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臺灣博碩士論文加值系統

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研究生:陳賀璞
研究生(外文):Ho-Pu Chen
論文名稱:徑向基與非良置問題
論文名稱(外文):Radial Bases and Ill-Posed Problems
指導教授:呂宗澤
指導教授(外文):Tzon-Tzer Lu
學位類別:碩士
校院名稱:國立中山大學
系所名稱:應用數學系研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2006
畢業學年度:94
語文別:英文
論文頁數:101
中文關鍵詞:非良置問題徑向基
外文關鍵詞:ill-posed problemsradial basis function
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在科學計算上,徑向基(RBFs)是一種非常有用的工具。在本文獻中,我們研究使插值矩陣產生奇異及病態的徑向基配置點與中心點的位置。 我們探索最佳的基底使得插值的誤差函數於最大範數與均方根是最小的。 我們也使用徑向基對受到劇烈擾動的資料點做插值,並設法找出所對應的最佳基底。

在第二個部分,我們使用不同的徑向基與不同的基底個數去解非良置問題。假如解是不唯一的,我們可以由不同的徑向基基底求出相異的數值解。我們選取一組數值解並加上不同的差函數的線性組合,可以建構所有的解。假如解是不存在的,我們可以顯示數值解並不符合原始的方程式。
RBFs are useful in scientific computing. In this thesis, we are interested in the positions of collocation points and RBF centers which causes the matrix for RBF interpolation singular and ill-conditioned. We explore the best bases by minimizing error function in supremum norm and root mean squares. We also use radial basis function to interpolate shifted data and find the best basis in certain sense.
In the second part, we solve ill-posed problems by radial basis collocation method with different radial basis functions and various number of bases. If the solution is not unique, then the numerical solutions are different for different bases. To construct all the solutions, we can choose one approximation solution and add the linear combinations of the difference functions for various bases. If the solution does not exist, we show the numerical solution always fail to satisfy the origin equation.
Contents
1 Interpolation by Radial Basis Functions 2
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Singular and Ill-Conditioned Matrices . . . . . . . . . . . . . . . . . . . 3
1.3 Best Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Interpolation of Shifted Data . . . . . . . . . . . . . . . . . . . . . . . . 37
1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2 Ill-posed Problems 43
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Well-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3 Ill-Posed Problems with Multiple Solutions . . . . . . . . . . . . . . . . 53
2.4 Ill-Posed Problems with No Solution . . . . . . . . . . . . . . . . . . . 75
2.5 Ill-Posed Laplace Boundary Value Problems . . . . . . . . . . . . . . . 87
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
References
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