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Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators. (English) Zbl 1195.65206

Summary: We combine the method of fundamental solutions with various regularization techniques to solve Cauchy problems of elliptic differential operators. The main idea is to approximate the unknown solution by a linear combination of fundamental solutions whose singularities are located outside the solution domain. To solve effectively the discrete ill-posed resultant matrix, we use three regularization strategies under three different choices for the regularization parameter. Several examples on problems with smooth and non-smooth geometries in 2D and 3D spaces using under-, equally, and over-specified Cauchy data on an accessible boundary are given. Numerical results indicate that the generalized cross-validation and \(L\)-curve choice rulers for Tikhonov regularization and damped singular value decomposition strategy are most effective when using the same numbers of collocation and source points. It has also been observed that the use of more Cauchy data will greatly improve the accuracy of the approximate solution.

MSC:

65N80 Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs
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[1] Bogomolny, A., Fundamental solutions method for elliptic boundary value problems, SIAM J numer anal, 22, 4, 644-669, (1985) · Zbl 0579.65121
[2] Katsurada, M.; Okamoto, H., The collocation points of the fundamental solution method for the potential problem, Comput math appl, 31, 1, 123-137, (1996) · Zbl 0852.65101
[3] Mathon, R.; Johnston, R.L., The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM J numer anal, 14, 4, 638-650, (1977) · Zbl 0368.65058
[4] Karageorghis, A.; Fairweather, G., The method of fundamental solutions for the numerical solution of the biharmonic equation, J comput phys, 69, 433-459, (1987) · Zbl 0618.65108
[5] Smyrlis, Y.-S.; Karageorghis, A., Some aspects of the method of fundamental solutions for certain biharmonic problems, CMES comput model eng sci, 4, 5, 535-550, (2003) · Zbl 1051.65110
[6] Poullikkas, A.; Karageorghis, A.; Georgiou, G., The method of fundamental solutions for three-dimensional elastostatics problems, Comput struct, 80, 3-4, 365-370, (2002)
[7] Kondapalli, P.S.; Shippy, D.J.; Fairweather, G., Analysis of acoustic scattering in fluids and solids by the method of fundamental solutions, J acoust soc am, 91, 4 Part 1, 1844-1854, (1992)
[8] Kondapalli, P.S.; Shippy, D.J.; Fairweather, G., The method of fundamental solutions for transmission and scattering of elastic waves, Comput methods appl mech eng, 96, 2, 255-269, (1992) · Zbl 0825.73150
[9] Balakrishnan, K.; Ramachandran, P.A., The method of fundamental solutions for linear diffusion-reaction equations, Math comput model, 31, 2-3, 221-237, (2000) · Zbl 1042.35569
[10] Balakrishnan, K.; Ramachandran, P.A., Osculatory interpolation in the method of fundamental solution for nonlinear possion problems, J comput phys, 172, 1, 1-18, (2001) · Zbl 0992.65131
[11] Golberg, M.A., The method of fundamental solutions for Poisson’s equation, Eng anal boundary elem, 16, 3, 205-213, (1995)
[12] Partridge, P.W.; Sensale, B., The method of fundamental solutions with dual reciprocity for diffusion and diffusion convection using subdomains, Eng anal boundary elem, 24, 9, 633-641, (2000) · Zbl 1005.76064
[13] Fairweather, G.; Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv comput math, 9, 1-2, 69-95, (1998) · Zbl 0922.65074
[14] Golberg, M.A.; Chen, C.S., The method of fundamental solutions for potential, Helmholtz and diffusion problems, (), 103-176 · Zbl 0945.65130
[15] Alessandrini, G., Stable determination of a crack from boundary measurements, Proc roy soc Edinburgh sect A, 123, 3, 497-516, (1993) · Zbl 0802.35151
[16] Al-Najem, N.M.; Osman, A.M.; El-Refaee, M.M.; Khanafer, K.M., Two dimensional steady-state inverse heat conduction problems, Int commun heat mass transfer, 25, 4, 541-550, (1998)
[17] Franzone, P.C.; Magenes, E., On the inverse potential problem of electrocardiology, Calcolo, 16, 4, 459-538, (1980) · Zbl 0452.92008
[18] Ang, D.D.; Nghia, N.H.; Tam, N.C., Regularized solutions of Cauchy problem for the Laplace equation in an irregular layer: a three dimensional case, Acta math Vietnam, 23, 65-74, (1998) · Zbl 0929.35169
[19] Berntsson, F.; Eldén, L., Numerical solution of a Cauchy problem for the Laplace equation, Inverse probl, 17, 4, 839-853, (2001) · Zbl 0993.65119
[20] Cheng, J.; Hon, Y.C.; Wei, T.; Yamamoto, M., Numerical computation of a Cauchy problem for Laplace’s equation, ZAMM Z angew math mech, 81, 10, 665-674, (2001) · Zbl 0999.65100
[21] Hào, D.N.; Lesnic, D., The Cauchy problem for Laplace’s equation via the conjugate gradient method, IMA J appl math, 65, 2, 199-217, (2000) · Zbl 0967.35032
