Fast reconstruction of harmonic functions from Cauchy data using the Dirichlet-to-Neumann map and integral equations

Johan Helsing, B. Tomas Johansson

Research output: Contribution to journalArticlepeer-review

Abstract

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.
Original languageEnglish
Pages (from-to)717-727
Number of pages11
JournalInverse Problems in Science and Engineering
Volume19
Issue number5
DOIs
Publication statusPublished - 2011
Event5th International Conference on Inverse Problems: Modeling and Simulation - Antalya, Turkey
Duration: 24 May 201029 May 2010

Keywords

  • alternating method
  • Cauchy problem
  • Dirichlet-to-Neumann map
  • Laplace equation
  • second kind boundary integral equation

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