Modified incomplete orthogonal factorization methods using Givens rotations

We present four numerical methods for computing the singular value decomposition (SVD) of large sparse matrices on a multiprocessor architecture. We emphasize Lanczos and subspace iteration-based methods for determining several of the largest singular triplets (singular values and corresponding left- and right-singular vectors) for sparse matrices arising from two practical applications: information retrieval and seismic reflection tomography. The target architectures for our implementations are the CRAY-2S/4–128 and Alliant FX/80. The sparse SVD problem is well motivated by recent information-retrieval techniques in which dominant singular values and their corresponding singular vectors of large sparse term-document matrices are desired, and by nonlinear inverse problems from seismic tomography applications which require approximate pseudo-inverses of large sparse Jacobian matrices. This research may help advance the development of future out-of-core sparse SVD methods, which can be used, for example, to handle extremely large sparse matrices 0 × (106) rows or columns associated with extremely large databases in query-based information-retrieval applications.

Supernode pivoting for unsymmetric matrices coupled with supernode partitioning and asynchronous computation can achieve high gigaflop rates for parallel sparse LU factorization on shared memory parallel computers. The progress in weighted graph matching algorithms helps to extend these concepts further and prepermutation of rows is used to place large matrix entries on the diagonal. Supernode pivoting allows dynamical interchanges of columns and rows during the factorization process. The BLAS-3 level efficiency is retained. An enhanced left-right looking scheduling scheme is uneffected and results in good speedup on SMP machines without increasing the operation count. These algorithms have been integrated into the recent unsymmetric version of the PARDISO solver. Experiments demonstrate that a wide set of unsymmetric linear systems can be solved and high performance is consistently achieved for large sparse unsymmetric matrices from real world applications.

A parallel method for large sparse generalized eigenvalue problems using a GridRPC system

This paper presents a novel hardware solution for solving sparse matrices using emulation technology. The proposed solution introduces a bio-inspired technique, which is Genetic Algorithm (GA) to solve the sparse system of linear equations (SLEs). The proposed solution can be used with any physical system that can be modeled as SLEs. Computations involved in Finite Element Method (FEM) consume too much time, which affects the final time-to-market value. Profiling shows that the most time-consuming part in the simulation process is the solver part which is responsible for solving the resultant SLEs generated from the FEM. The total number of equations may reach thousands or millions of linear equations. Hardware implementation of GA and making a good use of parallelism in implementing the architecture accelerate the time taken by the solver part. Solving the system by classical numerical methods, such as conjugate gradient (CG) implemented in software is performed sequentially, here the architecture allows the parallelism in performing GA operators, such as selection, crossover, and mutation. The parallelism in the proposed architecture is the superiority point which we rely on. The results show that our proposed method to solve the SLEs outperforms the classical CG software implementation on run time and number of equations.

For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to efficient and high-quality preconditioning matrices for some typical matrices from the real-world applications.

During the past few years, algorithmic improvements alone have reduced the time required for the direct solution of unsymmetric sparse systems of linear equations by almost an order of magnitude. This paper compares the performance of some well-known software packages for solving general sparse systems. In particular, it demonstrates the consistently high level of performance achieved by WSMP---the most recent of such solvers. It compares the various algorithmic components of these solvers and discusses their impact on solver performance. Our experiments show that the algorithmic choices made in WSMP enable it to run more than twice as fast as the best among similar solvers and that WSMP can factor some of the largest sparse matrices available from real applications in only a few seconds on a 4-CPU workstation. Thus, the combination of advances in hardware and algorithms makes it possible to solve those general sparse linear systems quickly and easily that might have been considered too large until recently.

Computer subroutines for solving large sparse systems of linear equations usually involve fairly complicated data structures and storage management. In most cases, the user of such subroutines simply wants top solve his problem and does not have to understand how the storage management is done, or how the matrix components are stored. Thus software for solving sparse systems of linear equations should be designed in such a way that it can insulate the user from these considerations, while still allowing the software to be used in a variety of ways.

A procedure for symmetric matrix updating subject to a linear equation and retaining any sparsity present in the original matrix is derived. The main feature of this procedure is the reduction of the problem to the solution of an n dimensional sparse system of linear equations. The matrix of this system is shown to be symmetric and positive definite. The method depends on the Frobenius matrix norm. Comments are made on the difficulties of extending the technique so that it uses more general norms, the main points being shown by a numerical example.

