Fast reconstruction of harmonic functions from Cauchy data using the DirichlettoNeumann map and integral equations
(2011) In Inverse Problems in Science and Engineering 19(5). p.717727 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 DirichlettoNeumann 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 DirichlettoNeumann 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)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/2028055
 author
 Helsing, Johan ^{LU} and Johansson, B. Tomas
 organization
 publishing date
 2011
 type
 Contribution to journal
 publication status
 published
 subject
 keywords
 * alternating method, * Cauchy problem, * DirichlettoNeumann 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
 17415985
 DOI
 10.1080/17415977.2011.590897
 language
 English
 LU publication?
 yes
 id
 d7ab7a4f7f6144cdab61e3884a35edce (old id 2028055)
 alternative location
 http://www.maths.lth.se/na/staff/helsing/BTomas3.pdf
 date added to LUP
 20110823 12:58:28
 date last changed
 20180107 03:26:52
@article{d7ab7a4f7f6144cdab61e3884a35edce, 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 DirichlettoNeumann 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 = {17415985}, keyword = {* alternating method,* Cauchy problem,* DirichlettoNeumann map,* Laplace equation,* second kind boundary integral equation}, language = {eng}, number = {5}, pages = {717727}, publisher = {Taylor & Francis}, series = {Inverse Problems in Science and Engineering}, title = {Fast reconstruction of harmonic functions from Cauchy data using the DirichlettoNeumann map and integral equations}, url = {http://dx.doi.org/10.1080/17415977.2011.590897}, volume = {19}, year = {2011}, }