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

Direct link
Cite
Citation style
  • apa
  • 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
Analysis of the Robin-Dirichlet iterative procedure for solving the Cauchy problem for elliptic equations with extension to unbounded domains
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
2020 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we study the Cauchy problem for elliptic equations. It arises in many areas of application in science and engineering as a problem of reconstruction of solutions to elliptic equations in a domain from boundary measurements taken on a part of the boundary of this domain. The Cauchy problem for elliptic equations is known to be ill-posed.

We use an iterative regularization method based on alternatively solving a sequence of well-posed mixed boundary value problems for the same elliptic equation. This method, based on iterations between Dirichlet-Neumann and Neumann-Dirichlet mixed boundary value problems was first proposed by Kozlov and Maz’ya [13] for Laplace equation and Lame’ system but not Helmholtz-type equations. As a result different modifications of this original regularization method have been proposed in literature. We consider the Robin-Dirichlet iterative method proposed by Mpinganzima et.al [3] for the Cauchy problem for the Helmholtz equation in bounded domains.

We demonstrate that the Robin-Dirichlet iterative procedure is convergent for second order elliptic equations with variable coefficients provided the parameter in the Robin condition is appropriately chosen. We further investigate the convergence of the Robin-Dirichlet iterative procedure for the Cauchy problem for the Helmholtz equation in a an unbounded domain. We derive and analyse the necessary conditions needed for the convergence of the procedure.

In the numerical experiments, the precise behaviour of the procedure for different values of k2 in the Helmholtz equation is investigated and the results show that the speed of convergence depends on the choice of the Robin parameters, μ0 and μ1. In the unbounded domain case, the numerical experiments demonstrate that the procedure is convergent provided that the domain is truncated appropriately and the Robin parameters, μ0 and μ1 are also chosen appropriately.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2020. , p. 10
Series
Linköping Studies in Science and Technology. Licentiate Thesis, ISSN 0280-7971 ; 1891
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-170835DOI: 10.3384/lic.diva-170835ISBN: 9789179297565 (print)OAI: oai:DiVA.org:liu-170835DiVA, id: diva2:1479114
Presentation
2020-12-03, Planck, F-building, entrance 57, Campus Valla, Linköping, 10:00 (English)
Opponent
Supervisors
Note

Funding Agencies: International Science Programme (ISP) and the Eastern Africa Universities Mathematics Programme (EAUMP).

Available from: 2020-10-26 Created: 2020-10-26 Last updated: 2020-10-28Bibliographically approved
List of papers
1. Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic Equations
Open this publication in new window or tab >>Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic Equations
2021 (English)In: Bulletin of the Iranian Mathematical Society, ISSN 1735-8515, Vol. 47, p. 1681-1699Article in journal (Refereed) Published
Abstract [en]

The Cauchy problem for general elliptic equations of second order is considered. In a previous paper (Berntsson et al. in Inverse Probl Sci Eng 26(7):1062–1078, 2018), it was suggested that the alternating iterative algorithm suggested by Kozlov and Maz’ya can be convergent, even for large wavenumbers k2, in the Helmholtz equation, if the Neumann boundary conditions are replaced by Robin conditions. In this paper, we provide a proof that shows that the Dirichlet–Robin alternating algorithm is indeed convergent for general elliptic operators provided that the parameters in the Robin conditions are chosen appropriately. We also give numerical experiments intended to investigate the precise behaviour of the algorithm for different values of k2 in the Helmholtz equation. In particular, we show how the speed of the convergence depends on the choice of Robin parameters.

Place, publisher, year, edition, pages
Springer, 2021
Keywords
Helmholtz equation, Cauchy problem, Inverse problem, Ill-posed problem
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:liu:diva-170834 (URN)10.1007/s41980-020-00466-7 (DOI)000575739300001 ()2-s2.0-85092146699 (Scopus ID)
Available from: 2020-10-26 Created: 2020-10-26 Last updated: 2024-02-22Bibliographically approved

Open Access in DiVA

fulltext(514 kB)364 downloads
File information
File name FULLTEXT01.pdfFile size 514 kBChecksum SHA-512
14f776098a9bbada328f855ff7fedf28352a41da0355c4b8785c72740940c1ed843aa3a485ae194c24605a3cbf685abd00b08f0994ca3fe2700169ce2774f01c
Type fulltextMimetype application/pdf
Order online >>

Other links

Publisher's full text

Authority records

Achieng, Pauline

Search in DiVA

By author/editor
Achieng, Pauline
By organisation
Mathematics and Applied MathematicsFaculty of Science & Engineering
Computational Mathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 364 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: 539 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • 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