Educating local radial basis functions using the highest gradient of interest in three dimensional geometries

Abstract

Summary We present a novel methodology to effectively localize radial basis function approximation methods in three dimensions. The local scheme requires shape parameter‐dependent functions that can be used to approximate gradients of scattered data and to solve partial differential operators. The optimum shape parameter is obtained from the highest gradient of interest, where a known analytical function, when boundary conditions are not present, or a shape parameter‐free global approximation are used to educate the localized scheme. The later option is applicable to problems where the operator needs to be solved multiple times, like in time evolution or stochastic integration. Past shape parameter's optimizations, for two‐dimensional domains, based on the condition number of the interpolant matrix, were unable to provide satisfactory approximations. The applicability of our method is illustrated in the context of an analytical expression interpolation and during a Ginzburg–Landau relaxation of a free energy functional. In general, the optimum shape parameter depends on geometry, node distribution, and density, whereas the approximation errors decrease as the node density and the local stencil size increase. The effective localization of radial basis functions motivates its use in moving boundary problems and accelerates solutions through sparse matrix solvers. Copyright © 2016 John Wiley & Sons, Ltd.