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The Plane Waves Method for Numerical Boundary Identification

  • A. Karageorghis (a1), D. Lesnic (a2) and L. Marin (a3)
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.

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Corresponding author
*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)
References
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[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.
[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. 371382.
[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. 15251550.
[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. 4562.
[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. 4460.
[6] Beskos, D. E., Boundary element method in dynamic analysis: Part II (1986–1996), ASME Appl. Mech. Rev., 50 (1997), pp. 149197.
[7] Bin-Mohsin, B. and Lesnic, D., Identification of a corroded boundary and its Robin coefficient, East Asian J. Appl. Math., 2 (2012), pp. 126149.
[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. 381404.
[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. 4454.
[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. 227238.
[11] Chen, C. S., Karageorghis, A. and Li, Y., On choosing the location of the sources in the MFS, Numer. Algor., 72 (2016), pp. 107130.
[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. 7595.
[13] Chen, W., Meshfree boundary particle method applied to Helmholtz problems, Eng. Anal. Bound. Elem., 26 (2002), pp. 577581.
[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. 6585.
[15] Debye, P. and Hückel, E., The theory of electrolytes. I. Lowering of freezing point and related phenomena, Phys. Z., 24 (1923), pp. 185206.
[16] Fairweather, G. and Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9 (1998), pp. 6995.
[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. 245252.
[18] Hansen, P. C., Rank-Defficient and Discrete Ill-Posed Problems: Numerical Aspects of Numerical Inversion, SIAM, Philadelphia, 1998.
[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. 11051131.
[20] Herrera, I., Boundary Methods: An Algebraic Theory, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984.
[21] Isakov, V., Inverse obstacle problems, Inverse Problems 25 (2009), 123002.
[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. 223240.
[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. 16361651.
[24] Jones, D. S., Methods in Electromagnetic Wave Propagation, Oxford University Press, New York, 1979.
[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. 5963.
[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. 871884.
[27] Kaltenbacher, B., Neubauer, A. and Scherzer, O., Iterative Regularization Methods for Nonlinear Problems, de Gruyter, Berlin, 2008.
[28] Kansa, E. J., Multiquadrics: A scattered data approximation scheme with applications to computational fluid dynamics, Comput. Math Appl., 19 (1990), pp. 147161.
[29] Karageorghis, A., The plane waves method for axisymmetric Helmholtz problems, Eng. Anal. Bound. Elem., 69 (2016), pp. 4656.
[30] Karageorghis, A. and Lesnic, D., The method of fundamental solutions for the inverse conductivity problem, Inverse Problems Sci. Eng., 18 (2010), pp. 567583.
[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. 309336.
[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. 191216.
[33] Kraus, A. D., Aziz, A. and Welty, J., Extended Surface Heat Transfer, John Wiley & Sons, New York, 2001.
[34] Lax, P. D. and Phillips, R. S., Scattering Theory, Academic Press, New York, 1967.
[35] Lian, J. and Subramanian, S., Computation of molecular electrostatics with boundary element methods, Biophys. J., 73 (1997), pp. 18301841.
[36] Li, X., On solving boundary value problems of modified Helmholtz equations by plane wave functions, J. Comput. Appl. Math., 195 (2006), pp. 6682.
[37] Marin, L., Numerical boundary identification for Helmholtz-type equations, Comput. Mech., 39 (2006), pp. 2540.
[38] Marin, L. and Karageorghis, A., Regularized MFS-based boundary identification in two-dimensional Helmholtz-type equations, CMC Comput. Mater. Continua, 10 (2009), pp. 259293.
[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. 367377.
[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. 10251034.
[41] The MathWorks, Inc., 3 Apple Hill Dr., Natick, MA, Matlab.
[42] Numerical Algorithms Group Library Mark 21 (2007), NAG (UK) Ltd, Wilkinson House, Jordan Hill Road, Oxford, UK.
[43] Tikhonov, A. N. and Arsenin, V. Y., Methods for Solving Ill-Posed Problems, Nauka, Moscow, 1986.
[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.
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Advances in Applied Mathematics and Mechanics
  • ISSN: 2070-0733
  • EISSN: 2075-1354
  • URL: /core/journals/advances-in-applied-mathematics-and-mechanics
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