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

J Helsing, Bjorn Johansson

Research output: Contribution to journalArticle

18 Citations (Scopus)


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 Nystrom 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
Issue number3
Publication statusPublished - 1 Jan 2010


  • Relaxation Procedure
  • Cauchy Problem
  • Inverse Problem
  • Boundary Element Method (BEM)
  • Alternating Iterative Algorithms
  • Helmholtz Equation


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