An analytic approximation to the cardinal functions of Gaussian radial basis functions on an infinite lattice

Abstract

Gaussian radial basis functions (RBFs) have been very useful in computer graphics and for numerical solutions of partial differential equations where these RBFs are defined, on a grid with uniform spacing h, as translates of the “master” function ϕ ( x ; α , h ) ≡ exp ( - [ α 2 / h 2 ] x 2 ) where α is a user-choosable constant. Unfortunately, computing the coefficients of ϕ ( x - jh ; α , h ) requires solving a linear system with a dense matrix. It would be much more efficient to rearrange the basis functions into the equivalent “Lagrangian” or “cardinal” basis because the interpolation matrix in the new basis is the identity matrix; the cardinal basis C j ( x ; α , h ) is defined by the set of linear combinations of the Gaussians such that C j ( kh ) = 1 when k = j and C j ( kh ) = 0 for all integers k ≠ j . We show that the cardinal functions for the uniform grid are C j ( x ; h , α ) = C ( x / h - j ; α ) where C ( X ; α ) ≈ ( α 2 / π ) sin ( π X ) / sinh ( α 2 X ) . The relative error is only about 4 exp ( - 2 π 2 / α 2 ) as demonstrated by the explicit second order approximation. It has long been known that the error in a series of Gaussian RBFs does not converge to zero for fixed α as h → 0 , but only to an “error saturation” proportional to exp ( - π 2 / α 2 ) . Because the error in our approximation to the master cardinal function C ( X ; α ) is the square of the error saturation, there is no penalty for using our new approximations to obtain matrix-free interpolating RBF approximations to an arbitrary function f ( x ) . The master cardinal function on a uniform grid in d dimensions is just the direct product of the one-dimensional cardinal functions. Thus in two dimensions C ( X , Y ; α ) ∼ ( α 4 / π 2 ) sin ( π X ) sin ( π Y ) / [ sinh ( α 2 X ) sinh ( α 2 Y ) ] . We show that the matrix-free interpolation can be extended to non-uniform grids by a smooth change of coordinates.