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Fast reconstruction of harmonic functions from Cauchy data using the Dirichlet-to-Neumann map and integral equations

Helsing, Johan LU and Johansson, B. Tomas (2011) In Inverse Problems in Science and Engineering 19(5). p.717-727
Abstract
We propose and investigate a method for the stable determination of a harmonic function from knowledge of its value and its normal derivative on a part of the boundary of the (bounded) solution domain (Cauchy problem). We reformulate the Cauchy problem as an operator equation on the boundary using the Dirichlet-to-Neumann map. To discretize the obtained operator, we modify and employ a method denoted as Classic II given in [J. Helsing, Faster convergence and higher accuracy for the Dirichlet–Neumann map, J. Comput. Phys. 228 (2009), pp. 2578–2576, Section 3], which is based on Fredholm integral equations and Nyström discretization schemes. Then, for stability reasons, to solve the discretized integral equation we use the method of... (More)
We propose and investigate a method for the stable determination of a harmonic function from knowledge of its value and its normal derivative on a part of the boundary of the (bounded) solution domain (Cauchy problem). We reformulate the Cauchy problem as an operator equation on the boundary using the Dirichlet-to-Neumann map. To discretize the obtained operator, we modify and employ a method denoted as Classic II given in [J. Helsing, Faster convergence and higher accuracy for the Dirichlet–Neumann map, J. Comput. Phys. 228 (2009), pp. 2578–2576, Section 3], which is based on Fredholm integral equations and Nyström discretization schemes. Then, for stability reasons, to solve the discretized integral equation we use the method of smoothing projection introduced in [J. Helsing and B.T. Johansson, Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques, Inverse Probl. Sci. Eng. 18 (2010), pp. 381–399, Section 7], which makes it possible to solve the discretized operator equation in a stable way with minor computational cost and high accuracy. With this approach, for sufficiently smooth Cauchy data, the normal derivative can also be accurately computed on the part of the boundary where no data is initially given. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
* alternating method, * Cauchy problem, * Dirichlet-to-Neumann map, * Laplace equation, * second kind boundary integral equation
in
Inverse Problems in Science and Engineering
volume
19
issue
5
pages
717 - 727
publisher
Taylor & Francis
external identifiers
  • wos:000299260400009
  • scopus:79960538963
ISSN
1741-5985
DOI
10.1080/17415977.2011.590897
language
English
LU publication?
yes
id
d7ab7a4f-7f61-44cd-ab61-e3884a35edce (old id 2028055)
alternative location
http://www.maths.lth.se/na/staff/helsing/BTomas3.pdf
date added to LUP
2011-08-23 12:58:28
date last changed
2017-03-15 09:39:35
@article{d7ab7a4f-7f61-44cd-ab61-e3884a35edce,
  abstract     = {We propose and investigate a method for the stable determination of a harmonic function from knowledge of its value and its normal derivative on a part of the boundary of the (bounded) solution domain (Cauchy problem). We reformulate the Cauchy problem as an operator equation on the boundary using the Dirichlet-to-Neumann map. To discretize the obtained operator, we modify and employ a method denoted as Classic II given in [J. Helsing, Faster convergence and higher accuracy for the Dirichlet–Neumann map, J. Comput. Phys. 228 (2009), pp. 2578–2576, Section 3], which is based on Fredholm integral equations and Nyström discretization schemes. Then, for stability reasons, to solve the discretized integral equation we use the method of smoothing projection introduced in [J. Helsing and B.T. Johansson, Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques, Inverse Probl. Sci. Eng. 18 (2010), pp. 381–399, Section 7], which makes it possible to solve the discretized operator equation in a stable way with minor computational cost and high accuracy. With this approach, for sufficiently smooth Cauchy data, the normal derivative can also be accurately computed on the part of the boundary where no data is initially given.},
  author       = {Helsing, Johan and Johansson, B. Tomas},
  issn         = {1741-5985},
  keyword      = {* alternating method,* Cauchy problem,* Dirichlet-to-Neumann map,* Laplace equation,* second kind boundary integral equation},
  language     = {eng},
  number       = {5},
  pages        = {717--727},
  publisher    = {Taylor & Francis},
  series       = {Inverse Problems in Science and Engineering},
  title        = {Fast reconstruction of harmonic functions from Cauchy data using the Dirichlet-to-Neumann map and integral equations},
  url          = {http://dx.doi.org/10.1080/17415977.2011.590897},
  volume       = {19},
  year         = {2011},
}