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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:[email protected] (A. Karageorghis), [email protected] (D. Lesnic), [email protected], [email protected] (L. Marin)
*Corresponding author. Email:[email protected] (A. Karageorghis), [email protected] (D. Lesnic), [email protected], [email protected] (L. Marin)
*Corresponding author. Email:[email protected] (A. Karageorghis), [email protected] (D. Lesnic), [email protected], [email protected] (L. Marin)
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Abstract

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 methods 65N21: Inverse problems 65N38: Boundary element methods
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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References

[1] Alves, C. J. S. and Valtchev, S. S., Numerical simulation of acoustic wave scattering using a meshfree plane waves method, International Workshop on Meshfree Methods, 2003, http://www.math.ist.utl.pt/meshfree/silen.pdf.Google Scholar
[2] Alves, C. J. S. and Valtchev, S. S., Numerical comparison of two meshfree methods for acoustic wave scattering, Eng. Anal. Bound. Elem., 29 (2005), pp. 371–382.CrossRefGoogle Scholar
[3] Antunes, P. R. S., Numerical calculation of eigensolutions of 3D shapes using the method of fundamental solutions, Numer. Methods Partial Differential Equations, 27 (2011), pp. 1525–1550.CrossRefGoogle Scholar
[4] Berntsson, F., Kozlov, V. A., Mpinganzima, L. and Turesson, B. O., An alternating iterative procedure for the Cauchy problem for the Helmholtz equation, Inverse Problems Sci. Eng., 22 (2014), pp. 45–62.CrossRefGoogle Scholar
[5] Berntsson, F., Kozlov, V. A., Mpinganzima, L. and Turesson, B. O., An accelerating alternating iterative procedure for the Cauchy problem for the Helmholtz equation, Comput. Math. Appl., 68 (2014), pp. 44–60.CrossRefGoogle Scholar
[6] Beskos, D. E., Boundary element method in dynamic analysis: Part II (1986–1996), ASME Appl. Mech. Rev., 50 (1997), pp. 149–197.CrossRefGoogle Scholar
[7] Bin-Mohsin, B. and Lesnic, D., Identification of a corroded boundary and its Robin coefficient, East Asian J. Appl. Math., 2 (2012), pp. 126–149.Google Scholar
[8] Borman, D., Ingham, D. B., Johansson, B. T. and Lesnic, D., The method of fundamental solutions for detection of cavities in EIT, J. Integral Equations Appl., 21 (2009), pp. 381–404.CrossRefGoogle Scholar
[9] Borsic, A., Graham, B. M., Adler, A. and Lionheart, W. R. B., In vivo impedance imaging with total variation regularization, IEEE Trans. Med. Inaging, 29 (2010), pp. 44–54.CrossRefGoogle ScholarPubMed
[10] Cessenat, O. and Després, B., Using plane waves as base functions for solving time harmonic equations with the ultra weak variational formulation, J. Comput. Acoust., 11 (2003), pp. 227–238.CrossRefGoogle Scholar
[11] Chen, C. S., Karageorghis, A. and Li, Y., On choosing the location of the sources in the MFS, Numer. Algor., 72 (2016), pp. 107–130.CrossRefGoogle Scholar
[12] Chen, J. T. and Wong, F. C., Dual formulation of multiple reciprocity method for the acoustic mode of a cavity with a thin partition, J. Sound Vibration, 217 (1998), pp. 75–95.CrossRefGoogle Scholar
[13] Chen, W., Meshfree boundary particle method applied to Helmholtz problems, Eng. Anal. Bound. Elem., 26 (2002), pp. 577–581.CrossRefGoogle Scholar
[14] Chen, W., Fu, Z. J. and Wei, X., Potential problems by singular boundary method satisfying moment condition, CMES Comput. Model. Eng. Sci., 54 (2009), pp. 65–85.Google Scholar
[15] Debye, P. and Hückel, E., The theory of electrolytes. I. Lowering of freezing point and related phenomena, Phys. Z., 24 (1923), pp. 185–206.Google Scholar
[16] Fairweather, G. and Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9 (1998), pp. 69–95.CrossRefGoogle Scholar
[17] Hall, W. S. and Mao, X. Q., A boundary element investigation of irregular frequencies in electromagnetic scattering, Eng. Anal. Bound. Elem., 16 (1995), pp. 245–252.CrossRefGoogle Scholar
[18] Hansen, P. C., Rank-Defficient and Discrete Ill-Posed Problems: Numerical Aspects of Numerical Inversion, SIAM, Philadelphia, 1998.CrossRefGoogle Scholar
[19] Harari, I., Barbone, P. E., Slavutin, M. and Shalom, R., Boundary infinite elements for the Helmholtz equation in exterior domains, Int. J. Numer. Meth. Eng., 41 (1998), pp. 1105–1131.3.0.CO;2-0>CrossRefGoogle Scholar
