A power sparse approximate inverse preconditioning procedure for large sparse linear systems

TLDR: The results indicate that the PSAI algorithm is at least comparable to and can be much more effective than the adaptive SPAI algorithm and it often outperforms the static SAI algorithms very considerably and is more robust and practical than the static ones for general problems.

Abstract: Motivated by the Cayley–Hamilton theorem, a novel adaptive procedure, called a Power Sparse Approximate Inverse (PSAI) procedure, is proposed that uses a different adaptive sparsity pattern selection approach to constructing a right preconditioner M for the large sparse linear system Ax=b. It determines the sparsity pattern of M dynamically and computes the n independent columns of M that is optimal in the Frobenius norm minimization, subject to the sparsity pattern of M . The PSAI procedure needs a matrix–vector product at each step and updates the solution of a small least squares problem cheaply. To control the sparsity of M and develop a practical PSAI algorithm, two dropping strategies are proposed. The PSAI algorithm can capture an effective approximate sparsity pattern of A−1 and compute a good sparse approximate inverse M efficiently. Numerical experiments are reported to verify the effectiveness of the PSAI algorithm. Numerical comparisons are made for the PSAI algorithm and the adaptive SPAI algorithm proposed by Grote and Huckle as well as for the PSAI algorithm and three static Sparse Approximate Inverse (SAI) algorithms. The results indicate that the PSAI algorithm is at least comparable to and can be much more effective than the adaptive SPAI algorithm and it often outperforms the static SAI algorithms very considerably and is more robust and practical than the static ones for general problems. Copyright q 2008 John Wiley & Sons, Ltd.