Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators

By Ting Wei, Yiu Chung Benny Hon, Leevan Ling. 2007

198 citations recorded by the source corpus.

Abstract In this paper we combine the method of fundamental solutions with various regularization techniques to solve Cauchy problems of elliptic differential operators. The main idea is to approximate the unknown solution by a linear combination of fundamental solutions whose singularities are located outside the solution domain. To solve effectively the discrete ill-posed resultant matrix, we use three regularization strategies under three different choices for the regularization parameter. Several examples on problems with smooth and non-smooth geometries in 2D and 3D spaces using under-, equally, and over-specified Cauchy data on an accessible boundary are given. Numerical results indicate that the generalized cross-validation and L -curve choice rulers for Tikhonov regularization and damped singular value decomposition strategy are most effective when using the same numbers of collocation and source points. It has also been observed that the use of more Cauchy data will greatly improve the accuracy of the approximate solution.

Subfields: Mathematical Physics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/da140830dff722a650a6d245199a11e1ccb46b1b

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In Mathematical Physics

In Mechanics of Materials

Citations in the corpus

A survey of applications of the MFS to inverse problems (2011, methodology): “In [58], the authors applied the MFS with various regularization techniques for the solution of the Cauchy problems (3.”
Two regularization methods for the Cauchy problems of the Helmholtz equation (2010, methodology): “Several numerical methods have been proposed to solve this problem, such as the boundary element method (BEM) [14], the conjugate gradient method [4,16], the Landweber method [15], the singular value decomposition (SVD) method [16], the method of fundamental solutions (MFS) [13,17,18], the boundary ”
Reconstruction of a moving boundary from Cauchy data in one-dimensional heat equation (2009, background): “In [33], the authors compared the effects of several regularization techniques and found that the Tikhonov regularization with a generalized cross-validation choice of the regularization parameter is more effective while using the MFS.”
BOUNDARY PARTICLE METHOD FOR INVERSE CAUCHY PROBLEMS OF INHOMOGENEOUS HELMHOLTZ EQUATIONS (2009, methodology): “To overcome mesh generation problem, facing the traditional mesh-based BEM, a variety of boundary-type meshless methods, in recent years, have been developed and attract a lot of attention, for example, method of fundamental solution (MFS) [4, 11, 22], boundary knot method (BKM) [10], plane wave met”
Applicability of the method of fundamental solutions (2009, methodology): “The method of fundamental solutions (MFS) introduced by Kupradze and Aleksidze [23] has become a popular meshless method for solving various partial differential equations (PDEs) of practical importance; for example, Poisson equations [13], nonlinear Poisson problems [2], nonlinear thermal explosion”