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Analysis of the Robin-Dirichlet iterative procedure for solving the Cauchy problem for elliptic equations with extension to unbounded domains
Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
2020 (Engelska)Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
##### 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.

##### Ort, förlag, år, upplaga, sidor
Linköping: Linköping University Electronic Press, 2020. , s. 10
##### Serie
Linköping Studies in Science and Technology. Licentiate Thesis, ISSN 0280-7971 ; 1891
##### Nationell ämneskategori
Beräkningsmatematik
##### Identifikatorer
ISBN: 9789179297565 (tryckt)OAI: oai:DiVA.org:liu-170835DiVA, id: diva2:1479114
##### Presentation
2020-12-03, Planck, F-building, entrance 57, Campus Valla, Linköping, 10:00 (Engelska)
##### Anmärkning

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

##### Delarbeten
1. Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic Equations
Öppna denna publikation i ny flik eller fönster >>Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic Equations
2021 (Engelska)Ingår i: Bulletin of the Iranian Mathematical Society, ISSN 1735-8515, Vol. 47, s. 1681-1699Artikel i tidskrift (Refereegranskat) 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.

Springer, 2021
##### Nyckelord
Helmholtz equation, Cauchy problem, Inverse problem, Ill-posed problem
##### Nationell ämneskategori
Matematisk analys
##### Identifikatorer
urn:nbn:se:liu:diva-170834 (URN)10.1007/s41980-020-00466-7 (DOI)000575739300001 ()2-s2.0-85092146699 (Scopus ID)

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Achieng, Pauline

#### Sök vidare i DiVA

Achieng, Pauline
##### Av organisationen
Matematik och tillämpad matematikTekniska fakulteten
##### I ämnet
Beräkningsmatematik

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