Time varying radial basis functions

• Autores: A. A. Jamshidi, C. William Gear, Ioannis G. Kevrekidis
• Localización: Journal of computational and applied mathematics, ISSN 0377-0427, Vol. 266, Nº 1, 2014, págs. 61-72
• Idioma: inglés
• DOI: 10.1016/j.cam.2014.01.018
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• Resumen

  • We introduce radial basis functions (RBFs) whose time-varying coefficients determine not only the amplitude and position of each RBF but also their shape. The intended use of these Time Varying-RBFs (TV-RBFs) is in the local-in-time representation of lowdimensional approximations of functions that arise in solving spatiotemporal evolution problems; in particular, for time-varying spatially localized solutions with a temporal translation component such as traveling waves, modulated pulses or soliton-like solutions of evolutionary differential equations. This paper is restricted to the one-dimensional spatial case. We also present an algorithm that places the Time Varying-RBFs (TV-RBFs) over spatiotemporal data that may come from experiments, from finely discretized PDE simulations, or even from multiscale, particle-based simulations. It first approximates the function at a single time instant (a temporal snapshot) as a sum of RBFs using a novel weighted minimization that causes each RBF to primarily approximate one of the localized parts of the function). It then extends that approximation to TV-RBFs over a sequence of snapshots of the function at different times. We conclude by discussing the potential uses of these TV-RBFs.