zbMATH — the first resource for mathematics

A meshless method for some inverse problems associated with the Helmholtz equation. (English) Zbl 1123.65111
Summary: A new numerical scheme based on the method of fundamental solutions is proposed for the numerical solution of some inverse boundary value problems associated with the Helmholtz equation, including the Cauchy problem. Since the resulting matrix equation is badly ill-conditioned, a regularized solution is obtained by employing truncated singular value decomposition, while the regularization parameter for the regularization method is provided by the L-curve method.
Numerical results are presented for problems on smooth and piecewise smooth domains with both exact and noisy data, and the convergence and stability of the scheme are investigated. These results show that the proposed scheme is highly accurate, computationally efficient, stable with respect to the noise in the data and convergent with respect to decreasing the amount of data noise and increasing the distance between the physical and fictitious boundaries, and could be considered as a competitive alternative to existing methods for these problems.

65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
Full Text: DOI
[1] Gladwell, G.M.L.; Willms, N.B., On the mode shapes of the Helmholtz equation, J. sound vib., 188, 419-433, (1995) · Zbl 1232.74048
[2] Wood, A.S.; Tupholme, G.E.; Bhatti, M.I.H.; Heggs, P.J., Steady-state heat transfer through extended plane surfaces, Int. commun. heat mass transfer, 22, 99-109, (1995)
[3] Bai, M.R., Application of BEM (boundary element method)-based acoustic holography to radiation analysis of sound sources with arbitrarily shaped geometries, J. acoust. soc. am., 92, 533-549, (1992)
[4] Wang, Z.; Wu, S.F., Helmholtz equation least-squares method for reconstructing the acoustic pressure field, J. acoust. soc. am., 102, 2020-2032, (1997)
[5] Wu, S.F.; Yu, J., Reconstructing interior acoustic pressure fields via Helmholtz equation-least-squares method, J. acoust. soc. am., 104, 2054-2060, (1998)
[6] Isakov, V.; Wu, S.F., On theory and application of the Helmholtz equation least squares method in inverse acoustics, Inv. prob., 18, 1147-1159, (2002) · Zbl 1022.35087
[7] Marin, L.; Elliott, L.; Heggs, P.J.; Ingham, D.B.; Lesnic, D.; Wen, X., An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation, Comput. methods appl. mech. engrg., 192, 709-722, (2003) · Zbl 1022.78012
[8] Gryazin, Y.A.; Klibanov, M.V.; Lucas, T.R., Two numerical methods for an inverse problem for the 2-D Helmholtz equation, J. comput. phys., 184, 122-148, (2003) · Zbl 1016.65078
[9] Marin, L.; Lesnic, D., The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations, Comput. struct., 83, 267-278, (2005) · Zbl 1088.35079
[10] Marin, L.; Lesnic, D., A meshless method for the numerical solution of the Cauchy problem associated with three-dimensional Helmholtz-type equations, Appl. math. comput., 165, 355-374, (2005) · Zbl 1070.65115
[11] Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent developments, Comput. methods appl. mech. engrg., 139, 3-47, (1996) · Zbl 0891.73075
[12] Duarte, C.A.; Oden, J.T., H-p clouds—an h-p meshless method, Numer. methods part. diff. eq., 12, 673-705, (1996) · Zbl 0869.65069
[13] Liu, W.K.; Chen, Y.; Jun, S.; Chen, J.S.; Belytschko, T.; Pan, C.; Uras, R.A.; Chang, C.T., Overview and applications of the reproducing kernel particle methods, Arch. comput. methods engrg., 3, 3-80, (1996)
[14] Atluri, S.N.; Shen, S., The meshless local Petrov-Galerkin (MLPG) method, (2002), Technical Science Press Encino · Zbl 1012.65116
[15] Kansa, E.J., Multiquadric—A scattered data approximation scheme with applications to computational fluid dynamics II. solutions to parabolic, hyperbolic, and elliptic partial differential equations, Comput. math. appl., 19, 147-161, (1990) · Zbl 0850.76048
