Time varying radial basis functions

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 low-dimensional 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.