Α Study on Radial Basis Function and Quasi-Monte Carlo Methods

The radial basis function (RBF) and quasi Monte Carlo (QMC) methods are two very promising schemes to handle high-dimension problems with complex and moving boundary geometry due to the fact that they are independent of dimensionality and inherently meshless. The two strategies are seemingly irrelevant and are so far developed independently The former is largely used to solve partial differential equations (PDE), neural network, geometry generation, scattered data processing with mathematical justifications of interpolation theory [1], while the latter is often employed to evaluate high-dimension integration with the Monte Carlo method (MCM) background [2], The purpose of this communication is to try to establish their intrinsic relationship on the grounds of numerical integral. The kernel function of integral equation is found the key to construct efficient RBFs. Some significant results on RBF construction, error bound and node placement are also presented. It is stressed that the RBF is here established on integral analysis rather than on the sophisticated interpolation and native space analysis. method, kernel function, edge effects, GS-RBF,

[1]  E. Kansa,et al.  Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations , 2000 .

[2]  Carsten Franke,et al.  Solving partial differential equations by collocation using radial basis functions , 1998, Appl. Math. Comput..

[3]  Alexander G. Ramm,et al.  Theory and Applications of Some New Classes of Integral Equations , 1980 .

[4]  Michael A. Golberg,et al.  The method of fundamental solutions and quasi‐Monte‐Carlo method for diffusion equations , 1998 .

[5]  E. Kansa,et al.  Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary , 2002 .

[6]  Joe F. Thompson,et al.  Automatic numerical generation of body-fitted curvilinear coordinate system for field containing any number of arbitrary two-dimensional bodies , 1974 .

[7]  Alexander I. Fedoseyev,et al.  Continuation for nonlinear Elliptic Partial differential equations discretized by the multiquadric Method , 2000, Int. J. Bifurc. Chaos.

[8]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[9]  Wen Chen,et al.  New Insights in Boundary-only and Domain-type RBF Methods , 2002, ArXiv.

[10]  Walter Schempp,et al.  Constructive Theory of Functions of Several Variables: Proceedings of a Conference Held at Oberwolfach, Germany, April 25 - May 1, 1976 , 1977, Constructive Theory of Functions of Several Variables.