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博碩士論文 etd-0815106-153718 詳細資訊
Title page for etd-0815106-153718
論文名稱
Title
徑向基與非良置問題
Radial Bases and Ill-Posed Problems
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
101
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
2006-06-07
繳交日期
Date of Submission
2006-08-15
關鍵字
Keywords
非良置問題、徑向基
ill-posed problems, radial basis function
統計
Statistics
本論文已被瀏覽 5760 次,被下載 1523
The thesis/dissertation has been browsed 5760 times, has been downloaded 1523 times.
中文摘要
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。
Abstract
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.
目次 Table of Contents
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
參考文獻 References
References
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