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

Helsing, Johan LU and Johansson, B. Tomas (2010) In Inverse Problems in Science and Engineering 18(3). p.381-399
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 Nystroumlm 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... (More)
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 Nystroumlm 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. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
alternating method, Cauchy problem, second kind boundary integral equation, Laplace equation, Nyström method
in
Inverse Problems in Science and Engineering
volume
18
issue
3
pages
381 - 399
publisher
Taylor & Francis
external identifiers
  • wos:000277542100005
  • scopus:77951435404
ISSN
1741-5985
DOI
10.1080/17415971003624322
language
English
LU publication?
yes
id
4ade195f-f535-4626-8eb8-94998ba6c7a8 (old id 1578930)
alternative location
http://www.maths.lth.se/na/staff/helsing/BTomas1.pdf
date added to LUP
2010-03-19 16:38:22
date last changed
2018-05-29 09:21:57
@article{4ade195f-f535-4626-8eb8-94998ba6c7a8,
  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 Nystroumlm 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.},
  author       = {Helsing, Johan and Johansson, B. Tomas},
  issn         = {1741-5985},
  keyword      = {alternating method,Cauchy problem,second kind boundary integral equation,Laplace equation,Nyström method},
  language     = {eng},
  number       = {3},
  pages        = {381--399},
  publisher    = {Taylor & Francis},
  series       = {Inverse Problems in Science and Engineering},
  title        = {Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques},
  url          = {http://dx.doi.org/10.1080/17415971003624322},
  volume       = {18},
  year         = {2010},
}