Radial Basis Functions

University Of Colorado At Boulder, Boulder CO

Investigators

Abstract

Numerical computations in multiple space dimensions have traditionally been based on structured grids (e.g. finite differences or spectral methods) or on unstructured grids (e.g. finite elements). Even in the latter case, there is structure in the sense that one needs to work out which subsets of nodes should be connected into local triangles, tetrahedra, etc. Aspects associated with such grid generations can at times consume as much or even more computer resources than do the subsequent computations on the grid. Furthermore, it has been all but impossible to achieve high (spectral) computational accuracy on such grids without resorting to extensive use of domain decompositions. Numerous mesh-free methods have been proposed recently. The Radial Basis Function (RBF) approach stands out in several respects, most notably in that it generalizes traditional spectral methods to entirely mesh-free settings. Furthermore, implementations are usually remarkably simple. For example, 20-30 lines of Matlab typically suffices for solving an elliptic PDE on an irregular 3-D domain, to spectral accuracy. Both pseudospectral (PS) methods and RBFs were first proposed in the early 1970's. The superior performance of PS methods for solving many PDEs was demonstrated early on. A NSF proposal by the present investigator about 5 years ago was the first time RBFs were presented in terms of being a direct replacement for the traditional basis functions in PS methods. Recently, significant progress has been made both on overcoming the high computer cost and the numerical ill-conditioning that earlier were thought to severely limit the RBF approach. Recent NSF-DMS supported work by the present investigator has opened up numerous further opportunities in this area, which will now be pursued. These include combining spectral accuracy with local node clustering in a fully stable way, a new stable algorithm in the extremely high accuracy flat basis function limit, developments towards faster algorithms, and the application of RBFs to PDEs in the geometries that are most relevant in astro/geophysics, etc. In the last several decades, computational methods have become an increasingly essential part of how science and engineering are conducted, partly because of a rapid evolution of computer hardware, but equally much thanks to improvements in computational algorithms. Very often, the task to be solved, when formulated in mathematical terms, require the solution of partial differential equations (PDEs), often in irregularly shaped regions in several space dimensions. The Radial Basis Function (RBF) methodology, which is the subject of the present grant, opens entirely new opportunities in this regard, combining very high accuracy with unsurpassed geometric flexibility. One of several application areas of particular interest (pursued in collaboration with scientists at NCAR and NOAA) concerns geophysical and astrophysical modeling in spherical geometries, which are critical issues for effective climate modeling and for studies of solar dynamics.