Multiscale Computation with Interpolating Wavelets

TL;DR

This paper explores the use of interpolating wavelets for multiscale computation in physical problems, presenting algorithms for accurate transforms on variable resolution grids and demonstrating efficient solutions for Poisson's equation.

Contribution

It introduces algorithms for multiresolution analysis with interpolating wavelets that work exactly on variable resolution grids without extra points, and generalizes the non-standard multiply for inhomogeneous grids.

Findings

Exact multiresolution representation from finite samples.

Efficient O(n) solution for Poisson's equation.

Generalized non-standard multiply for inhomogeneous grids.

Abstract

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 the multiresolution representation of a function from its sample values on a finite set of points in space. We present a detailed study of the application of wavelet concepts to physical problems expressed in such bases. The manuscript describes algorithms for the associated transforms which, for properly constructed grids of variable resolution, compute correctly without having to introduce extra grid points. We demonstrate that for the application of local homogeneous operators in such bases, the non-standard multiply of Beylkin, Coifman and Rokhlin also proceeds exactly for inhomogeneous grids of appropriate form. To obtain less stringent conditions on the grids, we generalize the non-standard multiply so that communication may proceed between non-adjacent levels. The manuscript concludes with timing comparisons against naive algorithms and an illustration of the scale-independence of the convergence rate of the conjugate gradient solution of Poisson's equation using a simple preconditioning, suggesting that this approach leads to an O(n) solution of this equation.