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

Fingerprint

Harmonic functions
Harmonic Functions
Cauchy
Integral equations
Integral Equations
Fixed-point Equation
Derivative
Regularization Technique
Discretization Scheme
Bounded Solutions
Fredholm Integral Equation
Approximation
Derivatives
Computational Cost
Minor
Smoothness
High Accuracy
Iteration
Numerical Examples
Knowledge

Keywords

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

Cite this

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Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques. / Helsing, Johan; Johansson, B. Tomas.

In: Inverse Problems in Science and Engineering, Vol. 18, No. 3, 2010, p. 381-399.

Research output: Contribution to journalArticle

TY - JOUR

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AU - Johansson, B. Tomas

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