[20] Herrera, I., Boundary Methods: An Algebraic Theory, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984.Google Scholar
[21] Isakov, V., Inverse obstacle problems, Inverse Problems 25 (2009), 123002.CrossRefGoogle Scholar
[22] Jin, B. and Marin, L., The plane wave method for inverse problems associated with Helmholtz-type equations, Eng. Anal. Bound. Elem., 32 (2008), pp. 223–240.CrossRefGoogle Scholar
[23] Jin, B. and Zheng, Z., Boundary knot method for some inverse problems associated with the Helmholtz equation, Int. J. Numer. Meth. Eng., 62 (2005), pp. 1636–1651.CrossRefGoogle Scholar
[24] Jones, D. S., Methods in Electromagnetic Wave Propagation, Oxford University Press, New York, 1979.Google Scholar
[25] Kabanikhin, S. I. and Shishlenin, M. A., Stability analysis of a continuation problem for the Helmholtz equation, Bull. Novosibirsk Comput. Center, 16 (2013), pp. 59–63.Google Scholar
[26] Kabanikhin, S. I., Gasimov, Y. S., Nurseitsov, D. B., Shishlenin, M. A., Sholpanbaev, B. B. and Kasenov, S., Regularization of the continuation problem for elliptic equations, J. Inverse Ill-Posed Problems, 21 (2013), pp. 871–884.CrossRefGoogle Scholar
[27] Kaltenbacher, B., Neubauer, A. and Scherzer, O., Iterative Regularization Methods for Nonlinear Problems, de Gruyter, Berlin, 2008.CrossRefGoogle Scholar
[28] Kansa, E. J., Multiquadrics: A scattered data approximation scheme with applications to computational fluid dynamics, Comput. Math Appl., 19 (1990), pp. 147–161.CrossRefGoogle Scholar
[29] Karageorghis, A., The plane waves method for axisymmetric Helmholtz problems, Eng. Anal. Bound. Elem., 69 (2016), pp. 46–56.CrossRefGoogle Scholar
[30] Karageorghis, A. and Lesnic, D., The method of fundamental solutions for the inverse conductivity problem, Inverse Problems Sci. Eng., 18 (2010), pp. 567–583.CrossRefGoogle Scholar
[31] Karageorghis, A., Lesnic, D. and Marin, L., A survey of applications of the MFS to inverse problems, Inverse Problems Sci. Eng., 19 (2011), pp. 309–336.CrossRefGoogle Scholar
[32] Karageorghis, A., Lesnic, D. and Marin, L., The MFS for inverse geometric problems, Inverse Problems and Computational Mechanics (Munteanu, L. Marin, L. and Chiroiu, V., eds.), vol. 1, Editura Academiei, Bucharest, 2011, pp. 191–216.Google Scholar
[33] Kraus, A. D., Aziz, A. and Welty, J., Extended Surface Heat Transfer, John Wiley & Sons, New York, 2001.Google Scholar
[34] Lax, P. D. and Phillips, R. S., Scattering Theory, Academic Press, New York, 1967.Google Scholar
[35] Lian, J. and Subramanian, S., Computation of molecular electrostatics with boundary element methods, Biophys. J., 73 (1997), pp. 1830–1841.CrossRefGoogle Scholar
[36] Li, X., On solving boundary value problems of modified Helmholtz equations by plane wave functions, J. Comput. Appl. Math., 195 (2006), pp. 66–82.CrossRefGoogle Scholar
[37] Marin, L., Numerical boundary identification for Helmholtz-type equations, Comput. Mech., 39 (2006), pp. 25–40.CrossRefGoogle Scholar
[38] Marin, L. and Karageorghis, A., Regularized MFS-based boundary identification in two-dimensional Helmholtz-type equations, CMC Comput. Mater. Continua, 10 (2009), pp. 259–293.Google Scholar
[39] Marin, L., Elliott, L., Heggs, P. J., Ingham, D. B., Lesnic, D. and Wen, X., Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations, Comput. Mech., 31 (2003), pp. 367–377.CrossRefGoogle Scholar
[40] Marin, L., Elliott, L., Heggs, P. J., Ingham, D. B., Lesnic, D. and Wen, X., BEM solution for the Cauchy problem associated with Helmholtz-type equations by the Landweber method, Eng. Anal. Bound. Elem., 28 (2004), pp. 1025–1034.CrossRefGoogle Scholar
[41] The MathWorks, Inc., 3 Apple Hill Dr., Natick, MA, Matlab.Google Scholar
[42] Numerical Algorithms Group Library Mark 21 (2007), NAG (UK) Ltd, Wilkinson House, Jordan Hill Road, Oxford, UK.Google Scholar
[43] Tikhonov, A. N. and Arsenin, V. Y., Methods for Solving Ill-Posed Problems, Nauka, Moscow, 1986.Google Scholar
[44] Valtchev, S. S., Numerical Analysis of Methods with Fundamental Solutions for Acoustic and Elastic Wave Propagation Problems, Ph.D. thesis, Department of Mathematics, Instituto Superior Téchnico, Universidade Técnica de Lisboa, Lisbon, 2008.Google Scholar