MULTISCALE COMPUTATION WITH INTERPOLATING WAVELETS

Multiresolution analyses based upon interpolets, interpolating scaling functions introduced by Deslauriers and Dubuc, are particularly well- suited to physical applications because they allow exact recovery of t he multiresolution representation of a function from its sample values on a finite set of points in space. We present a detailed study of th e application of wavelet concepts to physical problems expressed in su ch bases. The manuscript describes algorithms for the associated trans forms which for properly constructed grids of variable resolution comp ute correctly without having to introduce extra grid points, We demons trate that for the application of local homogeneous operators in such bases, the nonstandard multiply of Beylkin, Coifman, and Rokhlin also proceeds exactly for inhomogeneous grids of appropriate form. To obtai n less stringent conditions on the grids, we generalize the nonstandar d multiply so that communication may proceed between nonadjacent level s. The manuscript concludes with timing comparisons against naive algo rithms and an illustration of the scale-independence of the convergenc e rate of the conjugate gradient solution of Poisson's equation using a simple preconditioning. (C) 1998 Academic Press.