Date: Friday, March 01, 2013
Location: 1084 East Hall (3:00 PM to 4:00 PM)
Title: Hermite function interpolation on a finite interval and the interrelationships of polynomial and radial basis functions
Abstract: Radial basis functions (RBFs) are class of spectral basis functions that often succeed where polynomial interpolation fails. For example, RBFs can be successfully applied to irregular domains punctuated by islands and peninsulas on a "truncated uniform grid". This is constructed by embedding irregular domain in a rectangle, constructing uniform grid with a rectangle, then deleting all points which lie outside the chosen irregular domain. Multivariate polynomial interpolation is a complete disaster on such a grid because it usually diverges due to the Runge phenomenon. But what is the magic? What is the secret of RBFs?
In previous work, I showed that the RBF cardinal functions for five different species on an unbounded, uniform grid were, to a high degree of accuracy, the product of the sinc function, $\sin(\pi x)/(\pi x)$, with $(\pi x)(\rho \sinh([\pi/\rho]x)$. In this sense, Gaussian, sech, inverse quadratic, multiquadric and inverse multiquadric RBFs are really all the same. Equally significant, this analysis shows that RBF cardinal functions are *spatially localized*, decaying exponentially away from their peak, in contrast to the sinc cardinal functions, which decay only as $1/|x|$.
Extending these studies shows that RBF cardinal functions on a finite uniform grid are, to a high degree of approximation, the product of polynomial cardinal functions with a Gaussian. This suggests approximating functions on a finite interval by the product of a Gaussian with a polynomial. This is equivalent to a basis of Hermite functions, usually employed only for an unbounded domain. The Runge phenomenon, which is the divergence of interpolation on a finite, uniform grid, is greatly reduced. Although motivated by similarities to radial basis functions, the Hermite function interpolants are superior in accuracy, condition number and efficiency in much of the numerical parameter space. In particular, no matrix inversion is required.
Speaker: John Boyd
Institution: Atmospheric, Oceanic, and Space Sciences, University of Michigan
Event Organizer: Peter Miller millerpd@umich.edu
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