An Approach to Making SPAI and PSAI Preconditioning Effective for Large Irregular Sparse Linear Systems

Abstract

We investigate the SPAI and PSAI preconditioning procedures and shed light on two important features of them: (i) For the large linear system Ax = b with A irregular sparse, i.e., with A having s relatively dense columns, SPAI may be very costly to implement, and the resulting sparse approximate inverses may be ineffective for preconditioning. PSAI can be effective for preconditioning but may require excessive storage and be unacceptably time consuming; (ii) the situation is improved drastically when A is regular sparse, that is, all of its columns are sparse. In this case, both SPAI and PSAI are efficient. Moreover, SPAI and, especially, PSAI are more likely to construct effective preconditioners. Motivated by these features, we propose an approach to making SPAI and PSAI more practical for Ax = b with A irregular sparse. We first split A into a regular sparseà and a matrix of low rank s. Then exploiting the Sherman-Morrison-Woodbury formula, we transform Ax = b into s + 1 new linear systems with the same coefficient matrixÃ, use SPAI and PSAI to compute sparse approximate inverses ofà efficiently and apply Krylov iterative methods to solve the preconditioned linear systems. Theoretically, we consider the non-singularity and conditioning ofà obtained from some important classes of matrices. We show how to recover an approximate solution of Ax = b from those of the s + 1 new systems and how to design reliable stopping criteria for the s + 1 systems to guarantee that the approximate solution of Ax = b satisfies a desired accuracy. Given the fact that irregular sparse linear systems are common in applications, this approach widely extends the practicability of SPAI and PSAI. Numerical results demonstrate the considerable superiority of our approach to the direct application of SPAI and PSAI to Ax = b.

A transformation approach that makes SPAI, PSAI and RSAI procedures efficient for large double irregular nonsymmetric sparse linear systems

“…Following this standard, it is reported in [27] that column irregular linear problems have a wide range of applications and 34% of the square matrices in the University of Florida sparse matrix collection [14] are column irregular sparse; see [27] for further information on where column irregular sparse matrices come from and how dense irregular columns are, etc. For a column irregular sparse A, Jia and Zhang [27] have shown that SPAI and PSAI(tol) are costly and may be impractical; they have given theoretical arguments and numerical evidence that M obtained by SPAI may be ineffective for preconditioning (1.1), but M by PSAI(tol) is effective though its construction is costly. Their analysis has also revealed that SPAI and PSAI(tol) are costly when applied to A T for computing left preconditioners for A column irregular sparse, that is, we compute M by minimizing A T M T − I F with certain sparsity constraints on M .…”

“…As is known from [25,27,28], a common and attractive feature of the aforementioned three procedures is that they can construct preconditioners M efficiently for A double regular sparse, among of which PSAI(tol) is most effective and SPAI and RSAI(tol) are comparably effective for preconditioning double regular sparse linear systems. For the column irregular sparse (1.1), making use of the Sherman-Morrison-Woodbury formula, Jia and Zhang [27] have proposed an approach that transforms (1.1) into a small number of column regular sparse ones with the same coefficient matrix and multiple right-hand sides, so that SPAI and PSAI(tol) can construct preconditioners for the regular sparse problems much more efficiently than they do for (1.1) directly. An approximate solution of the original system with a prescribed accuracy ε is then recovered from those of the regular sparse ones with the accuracy determined by ε.…”

“…An approximate solution of the original system with a prescribed accuracy ε is then recovered from those of the regular sparse ones with the accuracy determined by ε. The numerical experiments in [27] have indicated that such transformation approach speeds up the computational efficiency of SPAI and PSAI(tol) very substantially, compared to them applied to the original problem. However, the transformation approach does not suit for row irregular sparse problems, for which SPAI, PSAI(tol) and RSAI(tol) are still costly, as has been pointed out above.…”

“…Proceed in such a way until r k ≤ η or l reaches the prescribed maximum l max , where η is a mildly small tolerance, usually, 0.1 ∼ 0.4; see [19,27,28]. Obviously, if the cardinality ofĴ k is very big, SPAI is costly, which corresponds to the case that the kth column of A is dense [27,28]. Precisely, suppose that the kth column of A is almost fully dense.…”

“…From the theory in [26], PSAI(tol) with the above dropping criterion will compute a preconditioner M that is as sparse as possible and meanwhile has comparable preconditioning quality to the one generated by BPSAI without dropping any nonzero entries. However, PSAI(tol) involves large sized LS problems for column irregular sparse matrices and is thus costly or simply faces out of memory, though M obtained by it is an effective preconditioner [27]. As a whole, if A is double irregular sparse, PSAI(tol) is costly and may be impractical.…”