Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques

Summary, in English

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: Johan Helsing; B. Tomas Johansson
• Department/s: Mathematics (Faculty of Engineering); Harmonic Analysis and Applications; eSSENCE: The e-Science Collaboration
• Publishing year: 2010
• Language: English
• Pages: 381-399
• Publication/Series: Inverse Problems in Science and Engineering
• Volume: 18
• Issue: 3
• Full text: - 266 kB; Download statistics
• Links: http://www.maths.lth.se/na/staff/helsing/BTomas1.pdf; http://dx.doi.org/10.1080/17415971003624322
• Document type: Journal article
• Publisher: Taylor & Francis
• Topic: Mathematics
• Keywords: alternating method; Cauchy problem; second kind boundary integral equation; Laplace equation; Nyström method
• Status: Published
• Research group: Harmonic Analysis and Applications; Harmonic Analysis and Applications
• ISBN/ISSN/Other: ISSN: 1741-5985