caa a22 4500 378913700 CHVBK 20180305123548.0 cr unu---uuuuu 161128e20040401xx s 000 0 eng 10.1515/156939804323089334 doi (NATIONALLICENCE)gruyter-10.1515/156939804323089334 Matrix approximations and solvers using tensor products and non-standard wavelet transforms related to irregular grids [Elektronische Daten] [J. M. Ford, I. V. Oseledets, E. E. Tyrtyshnikov] Dense large-scale matrices coming from integral equations and tensor-product grids can be approximated by a sum of Kronecker products with further sparsification 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 significantly, to a severe deterioration in efficiency of the preconditioners that have been considered previously (using a sparsification of the inverse to one Kronecker product or an incomplete factorization approach). 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 hypersingular integral equation modelling flow 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. Copyright 2004, Walter de Gruyter Ford J. M. Mathematics Department, UMIST, PO Box 88, Manchester M60 1QD, UK aut Oseledets I. V. Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow GSP-1, 119991, Russia aut Tyrtyshnikov E. E. Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow GSP-1, 119991, Russia aut Russian Journal of Numerical Analysis and Mathematical Modelling Walter de Gruyter 19/2(2004-04-01), 185-204 0927-6467 19:2<185 2004 19 rnam https://doi.org/10.1515/156939804323089334 text/html Onlinezugriff via DOI 1 research article jats NATIONALLICENCE 856 40 https://doi.org/10.1515/156939804323089334 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Ford J. M. Mathematics Department, UMIST, PO Box 88, Manchester M60 1QD, UK aut NATIONALLICENCE 700 1- Oseledets I. V. Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow GSP-1, 119991, Russia aut NATIONALLICENCE 700 1- Tyrtyshnikov E. E. Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow GSP-1, 119991, Russia aut NATIONALLICENCE 773 0- Russian Journal of Numerical Analysis and Mathematical Modelling Walter de Gruyter 19/2(2004-04-01), 185-204 0927-6467 19:2<185 2004 19 rnam CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-gruyter