Russian Journal of Numerical Analysis and Mathematical Modelling
Editor-in-Chief: Dymnikov, Valentin P. / Kuznetsov, Yuri
Managing Editor: Vassilevski, Yuri V.
Editorial Board: Agoshkov, Valeri I. / Amosov, Andrey A. / Kaporin, Igor E. / Kobelkov, Georgy M. / Mikhailov, Gennady A. / Repin, Sergey I. / Shaidurov, Vladimir V. / Shokin, Yuri I. / Tyrtyshnikov, Eugene E.
IMPACT FACTOR 2018: 0.779
CiteScore 2018: 0.92
SCImago Journal Rank (SJR) 2018: 0.512
Source Normalized Impact per Paper (SNIP) 2018: 1.275
Mathematical Citation Quotient (MCQ) 2018: 0.08
Matrix approximations and solvers using tensor products and non-standard wavelet transforms related to irregular grids
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.
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