An analysis of some approximate inverse preconditioning for linear system

The class of preconditioning that approximates the inverse of the matrix A is studied in the thesis. We emphasize on investigating some variants of the approximate inverse (APINV) preconditioning. They are formed basically from minimizing the norm of the residual matrix R = I - MA. If Frobenius nom is chosen, it can be decoupled into N independent least square problem. Hence the construction of this preconditioner has a high degree of parallelism, which is suitable for massively parallel computations. Exact solve and multi-color approaches to the APINV preconditioner are discussed in the thesis. We apply the conventional and two-color Fourier analyses to these two approaches on model problem. We find that both approaches give smaller condition number and numerical experiments confirm the results. Setup costs and parallel implementations of the APINV preconditioners will also be discussed and analysed. In addition we compare two variable APINV preconditioners proposed by Huckle[l5] and Chow[5]. These two algorithms impose a variable sparsity pattern to the preconditioner. Analysis on setup costs and comments on parallel implementations are given. Numerical results are also presented on the comparison of these two approaches