Chan, Tony F.; Chen, Ke On two variants of an algebraic wavelet preconditioner. (English) Zbl 1013.65040 SIAM J. Sci. Comput. 24, No. 1, 260-283 (2002). Summary: A recursive method of constructing preconditioning matrices for the nonsymmetric stiffness matrix in a wavelet basis is proposed for solving a class of integral and differential equations. It is based on a level-by-level application of the wavelet scales decoupling the different wavelet levels in a matrix form just as in the well-known nonstandard form. The result is a powerful iterative method with built-in preconditioning leading to two specific algebraic multilevel iteration algorithms: one with an exact Schur preconditioning and the other with an approximate Schur preconditioning. Numerical examples are presented to illustrate the efficiency of the new algorithms. Cited in 4 Documents MSC: 65F35 Numerical computation of matrix norms, conditioning, scaling 65Y20 Complexity and performance of numerical algorithms 65F10 Iterative numerical methods for linear systems 65N06 Finite difference methods for boundary value problems involving PDEs 65T60 Numerical methods for wavelets 35J25 Boundary value problems for second-order elliptic equations 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs Keywords:complexity of algorithm; multiresolution; level-by-level transforms; sparse approximate inverse; Schur complements; wavelets; Richardson iteration; GMRES iteration; preconditioning; algebraic multilevel iteration algorithms; numerical examples Software:ILUM PDFBibTeX XMLCite \textit{T. F. Chan} and \textit{K. Chen}, SIAM J. Sci. Comput. 24, No. 1, 260--283 (2002; Zbl 1013.65040) Full Text: DOI