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博碩士論文 etd-0621106-224436 詳細資訊
Title page for etd-0621106-224436
論文名稱
Title
用基本解法求解 Laplace 方程的穩定性分析
Stability Analysis of Method of Foundamental Solutions for Laplace's Equations
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
116
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2006-05-18
繳交日期
Date of Submission
2006-06-21
關鍵字
Keywords
截斷奇異值分解法、有效的condition number、基本解方法、傳統的condition number
truncated singular value decomposition, effective condition number, method of fundamental solutions, Tikhonov regularization, traditional condition number
統計
Statistics
本論文已被瀏覽 5722 次,被下載 1583
The thesis/dissertation has been browsed 5722 times, has been downloaded 1583 times.
中文摘要
這篇論文共包含兩部分,第一部分是說明解決齊次方程邊界問題,基本解方法是先選擇滿足方程的基本解,運用線性結合的技巧去滿足內外邊界條件,為了避免對數奇異的問題,將 source points 放置在解的範圍之外上。基本解方法最早是在1963年由 Kupradze 提出,之後陸續有數值結果發表出來,但相關的分析卻很少。本文的第一部分是推導在 Neumann 與 Robin 邊界條件下的特徵值與估計當非圓形區域下邊界條件為混合的情況下其 condition number 的上界。在第一部分的最後則是數值上的測試,當問題為 Motz’s 問題時,利用基本解加上奇異方程式與基本解使用局佈加密的方法,都可以得到傳統的condition number 會增加很大,但有效的condition number 是適度的增大。然而,其係數在基本解方法中有很大的擾動得到不穩定,造成原因是因為減法消去的誤差再有限齊次解中造成的,因此在實際的應用上需同時考慮誤差與 ill-condition。
在論文的第二部分則是如何改善不穩定情況利用截斷奇異值分解法與 Tikhonov Regularization,這兩種方法其傳統的condition number跟有效的condition number計算公式與誤差分析都推導出,最後透過數值結果可以發現係數與condition number都有明顯的下降。
Abstract
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 and
the interior boundary conditions. To avoid the logarithmic
singularity, 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 numerous
reports 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 of
Cond 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 scillatingly
large, to cause another kind of instability: subtraction
cancellation 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 of
TSVD and TR is proposed and called the truncated Tikhonov
regularization in this thesis, to better remove some effects of infinitesimal sigma_{min} and high frequency eigenvectors.
目次 Table of Contents
Stability Analysis of Method of Fundamental Solutions---{5}
{1}Introduction about Stability Analysis of MFS---{6}
{2}Algorithms of Method of Fundamental Solutions---{8}
{3}Effective Condition Number for Least Squares Methods with Rank Deficiency---{11}
{4}Dirichlet Problems in Disk Domains---{13}
{4.1}Eigenvalues of MFS---{13}
{4.2}General solutions by MFS for the Case of
ho leq R=1---{19}
{4.3}Linkage to Methods of Boundary Integral Equations of First Kind---{20}
{5} Neumann and Robin Problems on Disk Domains---{22}
{5.1}Eigenvalues of MFS for Neumann Problems---{22}
{5.2} Modification of MFS for Neumann Problems---{27}
{5.3} Eigenvalues of MFS for Robin Problems---{30}
{6}Mixed Problems in Bounded and Simply-Connected Domains---{31}
{6.1}Trefftz Methods---{31}
{6.2}The Collocation Trefftz Methods---{36}
{6.3}The Inverse Inequality (6.9)--- {37}
{6.4}Method of Fundamental Solutions---{43}
{7}Numerical Experiments---{48}
{7.1}MFS for Motz's problem by adding singular functions---{48}
{7.2}MFS by local refinements (6.77) of collocation nodes for Motz's problem---{52}
Truncated Singular Value Decomposition and Tikhonov Regularization---{63}
{8}Introduction about TSVD and TR---{64}
{9}Algorithms---{66}
{10}Effective Condition Number for Least Squares Methods with Rank Deficient---{68}
{11}Error Analysis---{76}
{12} Combinations of the TSVD and Tikhonov regularization---{80}
{13}Choice of Parameter lambda in the Tikhonov Regularization---{85}
{14}Numerical Experiments---{93}
{15}Concluding Remarks---{109}
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