Compact approximation stencils based on integrated flat radial basis functions

Abstract

This paper presents improved ways of constructing compact integrated radial basis function (CIRBF) stencils, based on extended precision, definite integrals, higher-order IRBFs and minimum number of derivative equations, to enhance their performance over large values of the RBF width. The proposed approaches are numerically verified through secondorder linear differential equations in one and two variables. Significant improvements in the matrix condition number, solution accuracy and convergence rate with grid refinement over the usual approaches are achieved.

Numerical Solution for the Extrapolation Problem of Analytic Functions

“…However, the digits studied were 100-200, while the precise computation of the derivatives was vital for the extrapolation, and so for a higher number of digits. Similarly, Mai-Duy et al [52] examined the compactly integrated radial basis functions, with errors for the derivatives ε ~ O (10 −10 ) for fifty digits accuracy.…”

“…The association of the condition number of Φ with ε ′ exhibits low R 2 (0.25595) with slightly negative slope (), although the high condition number is considered to increase instability [37, 41, 52] indicating the need for high precision to the calculations; however the majority of the values are 10 323 , that is, the maximum real number considered by the software [44]. Sensitivity analysis was selected instead of optimizing each method's parameters such as times of RBFs integration, the kernel function, or its shape parameter [40–42], as the interpolation errors in a number of the test cases were equal to zero at the nodes, eliminating any relevant objective function.…”

A symmetric integrated radial basis function method for solving differential equations

“…One can bypass this issue by using extended precision in computation. In , numerical investigations indicated that the condition number of the conversion matrix grows much faster than that of the final system matrix. Here, we only use extended precision for constructing and inverting small conversion matrices (other computational parts including the solving of the final system of equations are conducted using double precision).…”

“…As β increases, the computed errors of the two methods fluctuate due to their higher matrix condition numbers. Several algorithms to extend the working range of the RBF width have been proposed in the literature (see, e.g., [15][16][17][18][19]37]). In this work, we use the extended precision approach.…”

“…Numerical experiments showed that IRBF methods can yield an improved rate of convergence. In previous reports , we integrated RBFs with respect to the Cartesian coordinates (i.e., , and ). Through integration constants, nodal derivative values can be incorporated into the IRBF expressions.…”