Biorthogonal Wavelet Based Algebraic Multigrid Preconditioners for Large Sparse Linear Systems

A Padmanabha Reddy,Nagendrappa M Bujurke

doi:10.4236/am.2011.211194

Abstract

In this article algebraic multigrid as preconditioners are designed, with biorthogonal wavelets, as intergrid operators for the Krylov subspace iterative methods. Construction of hierarchy of matrices in algebraic multigrid context is based on lowpass filter version of Wavelet Transform. The robustness and efficiency of this new approach is tested by applying it to large sparse, unsymmetric and ill-conditioned matrices from Tim Davis collection of sparse matrices. Proposed preconditioners have potential in reducing cputime, operator complexity and storage space of algebraic multigrid V-cycle and meet the desired accuracy of solution compared with that of orthogonal wavelets.

Highlights

IntroductionWhere A is n n non-singular matrix and b is vector of size n arise while discretising various equations using finite difference, finite element and domain decomposition schemes etc
The linear system of algebraic equations Ax b (1.1)where A is n n non-singular matrix and b is vector of size n arise while discretising various equations using finite difference, finite element and domain decomposition schemes etc.One of the useful schemes to solve (1.1) is multigrid
Construction of hierarchy of matrices in algebraic multigrid context is based on lowpass filter version of Wavelet Transform

Summary

Introduction

Where A is n n non-singular matrix and b is vector of size n arise while discretising various equations using finite difference, finite element and domain decomposition schemes etc. Kumar and Mehra [2] developed matrix splitting based preconditioners, where as in [3] and [4] wavelet based algebraic multigrid as preconditioners for Krylov subspace iterative methods are developed. In these articles authors have shown that wavelet based algorithms are efficient and robust compared with that of classical schemes such as matrix splitting, algebraic multigrid and incomplete LU factorization(ilu) based preconditioners for Krylov subspace iterative methods. These studies motivate us to develop biorthogonal wavelet based preconditioners for Krylov subspace iterative methods

Discrete Biorthogonal Wavelet Transform

Wavelet-Based AMG

Cost of Orthogonal and Biorthogonal Wavelet Transforms

Numerical Experiments

Conclusions