This paper describes an efficient approach to sparse matrix factorization on vector supercomputers. The approach is suitable for application domains like circuit simulation that require the repeated direct solution of unsymmetric sparse linear systems of equations with identical zero-nonzero structure. An Overlap-Scatter data structure is used to represent the sparse matrix, enabling the use of multiple operand access modes to achieve higher performance than earlier proposed approaches. The superior performance of the new solver is demonstrated using a number of matrices derived from circuit simulation runs.

Image acquisition of magnetic resonance imaging (MRI) can be accelerated by using multiple receiving coils simultaneously. The problem of reconstructing an unaliased image from partially sampled k-space data can be formulated as a large system of sparse linear equations. The k-space sparse matrix (kSPA) algorithm proposes to solve the system of equations by finding a sparse approximate inverse. This algorithm has been shown to accelerate the image reconstruction for a large number of coils. The original kSPA algorithm requires knowledge of coil sensitivities. Here, we propose and demonstrate an auto-calibrated kSPA algorithm that does not require the explicit computation of the coil sensitivity maps. We have also shown that calibration data, in principle, can be acquired at any region of k-space. This property applies to arbitrary sampling trajectories and all reconstruction algorithms based on k-space. In practice, because of its higher SNR, calibration data acquired at the center of k-space performed more favorably. Such auto-calibration can be advantageous in cases where an accurate sensitivity map is difficult to obtain.

A projection method for solving nonsymmetric linear systems on multiprocessors

The DEFLATED-GMRES(m, k) method is one of the major iterative solvers for the large sparse linear systems of equations, x = b. This algorithm assembles a preconditioner adaptively for the GMRES(m) method based on eigencomponents gathered from the Arnoldi process during iterations. It is usually known that if a restarted GMRES(m) method is used to solve linear systems of equations, the information of the smallest eigencomponents is lost at each restart and the super-linear convergence may also be lost. In this paper, we propose an adaptive procedure that combines the DEFLATED-GMRES(m, k) algorithm and the determination of a restart frequency automatically. It is shown that a new algorithm combining elements of both will reduce the negative effects of the restarted procedure. The numerical experiments are presented on three test problems by using the MIMD parallel machine AP3000. From these numerical results, we show that the proposed algorithm leads to faster convergence than the conventional DEFLATED-GMRES(m, k) method. Copyright © 2000 John Wiley & Sons, Ltd.

The scheme [IEEE Trans. Big Data, 4 (1), 2018, 26-39] is flawed because the sparsity of coefficient matrix is neglected. In the discussed scenario, we find it is unnecessary to outsource such a problem because the client can solve it locally. Even if the matrix is not sparse, the proposed paradigm which requires a great number of interactions, is rarely adopted because of the possibly incurred massive communication cost.

We derive a new upper bound on the minimum rank of matrices belonging to an affine slice of the positive semidefinite cone, when the affine slice is defined according to a system of sparse linear matrix equations. It is shown that a feasible matrix whose rank is no greater than said bound can be computed in polynomial time. The bound depends on both the number of linear matrix equations and their underlying sparsity pattern. For certain problem families, this bound is shown to improve upon well known bounds in the literature. Several examples are provided to illustrate the efficacy of this bound.

In this work, we describe a multilevel preconditioning technique for solving sparse symmetric linear systems of equations. This “Multilevel Schur Low-Rank” (MSLR) preconditioner first builds a tree structure $\\mathcal{T}$ based on a hierarchical decomposition of the matrix and then computes an approximate inverse of the original matrix level by level. Unlike classical direct solvers, the construction of the MSLR preconditioner follows a top-down traversal of $\\mathcal{T}$ and exploits a low-rank property that is satisfied by the difference between the inverses of the local Schur complements and specific blocks of the original matrix. A few steps of the generalized Lanczos tridiagonalization procedure are applied to capture most of this difference. Numerical results are reported to illustrate the efficiency and robustness of the MSLR preconditioner with both two- and three-dimensional discretized PDE problems and with publicly available test problems.