Solving linear systems using wavelet compression combined with Kronecker product approximation
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Discrete wavelet transform approximation is an established means of approximating dense linear systems arising from discretization of differential and integral equations defined on a one-dimensional domain. For higher dimensional problems, approximation with a sum of Kronecker products has been shown to be effective in reducing storage and computational costs. We have combined these two approaches to enable solution of very large dense linear systems by an iterative technique using a Kronecker product approximation represented in a wavelet basis. Further approximation of the system using only a single Kronecker product provides an effective preconditioner for the system. Here we present our methods and illustrate them with some numerical examples. This technique has the potential for application in a range of areas including computational fluid dynamics, elasticity, lubrication theory and electrostatics.
KeywordsKronecker product wavelets dense matrices preconditioning
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- T.F. Chan, W.P. Tang and W.L. Wan, Wavelet sparse approximate inverse preconditioners, BIT 37 (1997) 644–660. Google Scholar
- K. Chen, Discrete wavelet transforms accelerated sparse preconditioners for dense boundary element systems, Elec. Trans. Numer. Anal. 8 (1999) 138–153. Google Scholar
- J. Ford, K. Chen and L. Scales, A new wavelet transform preconditioner for iterative solution of elastohydrodynamic lubrication problems, Internat. J. Comput. Math. 75 (2000) 497–513. Google Scholar
- J.M. Ford, An improved DWT-based preconditioner for dense matrix problems, Numerical Analysis Report No. 412, Manchester Centre for Computational Mathematics (September 2002). Google Scholar
- S.A. Goreinov and E.E. Tyrtyshnikov, The maximal-volume concept in approximation by low-rank matrices, Contemporary Math. 208 (2001) 47–51. Google Scholar
- S.A. Goreinov, E.E. Tyrtyshnikov and N.L. Zamarashkin, A theory of pseudo-skeleton approximations, Linear Algebra Appl. 261 (1997) 1–21. Google Scholar
- I.V. Oseledets, Application of B-splines and divided differences for construction of fast discrete wavelet transforms on irregular grids, Math. Notes (2004) in press. Google Scholar
- I.V. Oseledets and E.E. Tyrtyshnikov, Approximate inversion of matrices with solution of the hypersingular integral equation, J. Comput. Math. and Math. Physics (2004) to appear. Google Scholar
- E.E. Tyrtyshnikov, Mosaic-skeleton approximations, Calcolo 33 (1996) 47–57. Google Scholar
- E.E. Tyrtyshnikov, Tensor approximations of matrices associated with asymptotically smooth functions, Sbornik Math. 194(6) (2003) 147–160. Google Scholar
- C. Van Loan and N.P. Pitsianis, Approximation with Kronecker products, in: Linear Algebra for Large Scale and Real Time Applications, eds. M.S. Moonen and G.H. Golub (Kluwer, Dordrecht, 1993) pp. 293–314. Google Scholar