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Iterative Tikhonov regularization for the Cauchy problem for the Helmholtz equation
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-2681-8965
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
University of Rwanda, Rwanda.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
2017 (English)In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 73, no 1, 163-172 p.Article in journal (Refereed) Published
Abstract [en]

The Cauchy problem for the Helmholtz equation appears in various applications. The problem is severely ill-posed and regularization is needed to obtain accurate solutions. We start from a formulation of this problem as an operator equation on the boundary of the domain and consider the equation in (H-1/2)* spaces. By introducing an artificial boundary in the interior of the domain we obtain an inner product for this Hilbert space in terms of a quadratic form associated with the Helmholtz equation; perturbed by an integral over the artificial boundary. The perturbation guarantees positivity property of the quadratic form. This inner product allows an efficient evaluation of the adjoint operator in terms of solution of a well-posed boundary value problem for the Helmholtz equation with transmission boundary conditions on the artificial boundary. In an earlier paper we showed how to take advantage of this framework to implement the conjugate gradient method for solving the Cauchy problem. In this work we instead use the Conjugate gradient method for minimizing a Tikhonov functional. The added penalty term regularizes the problem and gives us a regularization parameter that can be used to easily control the stability of the numerical solution with respect to measurement errors in the data. Numerical tests show that the proposed algorithm works well. (C) 2016 Elsevier Ltd. All rights reserved.

Place, publisher, year, edition, pages
PERGAMON-ELSEVIER SCIENCE LTD , 2017. Vol. 73, no 1, 163-172 p.
Keyword [en]
Helmholtz equation; Cauchy problem; Adjoint method; Inverse problem; Ill-posed problem; Tikhonov regularization
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-134990DOI: 10.1016/j.camwa.2016.11.004ISI: 000393010400013OAI: oai:DiVA.org:liu-134990DiVA: diva2:1078811
Available from: 2017-03-06 Created: 2017-03-06 Last updated: 2017-03-23

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The full text will be freely available from 2018-12-01 14:25
Available from 2018-12-01 14:25

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