A numerical study of some radial basis function based solution methods for elliptic PDEs

By Elisabeth Larsson, Bengt Fornberg. 2003

411 citations recorded by the source corpus.

Subfields: Computational Mechanics, Mechanics of Materials, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/40a98d4cb7c814c6595a2adc4035cf16db39f6fe

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In Computational Mechanics

In Mechanics of Materials

In Modeling and Simulation

Citations in the corpus

Kernel techniques: From machine learning to meshless methods (2006, background): “…collocation can be found in the literature (Wu 1992, Iske 1995, Franke and Schaback 1998a, 1998b), while reports on applications are somewhat scattered (Power and Barraco 2002, Larsson and Fornberg 2003, Fasshauer 2004, Šarler 2005) and often limited to small problems with regular solutions.”
Solving PDEs with radial basis functions * (2015, methodology): “We summarize the study by Larsson and Fornberg (2003), since this also compared RBF-Direct with Contour-Padé (the only stable choice in 2003).”
Scattered node compact finite difference-type formulas generated from radial basis functions (2006, methodology): “In the past decade or so they have also received increased attention as a " mesh-free " method for numerically solving partial differential equations (PDEs) on irregular domains by a global collocation approach (see, for example, [1] [2] [3] [4]).”
A collocation method based on one-dimensional RBF interpolation scheme for solving PDEs (2007, neutral): “…equations has received a great deal of attention over the past 15 years (see, for example, Kansa, 1990; Fasshauer, 1997; Zerroukat et al, 1998; Mai-Duy and Tran-Cong, 2001; Fedoseyev et al, 2002; Power and Barraco, 2002; Cheng et al, 2003; Larsson and Fornberg, 2003; Sarler et al, 2004).”
On choosing “optimal” shape parameters for RBF approximation (2007, methodology): “, [2, 15, 16, 22, 23]) has shown that one may be able to overcome the conditioning problems of the traditional RBF approach by using other techniques such as the Contour–Padé algorithm of [15].”
The Runge phenomenon and spatially variable shape parameters in RBF interpolation (2007, background): “The benefits of computing in a very low ε-range were strikingly illustrated in [14].”
Using meshfree approximation for multi‐asset American options (2004, methodology): “The Contour-Padé algorithm has already been successfully applied to the solution of elliptic PDEs by collocation by Larsson and Fornberg (2003).”
Stable Computations with Gaussian Radial Basis Functions (2011, background): “This has previously been observed and discussed from different perspectives several times; see, for example, [3], [11], [12], [15], [18], [19].”