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An alternating iterative procedure for the Cauchy problem for the Helmholtz equation
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
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. Vol. 22, no 1, 45-62 p.
Mathematics
##### Identifiers
ISI: 000328245900005OAI: oai:DiVA.org:liu-77298DiVA: diva2:526240
##### 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: 2015-02-04Bibliographically approved
##### In thesis
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
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
urn:nbn:se:liu:diva-77300 (URN)LIU-TEK-LIC-2012:15 (Local ID)978-91-7519-890-3 (ISBN)LIU-TEK-LIC-2012:15 (Archive number)LIU-TEK-LIC-2012:15 (OAI)
##### 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
2. 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.

##### 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 (print) (ISBN)
##### 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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