Six strategies for defeating the Runge Phenomenon in Gaussian radial basis functions on a finite interval

Radial basis function (RBF) interpolation is a ‘‘meshless’’ strategy with great promise for adaptive approximation. Because it is meshless, there is no canonical grid to act as a starting point for function-adaptive grid modification. Uniform grids are therefore common with RBFs. Like polynomial interpolation, which is the limit of RBF interpolation when the RBF functions are very wide (the ‘‘flat limit’’), RBF interpolants are vulnerable to the Runge Phenomenon, whether the grid is uniform or irregular: the N-point interpolant may diverge as N → ∞on the interval spanned by the interpolation points even for a function f (x) that is analytic everywhere on and near this interval. In this work, we discuss six strategies for defeating the Runge Phenomenon, specializing to a uniform grid on a finite interval and one particular species of RBFs in one space dimension: φ(x) = exp(−[α2/h2]x2) where h is the grid spacing. Three strategies fail, but three are successful. One good strategy is to vary α with the number of interpolation points N as N−1/4. Unfortunately this gives both a subgeometric rate of convergence for the approximation (error falling as exp(−q √ N) for some constant q) and a matrix condition number that grows as exp(p √ N) for some p > 0. (In order to explain why fixed α, independent of N, is not a convergent strategy, we digress to discuss error saturation, and show experimentally that the saturation error on the finite interval is about 0.6 exp(−0.47/α2)‖f ‖∞; if a user-chosen error tolerance δ is acceptable, then the optimum choice is αoptimum(δ) = 1/ √ −2 log(δ/0.06).) The second good strategy is RBF Extension, which uses RBF functions with centers on a interval slightly larger than the target interval to approximate f (x). A third strategy is to split the interval into two ‘‘boundary layers’’ and a large middle interval and apply separate RBF approximations on each. The slow-decrease-of-α strategy is much cheaper, but RBF Extension is much more resistant to ill-conditioning and therefore can achieve much lower errors. The threeinterval method is the least accurate, but is robust and inexpensive.