[16] Golberg, M.A.; Chen, C.S., The method of fundamental solution for potential, Helmholtz and diffusion problems, (), 103-176 · Zbl 0945.65130
[17] Fairweather, G.; Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv. comput. math., 9, 69-95, (1998) · Zbl 0922.65074
[18] Balakrishnan, K.; Ramachandran, P.A., A particular solution Trefftz method for nonlinear Poisson problems in heat and mass transfer, J. comput. phys., 150, 239-267, (1999) · Zbl 0926.65121
[19] Balakrishnan, K.; Ramachandran, P.A., The method of fundamental solutions for linear diffusion-reaction equations, Math. comput. model., 31, 221-237, (2000) · Zbl 1042.35569
[20] Fairweather, G.; Karageorghis, A.; Martin, P.A., The method of fundamental solutions for scattering and radiation problems, Engrg. anal. bound. elem., 27, 759-769, (2003) · Zbl 1060.76649
[21] Kythe, P.K., Fundamental solutions for differential operators and applications, (1996), Birkháuser Boston · Zbl 0854.35118
[22] Mitic, P.; Rashed, Y.F., Convergence and stability of the method of meshless fundamental solutions using an array of randomly distributed source, Engrg. anal. bound. elem., 28, 143-153, (2004) · Zbl 1057.65091
[23] Kitagawa, T., On the numerical stability of the method of fundamental solutions applied to the Dirichlet problem, Jpn. J. appl. math., 35, 507-518, (1988)
[24] Kitagawa, T., Asymptotical stability of the fundamental solution method, J. comput. appl. math., 38, 263-269, (1991) · Zbl 0752.65077
[25] Kress, R., Linear integral equations, (1989), Springer-Verlag Berlin
[26] Hansen, P.C., The truncated SVD as a method for regularization, Bit, 27, 534-553, (1987) · Zbl 0633.65041
[27] Engl, H.W., Regularization methods for the stable solution of inverse problems, Surv. math. ind., 3, 71-142, (1993) · Zbl 0776.65043
[28] Ramachandran, P.A., Method of fundamental solutions: singular value decomposition analysis, Comm. numer. methods engrg., 18, 789-801, (2002) · Zbl 1016.65095
[29] Jin, B., A meshless method for the Laplace and biharmonic equations subjected to noisy boundary data, CMES-comput. model. engrg. sci., 6, 253-261, (2004) · Zbl 1081.65548
[30] Hansen, P.C., Analysis of discrete ill-posed problems by means of the L-curve, SIAM rev., 34, 561-580, (1992) · Zbl 0770.65026
[31] Hansen, P.C.; O’Leary, D.P., The use of the L-curve in the regularization of discrete ill-posed problems, SIAM J. sci. comput., 14, 1487-1503, (1993) · Zbl 0789.65030
[32] Golub, G.; Heath, M.; Wahba, G., Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21, 215-223, (1979) · Zbl 0461.62059
[33] Lawson, C.L.; Hanson, R.J., Solving least squares problems, (1974), Prentice-Hall Englewood Cliff · Zbl 0185.40701
[34] Engl, H.W.; Orever, W., Using the L-curve for determining optimal regularization parameters, Numer. math., 69, 25-31, (1994) · Zbl 0819.65090
[35] Vogel, C.R., Non-convergence of the L-curve regularization parameter selection method, Inv. prob., 12, 535-547, (1996) · Zbl 0867.65025
[36] Bogomolny, A., Fundamental solutions method for elliptic boundary value problems, SIAM J. numer. anal., 22, 644-669, (1985) · Zbl 0579.65121
[37] Kozlov, V.A.; Maz̀ya, V.G.; Fomin, A.V., An alternating method for solving the Cauchy problem for elliptic equations, USSR comput. math. math. phys., 31, 45-52, (1992) · Zbl 0774.65069
[38] Fenner, R.T., Source field superposition analysis of two-dimensional potential problems, Int. J. numer. methods engrg., 32, 1079-1091, (1991) · Zbl 0755.76060
[39] Nardini, D.; Brebbia, C.A., A new approach for free vibration analysis using boundary elements, (), 312-326 · Zbl 0541.73104
[40] Hanke, M.; Hansen, P.C., Regularization methods for large-scale problems, Surv. math. ind., 3, 253-315, (1993) · Zbl 0805.65058
[41] Saavedra, I.; Power, H., Multipole fast algorithm for the least squares approach of the method of fundamental solutions for three-dimensional harmonic problems, Numer. methods part. diff. eq., 19, 828-845, (2003) · Zbl 1038.65127
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.