A Radial Basis Function based Frames Strategy for Bypassing the Runge Phenomenon

Abstract

Similarly to polynomials, smooth radial basis function (RBF) interpolants converge exponentially fast to analytic functions on a one dimensional bounded domain but are also vulnerable to the Runge phenomenon [R. Platte, IMA J. Numer. Anal. 31 (2014), pp. 1578--1597]. A common topic in the study of RBFs has been to find and match an optimal node sets and an RBF shape parameter; the location of the nodes is critical to prevent the Runge phenomenon from occur ring, and small variations in the shape parameter can radically change the interpolation accuracy. However, even a finely tuned combination of shape parameter and node distribution leads to numerical unstability in finite precision arithmetic as the number of nodes increases. In [D. Huybrechs, SIAM J. Numer. Anal. 47 (2010), pp. 4326--4355], [B. Adcock, D. Huybrechs, and J. Martin-Vaquero, Found. Comput. Math., 14 (2014), pp. 635--687.], Huybrechs et al. showed that it is always possible, using the Fourier extensions method, to stably approximate an ana...