### Title page for etd-0621106-224436

URN etd-0621106-224436 Shiu-ling Huang huanghl@math.nsysu.edu.tw This thesis had been viewed 5207 times. Download 1514 times. Applied Mathematics 2005 2 Master English Stability Analysis of Method of Foundamental Solutions for Laplace's Equations 2006-05-18 116 truncated singular value decomposition effective condition number method of fundamental solutions Tikhonov regularization traditional condition number This thesis consists of two parts. In the first part, to solve the boundary value problems of homogeneous equations, the fundamental solutions (FS) satisfying the homogeneous equations are chosen, and their linear combination is forced to satisfy the exterior andthe interior boundary conditions. To avoid the logarithmicsingularity, the source points of FS are located outside of the solution domain S. This method is called the method of fundamental solutions (MFS). The MFS was first used in Kupradze in 1963. Since then, there have appeared numerousreports of MFS for computation, but only a few for analysis. The part one of this thesis is to derive the eigenvalues for the Neumann and the Robin boundary conditions in the simple case, and to estimate the bounds of condition number for the mixed boundary conditions in some non-disk domains. The same exponential rates ofCond are obtained. And to report numerical results for two kinds of cases. (I) MFS for Motz's problem by adding singular functions. (II) MFS for Motz's problem by local refinements of collocation nodes. The values of traditional condition number are huge, and those of effective condition number are moderately large. However,the expansion coefficients obtained by MFS are scillatinglylarge, to cause another kind of instability: subtractioncancellation errors in the final harmonic solutions. Hence, for practical applications, the errors and the ill-conditioning must be balanced each other. To mitigate the ill-conditioning, it is suggested that the number of FS should not be large, and the distance between the source circle and the partial S should not be far, either.In the second part, to reduce the severe instability of MFS, the truncated singular value decomposition(TSVD) and Tikhonov regularization(TR) are employed. The computational formulas of the condition number and the effective condition number are derived, and their analysis is explored in detail. Besides, the error analysis of TSVD and TR is also made. Moreover, the combination ofTSVD and TR is proposed and called the truncated Tikhonovregularization in this thesis, to better remove some effects of infinitesimal sigma_{min} and high frequency eigenvectors. Jeng-Tzong Chen - chair Leevan Ling - co-chair Song Wang - co-chair none - co-chair Zi-Cai Li - advisor indicate accessible in a year 2006-06-21

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