Biorthogonal Wavelets for Fast Matrix Computations

In [1], Beylkin et al. introduced a wavelet-based algorithm that approximates integral or matrix operators of a certain type by highly sparse matrices, as the basis for efficient approximate calculations. The wavelets best suited for achieving the highest possible compression with this algorithm are Daubechies wavelets, while Coiflets lead to a faster decomposition algorithm at slightly lesser compression. We observe that the same algorithm can be based on biorthogonal instead of orthogonal wavelets, and derive two classes of biorthogonal wavelets that achieve high compression and high decomposition speed, respectively. In numerical experiments, these biorthogonal wavelets achieved both higher compression and higher speed than their wavelet counterparts, at comparable accuracy.