[22] Hon, Y.C.; Wei, T., Backus – gilbert algorithm for the Cauchy problem of the Laplace equation, Inverse probl, 17, 2, 261-271, (2001) · Zbl 0980.35167
[23] Hon, Y.C.; Wu, Z., A numerical computation for inverse boundary determination problem, Eng anal boundary elem, 24, 7-8, 599-606, (2000) · Zbl 0972.65085
[24] Reinhardt, H.-J.; Han, H.; Hào, D.N., Stability and regularization of a discrete approximation to the Cauchy problem for Laplace’s equation, SIAM J numer anal, 36, 3, 890-905, (1999), (electronic) · Zbl 0928.35184
[25] Jin B, Zheng Y. A meshless method for some inverse problems associated with the helmholtz equation. Comput Methods Appl Mech Eng 2006; 195 (19-22): 2270-88.
[26] Marin, L., A meshless method for the numerical solution of Cauchy problem associated with three-dimensional Helmholtz-type equations, Appl math comput, 165, 355-374, (2005) · Zbl 1070.65115
[27] 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
[28] Engl, H.W.; Hanke, M.; Neubauer, A., Regularization of inverse problems. mathematics and its applications, vol. 375, (1996), Kluwer Academic Publishers Dordrecht
[29] Tikhonov AN, Arsenin VY. Solutions of ill-posed problems, Washington, DC, New York: V.H. Winston, Wiley; 1977. Translated from the Russian, Preface by translation editor Fritz J, Scripta Series in Mathematics.
[30] Cheng, J.; Yamamoto, M., Unique continuation on a line for harmonic functions, Inverse probl, 14, 4, 869-882, (1998) · Zbl 0905.31001
[31] Payne, L.E., Bounds in the Cauchy problem for the Laplace equation, Arch ration mech anal, 5, 35-45, (1960) · Zbl 0094.29801
[32] Klibanov, M.V.; Santosa, F., A computational quasi-reversibility method for Cauchy problems for Laplace’s equation, SIAM J appl math, 51, 6, 1653-1675, (1991) · Zbl 0769.35005
[33] Lattès R, Lions J-L. The method of quasi-reversibility. Applications to partial differential equations. Translated from the French edition and edited by Bellman R. Modern analytic and computational methods in science and mathematics, vol. 18. New York: Elsevier; 1969.
[34] Falk, R.S.; Monk, P.B., Logarithmic convexity for discrete harmonic functions and the approximation of the Cauchy problem for Poisson’s equation, Math comput, 47, 175, 135-149, (1986) · Zbl 0623.65095
[35] Wang, Z.; Wu, S.R., Helmholtz equation-least-squares method for reconstructing the acoustic pressure field, J acoust soc am, 102, 2020-2032, (1997)
[36] Wu, S.R.; Yu, J., Application of BEM-based holography to radiation analysis of sound sources with arbitrary geometries, J acoust soc am, 104, 1054-1060, (1998)
[37] Validivia, I.V.N.; Delillo, W.L.T., The detect of the source of acoustical noise in two dimensions, SIAM J appl math, 61, 2104-2121, (2001) · Zbl 0983.35149
[38] Validivia, I.V.N.; Delillo, W.L.T., The detect of surface vibrations from interior acoustical pressure, Inverse probl, 19, 507-524, (2003)
[39] Elliott, L.; Heggs, P.J.; Ingham, D.B.; Lesnic, D.; Marin, L.; Wen, X., An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation, Comput methods appl mech eng, 192, 709-722, (2003) · Zbl 1022.78012
[40] Elliott, L.; Heggs, P.J.; Ingham, D.B.; Lesnic, D.; Marin, L.; Wen, X., Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations, Comput mech, 31, 367-372, (2003) · Zbl 1047.65097
[41] Jin, B.; Zheng, Y., Boundary knot method for some inverse problems associated with the Helmholtz equation, J numer methods eng, 62, 1636-1651, (2005) · Zbl 1085.65104
[42] Jin, B.; Zheng, Y., Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation, Eng anal boundary elem, 29, 925-935, (2005) · Zbl 1182.65179
[43] Elliott, L.; Heggs, P.J.; Ingham, D.B.; Lesnic, D.; Marin, L.; Wen, X., BEM solution for the Cauchy problem associated with Helmholtz-type equations by the Landweber method, Eng anal boundary elem, 28, 1025-1034, (2004) · Zbl 1066.80009
[44] Elliott, L.; Heggs, P.J.; Ingham, D.B.; Lesnic, D.; Marin, L.; Wen, X., Comparison of regularization methods for solving the Cauchy problem associated with the Helmholtz equation, Int J numer mech eng, 60, 1933-1947, (2004) · Zbl 1062.78015
[45] Kythe, P.K., Fundamental solutions for differential operators and applications, (1996), Birkhäuser Boston, MA · Zbl 0854.35118
[46] Ramachandran, P.A., Method of fundamental solutions: singular value decomposition analysis, Commun numer methods eng, 18, 11, 789-801, (2002) · Zbl 1016.65095
[47] Hansen, P.C., Analysis of discrete ill-posed problems by means of the L-curve, SIAM rev, 34, 4, 561-580, (1992) · Zbl 0770.65026
[48] Hansen, P.C., Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems, Numer algorithms, 6, 1-2, 1-35, (1994) · Zbl 0789.65029
[49] 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, 6, 1487-1503, (1993) · Zbl 0789.65030
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