Wavelet based preconditioners for sparse linear systems

A class of efficient preconditioners based on Daubechies family of wavelets for sparse, unsymmetric linear systems that arise in numerical solution of Partial Differential Equations (PDEs) in a wide variety of scientific and engineering disciplines are introduced. Complete and Incomplete Discrete Wavelet Transforms in conjunction with row and column permutations are used in the construction of these preconditioners. With these Wavelet Transform, the transformed matrix is permuted to band forms. The efficiency of our preconditioners with several Krylov subspace methods is illustrated by solving matrices from Harwell Boeing collection and Tim Davis collection. Also matrices resulting in the solution of Regularized Burgers Equation, free convection in porous enclosure are tested. Our results indicate that the preconditioner based on Incomplete Discrete Haar Wavelet Transform is both cheaper to construct and gives good convergence.