A Radial basis function based frames strategy for bypassing the Runge phenomenon

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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, SIAMJ. 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 analytic nonperiodic function on abounded domain with at least a superalgebraic accuracy, even on an equispaced grid, despite a crippling ill-conditioning. In this study, we develop an RBF-based frames method, which builds upon the Fourier extensions method. Minute modifications to the standard RBF algorithm, including a clear methodology for determining the RBF shape parameter, are provided. We observe identical convergence and stability patterns with our and the Fourier extensions methods and expose their fundamental connection by showing that the Fourier extensions method is a limiting case of our algorithm.

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© 2016 Society for Industrial and Applied Mathematics. Publisher's version of record: https://doi.org/10.1137/15M1040943

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Society for International and Applied Mathematics Journal on Scientific Computing