Error saturation in Gaussian radial basis functions on a finite interval

• Authors: John P. Boyd
• Venue: Journal of Computational and Applied Mathematics
• Year: 2010

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 @f(x;@a,h)=exp(-@a^2(x/h)^2): in the limit h-0 for fixed @a, the error does not converge to zero, but rather to E"S(@a). 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(@a).) We show experimentally that the saturation error on the unit interval, x@?[-1,1], is about 0.06exp(-0.47/@a^2)@?f@?"~ - huge compared to the O(2@p/@a^2)exp(-@p^2/[4@a^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 @a@?1, the ''flat limit'', but the condition number of the interpolation matrix explodes as O(exp(@p^2/[4@a^2])). The best strategy is to choose the largest @a which yields an acceptably small saturation error: If the user chooses an error tolerance @d, then @a"o"p"t"i"m"u"m(@d)=1/-2log(@d/0.06).