Error saturation in Gaussian radial basis functions on a finite interval

Radial basis function (RBF) interpolation is a “meshless” strategy with great promise for adaptive approximation. One restriction is “error saturation” which occurs for many types of RBFs including Gaussian RBFs of the form ϕ ( x ; α , h ) = exp ( − α 2 ( x / h ) 2 ) : in the limit h → 0 for fixed α , the error does not converge to zero, but rather to E S ( α ) . Previous studies have theoretically determined the saturation error for Gaussian RBF on an infinite, uniform interval and for the same with a single point omitted. (The gap enormously increases E S ( α ) .) We show experimentally that the saturation error on the unit interval, x ∈ [ − 1 , 1 ] , is about 0.06 exp ( − 0.47 / α 2 ) ‖ f ‖ ∞ — huge compared to the O ( 2 π / α 2 ) exp ( − π 2 / [ 4 α 2 ] ) saturation error for a grid with one point omitted. We show that the reason the saturation is so large on a finite interval is that it is equivalent to an infinite grid which is uniform except for a gap of many points. The saturation error can be avoided by choosing α ≪ 1 , the “flat limit”, but the condition number of the interpolation matrix explodes as O ( exp ( π 2 / [ 4 α 2 ] ) ) . The best strategy is to choose the largest α which yields an acceptably small saturation error: If the user chooses an error tolerance δ , then α o p t i m u m ( δ ) = 1 / − 2 log ( δ / 0.06 ) .