Kernel based meshless method for solving ill-posed and large scale problems


Student thesis: Doctoral Thesis

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  • Ming LI

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Awarding Institution
Award date4 Oct 2010


During the past decade, various types of meshless methods have emerged as effective numerical methods for solving problems in science and engineering which involve partial differential equations(PDEs). In contrast to mesh dependent methods (for instances, finite difference method; finite element method; and boundary element method), the meshless methods require neither domain nor boundary meshing. As a result, meshless methods have the potential for solving higher dimensional problems with complicated domain under more general partial differential equations with various coefficients. The definition of a well-posed problem was given by Hadamard in 1902 who stated that a well posed mathematical model of physical phenomena should have the properties: 1. A solution exists; 2. The solution is unique; and 3. The solution depends continuously on the data, in some reasonable topology. Problems that are not well-posed in the sense of Hadamard are termed ill-posed. In other words, a mathematical model that was built based on physical phenomena is ill-posed if one of the above properties is not satisfied. In this thesis, we investigate several types of ill-posed problems, such as inverse boundary determination problems, backward heat conduction problems and inverse source determination problems, by using kernel-based functions. Because of the difficulty in generating mesh in high dimension, the proposed kernel based meshless methods demonstrate the advantage over traditional mesh-dependent methods. For illustration, several kernel based meshless methods, for examples, radial basis functions, method of fundamental solutions and method of Green’s functions, are used for solving several typical ill-posed problems. High dimensional PDEs are ubiquitous: kinetic models, molecular dynamics, quantum mechanics, uncertainty quantification using computer expansions, finance, etc. Finally, a local radial basis function is firstly introduced in this thesis for solving a six-dimensional elliptic equation as an demonstration on the potential applicability of the kernel-based method for solving large scale problems. Compared with meshdependent methods, kernel based meshless methods could solve ill-posed problems directly without iteration and guessed initial value. Keywords: Meshless methods, Method of fundamental solutions, Radial basis functions, Green’s function

    Research areas

  • Meshfree methods (Numerical analysis), Kernel functions, Numerical analysis, Improperly posed problems, Differential equations, Partial