Sparse Approximate Inverse Preconditioner Research Articles

A sparse approximate inverse preconditioner for parallel preconditioning of general sparse matrices

A factored sparse approximate inverse is computed and used as a preconditioner for solving general sparse matrices. The algorithm is derived from a matrix decomposition algorithm for inverting dense nonsymmetric matrices. Several strategies and special data structures are proposed to implement the algorithm efficiently. Sparsity patterns of the factored inverse are exploited to reduce computational cost. The preconditioner possesses greater inherent parallelism than traditional preconditioners based on incomplete LU factorizations. Numerical experiments are used to show the effectiveness and efficiency of the proposed sparse approximate inverse preconditioner.

Multigrid treatment and robustness enhancement for factored sparse approximate inverse preconditioning

We investigate the use of sparse approximate inverse techniques (SAI) in a grid based multilevel ILU preconditioner (GILUM) to design a robust and parallelizable preconditioner for solving general sparse matrices. Taking the advantages of grid based multilevel methods, the resulting preconditioner outperforms sparse approximate inverse in robustness and efficiency. Conversely, taking the advantages of sparse approximate inverse, it affords an easy and convenient way to introduce parallelism within multilevel structure. Moreover, an independent set search strategy with automatic diagonal thresholding and a relative threshold dropping strategy are proposed to improve preconditioner performance. Numerical experiments are used to show the effectiveness and efficiency of the proposed preconditioner, and to compare it with some single and multilevel preconditioners.

Parallel Implementation and Practical Use of Sparse Approximate Inverse Preconditioners with a Priori Sparsity Patterns

This paper describes and tests a parallel message-passing code for constructing sparse approximate inverse preconditioners using Frobenius norm minimization. The sparsity patterns of the preconditioners are chosen as patterns of powers of sparsified matrices. Sparsification is necessary when powers of a matrix have a large number of nonzeros, making the approximate inverse computation expensive. For our test problems, the minimum solution time is achieved with approximate inverses with less than twice the number of nonzeros of the original matrix. Additional accuracy is not compensated by the increased cost per iteration. The results lead to further understanding of how to use these methods and how well these methods work in practice. In addition, this paper describes programming techniques required for high performance, including one-sided communication, local coordinate numbering, and load repartitioning.

A robust AINV-type method for constructing sparse approximate inverse preconditioners in factored form

This paper suggests a new method, called AINV-A, for constructing sparse approximate inverse preconditioners for positive-definite matrices, which can be regarded as a modification of the AINV method proposed by Benzi and Túma. Numerical results on SPD test matrices coming from different applications demonstrate the robustness of the AINV-A method and its superiority to the original AINV approach. Copyright © 2001 John Wiley & Sons, Ltd.

Comparisons of the ILU(0), Point-SSOR, and SPAI Preconditioners on the CRAY-T3E for Nonsymmetric Sparse Linear Systems Arising from PDEs on Structured Grids

When solving general sparse linear systems, multicoloring techniques can yield a parallelism of order N, where N is the dimension of the matrix. However, this strategy often suffers from the deterioration of the rate of convergence, as in the preconditioned conjugate gradient method. Another technique that does not suffer from this deterioration of the rate of convergence is the wavefront technique, which exploits parallelism that is available in the original matrix. The rate of convergence remains the same, but the maximum parallelism is limited by the length of the wavefronts, which are often nonuniform. Another popular technique is the sparse approximate inverse (SPAI) technique, capturing the inverse in the sparse form, which is very nice for parallel computation. In this paper, the author compares these two approaches of ILU(0), SPAI, and point-SSOR preconditioners using the BICGSTAB method as the outer iterative method on the CRAY-T3E. It was found that for the problems tested, ILU(0) with multicoloring outperforms the other alternatives.

Sparse pattern selection strategies for robust Frobenius-norm minimization preconditioners in electromagnetism

We consider preconditioning strategies for the iterative solution of dense complex symmetric non-Hermitian systems arising in computational electromagnetics. We consider in particular sparse approximate inverse pre-conditioners that use static non-zero pattern selection. The novelty of our approach comes from using a different non-zero pattern selection procedure for the original matrix from that for the preconditioner and from exploiting geometric or topological information from the underlying meshes instead of using methods based on the magnitude of the entries. The numerical and computational efficiency of the proposed preconditioners are illustrated on a set of model problems arising both from academic and from industrial applications. The results of our numerical experiments suggest that the new strategies are viable approaches for the solution of large-scale electromagnetic problems using preconditioned Krylov methods. In particular, our strategies are applicable when fast multipole techniques are used for the matrix-vector product on parallel distributed memory computers. Copyright © 2000 John Wiley & Sons, Ltd.

Orderings for Factorized Sparse Approximate Inverse Preconditioners

The influence of reorderings on the performance of factorized sparse approximate inverse preconditioners is considered. Some theoretical results on the effect of orderings on the fill-in and decay behavior of the inverse factors of a sparse matrix are presented. It is shown experimentally that certain reorderings, like minimum degree and nested dissection, can be very beneficial. The benefit consists of a reduction in the storage and time required for constructing the preconditioner, and of faster convergence of the preconditioned iteration in many cases of practical interest.

