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Please use this identifier to cite or link to this item: http://hdl.handle.net/2031/4605

Title: Numerical methods for inverse problems
Other Titles: Fan wen ti de shu zhi fang fa
反問題的數值方法
Authors: Wei, Ting (魏婷)
Department: Dept. of Mathematics
Degree: Doctor of Philosophy
Issue Date: 2005
Publisher: City University of Hong Kong
Subjects: Inverse problems (Differential equations) -- Numerical solutions
Notes: CityU Call Number: QA377.W43 2005
Includes bibliographical references (leaves [125]-135)
Thesis (Ph.D.)--City University of Hong Kong, 2005
v, 138 leaves : ill. (some col.) ; 30 cm.
Type: Thesis
Abstract: In this thesis, we develop three meshless numerical methods for solving three classical inverse problems. The waiver of interior or surface meshing makes these methods extremely attractive for problems with complicated geometries in high dimensions. The efficiencies of the proposed methods have been demonstrated by both mathematical analysis and numerical experiments. The main contents consist of three chapters on each method respectively. The second chapter devises a computational method for solving the Cauchy problem of Laplace equation in multi-dimensional space. By using the Green’s formula, the Cauchy problem is transformed to a moment problem from which the normal derivative on an inaccessible boundary can be obtained. Numerical solutions to the moment problem are then obtained respectively by using Talenti method and Backus-Gilbert method. Convergence estimates under the suitable choices of regularization parameters are also given. For verification, some numerical examples in both two-dimensional and three-dimensional cases are presented. In the third chapter we combine the method of fundamental solutions with some discrete regularization techniques to solve Cauchy problems of various elliptic operators and an inverse heat conduction problem. The main idea is to approximate the solution by a linear combination of fundamental solutions whose singularities are located outside the solution domain. Based on the collocating on the given conditions, a discrete problem is easy to be resulted in. To solve effectively the resultant ill-conditioned system, various regularization strategies with different choice rules for the determination of regularization parameters are used. Numerical examples for the Cauchy problems of Laplace, Helmholtz, modified Helmholtz equations respectively and an inverse heat conduction problem with smooth or non-smooth geometries in two- and three-dimensional spaces are given. The numerical results indicate that the proposed methods are effective and stable. We finally propose in the fourth chapter an effective regularization method based on radial basis functions approximation for the reconstruction of partial derivatives from scattered noisy data in multi-dimensional case. Under an a priori and an a posteriori choice rules for the regularization parameter, convergence results for approximating partial derivatives by varous radial basis functions can be obtained. Numerical examples verify that the a posteriori choice rule for the proposed regularization strategy is mostly effective and stable.
Online Catalog Link: http://lib.cityu.edu.hk/record=b1988575
Appears in Collections:MA - Doctor of Philosophy

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