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The Plane Waves Method for Numerical Boundary Identification

Part of:Partial differential equations, boundary value problems

Published online by Cambridge University Press:  28 November 2017

A. Karageorghis,
D. Lesnic and
L. Marin
A. Karageorghis*
Affiliation:
Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
D. Lesnic*
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
L. Marin*
Affiliation:
Department of Mathematics, Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei, 010014 Bucharest, and Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, 13 Calea 13 Septembrie, 050711 Bucharest, Romania
*
*Corresponding author. Email:andreask@ucy.ac.cy (A. Karageorghis), amt5ld@maths.leeds.ac.uk (D. Lesnic), marin.liviu@gmail.com, liviu.marin@fmi.unibuc.ro (L. Marin)
*Corresponding author. Email:andreask@ucy.ac.cy (A. Karageorghis), amt5ld@maths.leeds.ac.uk (D. Lesnic), marin.liviu@gmail.com, liviu.marin@fmi.unibuc.ro (L. Marin)
*Corresponding author. Email:andreask@ucy.ac.cy (A. Karageorghis), amt5ld@maths.leeds.ac.uk (D. Lesnic), marin.liviu@gmail.com, liviu.marin@fmi.unibuc.ro (L. Marin)

Abstract

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We study the numerical identification of an unknown portion of the boundary on which either the Dirichlet or the Neumann condition is provided from the knowledge of Cauchy data on the remaining, accessible and known part of the boundary of a two-dimensional domain, for problems governed by Helmholtz-type equations. This inverse geometric problem is solved using the plane waves method (PWM) in conjunction with the Tikhonov regularization method. The value for the regularization parameter is chosen according to Hansen's L-curve criterion. The stability, convergence, accuracy and efficiency of the proposed method are investigated by considering several examples.

Keywords

Plane waves methodcollocationinverse problemregularization

MSC classification

Secondary:65N35: Spectral, collocation and related methods65N21: Inverse problems65N38: Boundary element methods

Information

Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 6 , December 2017, pp. 1312 - 1329
Copyright
Copyright © Global-Science Press 2017 

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