Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques

Johan Helsing, B. Tomas Johansson

Research output: Contribution to journalArticle

Abstract

We consider the problem of stable determination of a harmonic function from knowledge of the solution and its normal derivative on a part of the boundary of the (bounded) solution domain. The alternating method is a procedure to generate an approximation to the harmonic function from such Cauchy data and we investigate a numerical implementation of this procedure based on Fredholm integral equations and Nyström discretization schemes, which makes it possible to perform a large number of iterations (millions) with minor computational cost (seconds) and high accuracy. Moreover, the original problem is rewritten as a fixed point equation on the boundary, and various other direct regularization techniques are discussed to solve that equation. We also discuss how knowledge of the smoothness of the data can be used to further improve the accuracy. Numerical examples are presented showing that accurate approximations of both the solution and its normal derivative can be obtained with much less computational time than in previous works.
Original languageEnglish
Pages (from-to)381-399
Number of pages19
JournalInverse Problems in Science and Engineering
Volume18
Issue number3
DOIs
Publication statusPublished - 2010

Keywords

  • alternating method
  • second kind boundary integral equation
  • Nyström method
  • Laplace equation
  • Cauchy problem

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