Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques
(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:
https://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
- 1741-5985
- DOI
- 10.1080/17415971003624322
- language
- English
- LU publication?
- yes
- additional info
- The information about affiliations in this record was updated in December 2015. The record was previously connected to the following departments: Numerical Analysis (011015004)
- 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
- 2016-04-01 09:59:38
- date last changed
- 2022-01-25 18:43:40
@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}},
keywords = {{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 = {{https://lup.lub.lu.se/search/files/1458763/4254516.pdf}},
doi = {{10.1080/17415971003624322}},
volume = {{18}},
year = {{2010}},
}