A new class of oscillatory radial basis functions

Authors:
B. Fornberg;E. Larsson;G. Wright
Affiliations:
University of Colorado Department of Applied Mathematics 526 UCB Boulder, CO 80309, U.S.A.;Uppsala University Department of Information Technology Scientific Computing, Box 337 SE-751 05 Uppsala, Sweden;University of Utah Department of Mathematics 155 South 1400 East, JWB 233 Salt Lake City, UT 84112-0090, U.S.A.
Venue:
Computers & Mathematics with Applications
Year:
2006

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Abstract

Radial basis functions (RBFs) form a primary tool for multivariate interpolation, and they are also receiving increased attention for solving PDEs on irregular domains. Traditionally, only nonoscillatory radial functions have been considered. We find here that a certain class of oscillatory radial functions (including Gaussians as a special case) leads to nonsingular interpolants with intriguing features especially as they are scaled to become increasingly flat. This flat limit is important in that it generalizes traditional spectral methods to completely general node layouts. Interpolants based on the new radial functions appear immune to many or possibly all cases of divergence that in this limit can arise with other standard types of radial functions (such as multiquadrics and inverse multiquadratics).