Cauchy problem for the Helmholtz equation
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
Keyword
Cauchy problem; alternating iterative method; conjugate gradient methods; inverse problem; ill–posed problem
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-105877 (URN)10.1016/j.camwa.2014.05.002 (DOI)000338816300004 ()
Available from: 2014-04-11 Created: 2014-04-11 Last updated: 2017-12-05Bibliographically approved
4. Robin–Dirichlet algorithms for the Cauchy problem for the Helmholtz equation
Open this publication in new window or tab >>Robin–Dirichlet algorithms for the Cauchy problem for the Helmholtz equation
2014 (English)Manuscript (preprint) (Other academic)
Abstract [en]

The Cauchy problem for the Helmholtz equation is considered. It was demonstrated in a previous paper by the authors that the alternating algorithm suggested by V.A. Kozlov and V.G. Maz’ya does not converge for large wavenumbers in the Helmholtz equation. We prove here that if we alternate Robin and Dirichlet boundary conditions instead of Neumann and Dirichlet boundary conditions, then the algorithm will converge. We present also another algorithm based on the same idea, which converges for large wavenumbers. Numerical implementations obtained using the finite difference method are presented. Numerical results illustrate that the algorithms suggested in this paper, produce a convergent iterative sequences.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-105876 (URN)
Available from: 2014-04-11 Created: 2014-04-11 Last updated: 2014-04-11Bibliographically approved

Open Access in DiVA

Iterative Methods for Solving the Cauchy Problem for the Helmholtz Equation(183 kB)956 downloads
File information
File name FULLTEXT01.pdfFile size 183 kBChecksum SHA-512
5b52ab77b48762fb1a8dba003841198b5124ff57a25204f1f9b5ce2d23c2621e8c7a3f0bc0cdf0c81ac445daeaabad0eac984bf093cab3d03109b1a8de665a65
Type fulltextMimetype application/pdf
omslag(48 kB)25 downloads
File information
File name COVER01.pdfFile size 48 kBChecksum SHA-512
9c8df3aa5c0645b37d4988b58934e0add8be5e896d9e88f48246e3bfbc8de7b0bddc0faa5db03ac9aa0ad90a8d424fc5ff335c036d267aeec9915947a4c6f285
Type coverMimetype application/pdf

Other links

Publisher's full text

Authority records BETA

Mpinganzima, Lydie

Search in DiVA

By author/editor
Mpinganzima, Lydie
By organisation
Department of MathematicsThe Institute of Technology
Engineering and Technology

Search outside of DiVA

GoogleGoogle Scholar
Total: 956 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

doi
isbn
urn-nbn

Altmetric score

doi
isbn
urn-nbn
Total: 1176 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf