Abstract. We develop a class of parallel multistep successive preconditioning strategies to enhance efficiency and robustness of standard sparse approximate inverse preconditioning techniques. The key idea is to compute a series of simple sparse matrices to approximate the inverse of the original matrix. Studies are conducted to show the advantages of such an approach in terms of both improving preconditioning accuracy and reducing computational cost, compared to the standard sparse approximate inverse preconditioners. Numerical experiments using one prototype implementation to solve a few sparse matrices on a distributed memory parallel computer are reported.

We compare dierent preconditioning techniques in connection with Krylov methods for the solution of large dense complex symmetric nonHermitian systems arising in computational electromagnetics. Both implicit and explicit preconditioners are considered, and we emphasize sparse approximate inverse methods. We also investigate simple strategies suggested by the underlying problems, showing their eciency on this class of applications. Keywords : Krylov subspace methods, preconditioning techniques, sparse approximate inverse, electromagnetic scattering, boundary element method. 1

In this review paper, we consider some important developments and trends in algorithm design for the solution of linear systems concentrating on aspects that involve the exploitation of parallelism. We briefly discuss the solution of dense linear systems, before studying the solution of sparse equations by direct and iterative methods. We consider preconditioning techniques for iterative solvers and discuss some of the present research issues in this field. Keywords: linear systems, dense matrices, sparse matrices, tridiagonal systems, parallelism, direct methods, iterative methods, Krylov methods, preconditioning. AMS(MOS) subject classifications: 65F05, 65F50. 1 Introduction Solution methods for systems of linear equations Ax = b; (1) where A is a coefficient matrix of order n and x and b are n-vectors, are usually grouped into two distinct classes: direct methods and iterative methods. However, CCLRC - Rutherford Appleton Laboratory, Oxfordshire, England and CERFACS, Toulouse,...

We design a grid based multilevel incomplete LU preconditioner (GILUM) for solving general sparse matrices. This preconditioner combines a high accuracy ILU factorization with an algebraic multilevel recursive reduction. The GILUM preconditioner is a compliment to the domain based multilevel block ILUT preconditioner. A major difference between these two preconditioners is the way that the coarse level nodes are chosen. In this sense the approach of GILUM is analogous to that of algebraic multigrid method. However, the GILUM construction is completely different from the algebraic multigrid construction. A partial ILUT factorization is applied to the reordered matrix and the coarse level system is obtained implicitly. The incomplete factorization process is repeated with the coarse level systems recursively. The GILUM approach avoids some controversial issues in algebraic multigrid method such as how to construct the interlevel transfer operators and how to compute the coarse level oper...

Sparse approximate inverse (SAI) techniques have recently emerged as a new class of parallel preconditioning techniques for solving large sparse linear systems on high performance computers. The choice of the sparsity pattern of the SAI matrix is probably the most important step in constructing an SAI preconditioner. Both dynamic and static sparsity pattern selection approaches have been proposed by researchers.

We introduce a general framework for constructing multilevel recursive incomplete LU preconditioning techniques (RILUM) for solving general sparse matrices. This technique is based on a recursive two by two block incomplete LU factorization on the coefficient matrix. The coarse level system is constructed as an (approximate) Schur complement. A dynamic preconditioner is obtained by solving the Schur complement matrix approximately. The novelty of the proposed techniques is to solve the Schur complement matrix by a preconditioned Krylov subspace method. The very preconditioner for this secondary iteration is constructed by considering the Schur complement matrix as a general sparse matrix and by applying to it the block ILU factorization process that was applied to the original matrix. This recursive procedure continues for a few times and results in a multilevel preconditioner. Different implementation strategies are discussed. We conduct numerical experiments with two particular RILUM...

Abstract. We investigate the use of the multistep successive preconditioning strategies (MSP) to construct a class of parallel multilevel sparse approximate inverse (SAI) preconditioners. We do not use independent set ordering, but a diagonal dominance based matrix permutation to build a multilevel structure. The purpose of introducing multilevel structure into SAI is to enhance the robustness of SAI for solving difficult problems. Forward and backward preconditioning iteration and two Schur complement preconditioning strategies are proposed to improve the performance and to reduce the storage cost of the multilevel preconditioners. One version of the parallel multilevel SAI preconditioner based on the MSP strategy is implemented. Numerical experiments for solving a few sparse matrices on a distributed memory parallel computer are reported. Key words. Sparse matrices, parallel preconditioning, sparse approximate inverse, multilevel preconditioning, multistep successive preconditioning. 1. Introduction. Large

e inner products, vector updates and matrix vector product are easily parallelized and vectorized. The more successful preconditionings, i.e, based upon incomplete LU decomposition, are not easily parallelizable. For that reason one is often satisfied with the use of only diagonal scaling as a preconditioner on highly parallel computers, such as the CM2 [24]. On distributed memory computers we need large grained parallelism in order to reduce synchronization overhead. This can be achieved by combining the work required for a successive number of iteration steps. The idea is to construct first in parallel a straight forward Krylov basis for the search subspace in which an update for the current solution will be determined. Once this basis has been computed, the vectors are orthogonalized, as is done in Krylov subspace methods. The construction as well as the orthogonalization can be done with large grained parallelism, and has su#cient degree of parallelism in it. This approach has be

Abstract We develop new concepts and parallel algorithms of multistep successive preconditioning strategies to enhance efficiency and robustness of standard sparse approximate inverse preconditioning techniques. The key idea is to compute a series of simple sparse matrices to approximate the inverse of the original matrix. Studies are conducted to show the advantages of such an approach in terms of both improving preconditioning accuracy and reducing computational cost. Numerical experiments using one prototype implementation to solve a few general sparse matrices on a distributed memory parallel computer are reported.