Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques
(2010) In Inverse Problems in Science and Engineering 18(3). p.381399 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:
http://lup.lub.lu.se/record/1578930
 author
 Helsing, Johan ^{LU} and Johansson, B. Tomas
 organization
 publishing date
 2010
 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
 17415985
 DOI
 10.1080/17415971003624322
 language
 English
 LU publication?
 yes
 id
 4ade195ff53546268eb894998ba6c7a8 (old id 1578930)
 alternative location
 http://www.maths.lth.se/na/staff/helsing/BTomas1.pdf
 date added to LUP
 20100319 16:38:22
 date last changed
 20180529 09:21:57
@article{4ade195ff53546268eb894998ba6c7a8, 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 = {17415985}, keyword = {alternating method,Cauchy problem,second kind boundary integral equation,Laplace equation,Nyström method}, language = {eng}, number = {3}, pages = {381399}, 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}, }