Matrix approximations and solvers using tensor products and non-standard wavelet transforms related to irregular grids

Abstract

Dense large-scale matrices coming from integral equations and tensor-product grids can be approximated by a sum of Kronecker products with further sparsi cation of the factors via discrete wavelet transforms, which results in reduced storage and computational costs and also in good preconditioners in the case of uniform one-dimensional grids. However, irregular grids lead to a loss of approximation quality and, more signi cantly, to a severe deterioration in ef ciency of the preconditioners that have been considered previously (using a sparsi cation of the inverse to one Kronecker product or an incomplete factorizationapproach). In this paper we propose to use non-standard wavelet transforms related to the irregular grids involved and, using numerical examples, we show that the new transforms provide better compression than the Daubechies wavelets. A further innovation is a scaled two-level circulant preconditioner that performs well on irregular grids. The proposed approximation and preconditioning techniques have been applied to a hypersingularintegral equation modelling ow around a thin aerofoil and made it possible to solve linear systems with more than 1 million unknowns in 15-20 minutes even on a personal computer.

Approximate iterations for structured matrices

“…There are only rare examples, for which A and B = f (A) can simultaneously be approximated by sparse matrices from S := {X ∈ R I ×I : R(X ) = X }. However, it is well-known that after a discrete wavelet transform X → L(X ) := T −1 X T one can apply a matrix compression (see [12,13,31,32]). Such a matrix compression is of the form (3.1) and will be denoted by instead of R. Then, the trunction R applied to the original matrix X is the composition of the wavelet transform L, the pattern projection and the back-transformation L −1 :…”

“…The very idea of iterations with truncation has been already advocated in several papers, chiefly for Toeplitz-like matrices [8,9,[31][32][33], rank-structured matrices [4,5,12,13,21,27,32] (see also [22][23][24][25][26]) and wavelet-based sparsification [1,6,12,13]. However, the proofs provided so far only for some particular cases of structures have appeared as different individual proofs.…”