An MPI Implementation of the SPAI Preconditioner on the T3E

The authors describe and test spai_1.1, a parallel MPI implementation of the sparse approximate inverse (SPAI) preconditioner. They show that SPAI can be very effective for solving a set of very large and difficult problems on a Cray T3E. The results clearly show the value of SPAI (and approximate inverse methods in general) as the viable alternative to ILU-type methods when facing very large and difficult problems. The authors strengthen this conclusion by showing that spai_1.1 also has very good scaling behavior.

A comparative study of sparse approximate inverse preconditioners

A number of recently proposed preconditioning techniques based on sparse approximate inverses are considered. A description of the preconditioners is given, and the results of an experimental comparison performed on one processor of a Cray C98 vector computer using sparse matrices from a variety of applications are presented. A comparison with more standard preconditioning techniques, such as incomplete factorizations, is also included. Robustness, convergence rates, and implementation issues are discussed.

Toward an Effective Sparse Approximate Inverse Preconditioner

Sparse approximate inverse preconditioners have attracted much attention recently, because of their potential usefulness in a parallel environment. In this paper, we explore several performance issues related to effective sparse approximate inverse preconditioners (SAIPs) for the matrices derived from PDEs. Our refinements can significantly improve the quality of existing SAIPs and/or reduce the cost of computing them. For the test problems from the Harwell--Boeing collection and some other applications, the performance of our preconditioners can be comparable or superior to incomplete LU (ILU) preconditioners with similar preconditioning cost.

Ordering, Anisotropy, and Factored Sparse Approximate Inverses

We consider ordering techniques to improve the performance of factored sparse approximate inverse preconditioners, concentrating on the AINV technique of M. Benzi and M. T\r{u}ma. Several practical existing unweighted orderings are considered along with a new algorithm, minimum inverse penalty (MIP), that we propose. We show how good orderings such as these can improve the speed of preconditioner computation dramatically and also demonstrate a fast and fairly reliable way of testing how good an ordering is in this respect. Our test results also show that these orderings generally improve convergence of Krylov subspace solvers but may have difficulties particularly for anisotropic problems. We then argue that weighted orderings, which take into account the numerical values in the matrix, will be necessary for such systems. After developing a simple heuristic for dealing with anisotropy we propose several practical algorithms to implement it. While these show promise, we conclude a better heuristic is required for robustness.

Numerical experiments with two approximate inverse preconditioners

We present the results of numerical experiments aimed at comparing two recently proposed sparse approximate inverse preconditioners from the point of view of robustness, cost, and effectiveness. Results for a standard ILU preconditioner are also included. The numerical experiments were carried out on a Cray C98 vector processor.

Wavelet sparse approximate inverse preconditioners

There is an increasing interest in using sparse approximate inverses as preconditioners for Krylov subspace iterative methods. Recent studies of Grote and Huckle and Chow and Saad also show that sparse approximate inverse preconditioner can be effective for a variety of matrices, e.g. Harwell-Boeing collections. Nonetheless a drawback is that it requires rapid decay of the inverse entries so that sparse approximate inverse is possible. However, for the class of matrices that, come from elliptic PDE problems, this assumption may not necessarily hold. Our main idea is to look for a basis, other than the standard one, such that a sparse representation of the inverse is feasible. A crucial observation is that the kind of matrices we are interested in typically have a piecewise smooth inverse. We exploit this fact, by applying wavelet techniques to construct a better sparse approximate inverse in the wavelet basis. We shall justify theoretically and numerically that our approach is effective for matrices with smooth inverse. We emphasize that in this paper we have only presented the idea of wavelet approximate inverses and demonstrated its potential but have not yet developed a highly refined and efficient algorithm.

Sparse approximate inverse preconditioning for dense linear systems arising in computational electromagnetics

We investigate the use of sparse approximate inverse preconditioners for the iterative solution of linear systems with dense complex coefficient matrices arising in industrial electromagnetic problems. An approximate inverse is computed via a Frobenius norm approach with a prescribed nonzero pattern. Some strategies for determining the nonzero pattern of an approximate inverse are described. The results of numerical experiments suggest that sparse approximate inverse preconditioning is a viable approach for the solution of large-scale dense linear systems on parallel computers.

FACTORIZED SPARSE APPROXIMATE INVERSE PRECONDITIONING II: SOLUTION OF 3D FE SYSTEMS ON MASSIVELY PARALLEL COMPUTERS

An iterative method for solving large linear systems with sparse symmetric positive definite matrices on massively parallel computers is suggested. The method is based on the Factorized Sparse Approximate Inverse (FSAI) preconditioning of ‘parallel’ CG iterations. Efficiency of a concurrent implementation of the FSAI-CG iterations is analyzed for a model hypercube, and an estimate of the optimal hypercube dimension is derived. For finite element applications, two strategies for selecting the preconditioner sparsity pattern are suggested. A high convergence rate of the resulting iterations is demonstrated numerically for the 3D equilibrium equations for linear elastic orthotropic materials approximated using both h- and p-versions of the FEM.