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Some comments on the ill-conditioning of the method of fundamental solutions. (English) Zbl 1187.65136
Summary: We consider the accuracy and stability of implementing the method of fundamental solutions. In contrast to the results shown by P. A. Ramachandran, Commun. Numer. Methods Eng. 18, No. 11, 789–801 (2002; Zbl 1016.65095)], we find that Gaussian elimination can be used reliably to solve the MFS equations and the use of the singular value decomposition shows no improvement over Gaussian elimination provided that the boundary condition is non-noisy. However, for noisy boundary conditions, there is evidence that the singular value decomposition with truncation is more accurate than Gaussian elimination.

MSC:
65N80 Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs
65N20 Numerical methods for ill-posed problems for boundary value problems involving PDEs
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[1] Cho, H.A.; Golberg, M.A.; Muleshkov, A.S.; Li, X., Trefftz methods for time dependent partial differential equations, Cmc, 1, 1-37, (2004)
[2] Fairweather, G.; Karageorghis, A., The method of fundamental solution for elliptic boundary value problems, Adv comput math, 9, 69-95, (1998) · Zbl 0922.65074
[3] Golberg, M.A.; Chen, C.S., The method of fundamental solutions for potential, Helmholtz and diffusion problems, (), 103-176 · Zbl 0945.65130
[4] Kitagawa, T., On the numerical stability of the method of fundamental solution applied to Dirichlet problem, Jpn J appl math, 5, 123-133, (1988) · Zbl 0644.65060
[5] Ramachandran, P.A., Method of fundamental solutions: singular value decomposition analysis, Commun numer methods eng, 18, 789-801, (2002) · Zbl 1016.65095
[6] Kupradze, V.D.; Aleksidze, M.A., The method of functional equations for the approximate solution of certain boundary value problems, USSR comput math math phys, 4, 82-126, (1964) · Zbl 0154.17604
[7] Bogomolny, A., Fundamental solutions method for elliptic boundary value problems, SIAM J numer anal, 22, 644-669, (1985) · Zbl 0579.65121
[8] Cheng RSC. Delta-trigonometric and spline methods using the single-layer potential representation. PhD Dissertation. University of Maryland; 1987.
[9] Katsurada, M., Asymptotic error analysis of the charge simulation method in Jordan regions with an analytic boundary, J fac sci univ Tokyo sect 1A math, 37, 635-657, (1990) · Zbl 0723.65093
[10] Tikhonov, A.N.; Arsenin, V.Y., On the solution of ill-posed problems, (1977), Wiley New York
[11] Hansen, P.C., Analysis of discrete ill-posed problems by means of the L-curve, SIAM rev, 34, 561-580, (1992) · Zbl 0770.65026
[12] 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
[13] Balakrishnan, K.; Ramachandran, P.A., A particular solution Trefftz method for non-linear Poisson problems in heat and mass transfer, J comput phys, 150, 239-267, (1999) · Zbl 0926.65121
[14] Hon, Y.C.; Wei, T., The method of fundamental solutions for solving multidimensional inverse heat conduction problems, Cmes, 7, 119-132, (2004) · Zbl 1114.80004
[15] Jin, B., A meshless method for the Laplace and biharmonic equations subjected to noisy boundary data, Cmes, 6, 253-262, (2005) · Zbl 1081.65548
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