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Solving an Ill-Posed Cauchy Problem for a Two-Dimensional Parabolic PDE with Variable Coefficients Using a Preconditioned GMRES Method
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.ORCID iD: 0000-0003-2281-856X
2014 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 36, no 5, B868-B886 p.Article in journal (Refereed) Published
Abstract [en]

The sideways parabolic equation (SPE) is a model of the problem of determiningthe temperature on the surface of a body from the interior measurements. Mathematically it can beformulated as a noncharacteristic Cauchy problem for a parabolic partial differential equation. Thisproblem is severely ill-posed in an L2 setting. We use a preconditioned generalized minimum residualmethod (GMRES) to solve a two-dimensional SPE with variable coefficients. The preconditioner issingular and chosen in a way that allows efficient implementation using the FFT. The preconditioneris a stabilized solver for a nearby problem with constant coefficients, and it reduces the numberof iterations in the GMRES algorithm significantly. Numerical experiments are performed thatdemonstrate the performance of the proposed method.

Place, publisher, year, edition, pages
Philadelphia: Society for Industrial and Applied Mathematics , 2014. Vol. 36, no 5, B868-B886 p.
Keyword [en]
Cauchy problem, inverse problem, ill-posed, iterative methods, GMRES precondi- tioning, singular preconditioner, parabolic PDE, FFT
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-111837DOI: 10.1137/130951166ISI: 000346123200021OAI: oai:DiVA.org:liu-111837DiVA: diva2:760904
Available from: 2014-11-05 Created: 2014-11-05 Last updated: 2017-12-05Bibliographically approved

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Ranjbar, ZohrehEldén, Lars
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