Fast reconstruction of harmonic functions from Cauchy data using the Dirichlet-to-Neumann map and integral equations
Author
Summary, in English
We propose and investigate a method for the stable determination of a harmonic function from knowledge of its value and its normal derivative on a part of the boundary of the (bounded) solution domain (Cauchy problem). We reformulate the Cauchy problem as an operator equation on the boundary using the Dirichlet-to-Neumann map. To discretize the obtained operator, we modify and employ a method denoted as Classic II given in [J. Helsing, Faster convergence and higher accuracy for the Dirichlet–Neumann map, J. Comput. Phys. 228 (2009), pp. 2578–2576, Section 3], which is based on Fredholm integral equations and Nyström discretization schemes. Then, for stability reasons, to solve the discretized integral equation we use the method of smoothing projection introduced in [J. Helsing and B.T. Johansson, Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques, Inverse Probl. Sci. Eng. 18 (2010), pp. 381–399, Section 7], which makes it possible to solve the discretized operator equation in a stable way with minor computational cost and high accuracy. With this approach, for sufficiently smooth Cauchy data, the normal derivative can also be accurately computed on the part of the boundary where no data is initially given.
Department/s
- Mathematics (Faculty of Engineering)
- Harmonic Analysis and Applications
- eSSENCE: The e-Science Collaboration
Publishing year
2011
Language
English
Pages
717-727
Publication/Series
Inverse Problems in Science and Engineering
Volume
19
Issue
5
Full text
- Available as PDF - 182 kB
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Document type
Journal article
Publisher
Taylor & Francis
Topic
- Mathematics
Keywords
- * alternating method
- * Cauchy problem
- * Dirichlet-to-Neumann map
- * Laplace equation
- * second kind boundary integral equation
Status
Published
Research group
- Harmonic Analysis and Applications
- Harmonic Analysis and Applications
ISBN/ISSN/Other
- ISSN: 1741-5985