Fast reconstruction of harmonic functions from Cauchy data using the Dirichlet-to-Neumann map and integral equations
(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)
Please use this url to cite or link to this publication:
https://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, * 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
- 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
- d7ab7a4f-7f61-44cd-ab61-e3884a35edce (old id 2028055)
- alternative location
- http://www.maths.lth.se/na/staff/helsing/BTomas3.pdf
- date added to LUP
- 2016-04-01 10:06:42
- date last changed
- 2022-01-25 19:51:26
@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}},
keywords = {{* 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 = {{https://lup.lub.lu.se/search/files/1571107/3878563.pdf}},
doi = {{10.1080/17415977.2011.590897}},
volume = {{19}},
year = {{2011}},
}