A Robust Iterative Scheme for Symmetric Indefinite Systems

To analyze the behavior of semiconductor devices numerically, we need to solve extremely large and sparse linear systems of equations as fast as possible. In this paper, parallel processing architectures using systolic arrays are proposed as a rapid solver of linear equations. The architectures are designed to reduce the number of processing elements, utilizing the regularity and the sparseness of coefficient matrices.The proposed systolic arrays consist of sixty‐four cells for incomplete matrix decomposition and of seven cells for matrix‐vector multiplication. This number is independent from the number N of grid points discretized for a three‐dimensional device. The processing time of the array to solve a system of equations is proportional to N. Furthermore, the architectures are revised for faster parallel computation. The acceleration is based on the fact that a coefficient matrix is decomposed of smaller matrices. By the acceleration for matrix‐vector multiplication, required cells are 8 × N1/3, but the processing time is proportional to N2/3.

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.

In this paper, an FPGA-based single precision floating point hybrid iterative architecture for solving a linear system of equations is proposed. The novel architecture implements Gauss-Seidel method using a Jacobi method based building block. The design takes advantage of the fast convergence of Gauss-Seidel Method conflated with parallel, pipelined architecture of Jacobi Iterative solver resulting in a much efficient architecture with acceptable hardware augmentation. The whole design has been implemented in Verilog HDL, having Virtex 7 XCV2000T as targeted device. Other design optimizations include using modified high-speed radix 4 multiplier and optimized high-speed 2's complementer. The efficacy of the design is tested and implemented in solving nonsingular, exactly determined and strictly diagonally dominant coefficient matrix based dense and sparse linear system of equations with different number of variables. The design results in reduced number of iterations and equivalent speedups are presented, which are calculated factoring in, the increased delay effects.

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.

This paper proposes an image fusion algorithm based on a sparse linear equation system, which uses local extreme of high resolution image and intensity of multi-spectral image to construct the system. Based on this sparse system, the multi-scale image decomposition algorithm can be implemented. This algorithm extracts the details of the high-resolution image and integrates with the multi-spectral information to derive the fused image.

Element-by-element iterative solution methods have been shown to be effective for solving large, sparse linear systems of equations with a positive definite stiffness matrix. In this paper the iterative method GMRES will be described. This solution method is also suited for solving nonsymmetric systems of equations. As acceleration technique an element-by-element preconditioner has been employed. Attention is focused mainly on the convergence behaviour of the method. The calculations described in this paper have been performed with the finite element package DIANA.

In recent years a number of solvers for the direct solution of large sparse symmetric linear systems of equations have been developed. These include solvers that are designed for the solution of positive definite systems as well as those that are principally intended for solving indefinite problems. In this study, we use performance profiles as a tool for evaluating and comparing the performance of serial sparse direct solvers on an extensive set of symmetric test problems taken from a range of practical applications.

A projection method for solving nonsymmetric linear systems on multiprocessors

In this paper, a class of parallel blockwise matrix multisplitting block relaxation methods, including the blockwise matrix multisplitting block symmetric accelerated overrelaxation method, the blockwise matrix multisplitting block unsymmetric and symmetric successive overrelaxation methods and the blockwise matrix multisplitting block unsymmetric and symmetric Gauss-Seidel methods, etc., is established for the large sparse block system of linear equations, and its convergence theory is set up thorouthly when the coefficient matrix is a block H-matrix. Also, the new methods are further extended by relaxing different block elements of the iterations with different relaxation parameters and, therefore, general frameworks of parallel blockwise matrix multisplitting block relaxation methods for solving the block system of linear equations are naturally obtained.

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.

It is known that the main computational costs in the FEM implementation are computing the global system of linear equations (material response computation and stiffness matrix ensemble) and its solution. With the aim of speeding up the total solving boundary value problem time with the help of the developed solver, algorithms of the parallel stiffness matrix ensemble and the system of linear equations solution were implemented. In this paper the results of using GPU to accelerate computations related to the solution of direct boundary value problems of crystal plasticity are present. The existing algorithms for solving the sparse systems of linear equations and parallel ensemble of the global stiffness matrix were implemented on both the CPU and GPU. As a result, a total acceleration of up to 10 times was obtained compared to the software implementation on the CPU.

We propose a new algorithm to find a rational solution to a sparse system of linear equations over the integers. This algorithm is based on a p-adic lifting technique combined with the use of block matrices with structured blocks. It achieves a sub-cubic complexity in terms of machine operations subject to a conjecture on the effectiveness of certain sparse projections. A LinBox-based implementation of this algorithm is demonstrated, and emphasizes the practical benefits of this new method over the previous state of the art.

We present a unified anisotropic geometric diffusion PDE model for smoothing (fairing) out noise both in triangulated two-manifold surface meshes inIR3and functions defined on these surface meshes, while enhancing curve features on both by careful choice of an anisotropic diffusion tensor. We combine theC1limit representation of Loop's subdivision for triangular surface meshes and vector functions on the surface mesh with the established diffusion model to arrive at a discretized version of the diffusion problem in the spatial direction. The time direction discretization then leads to a sparse linear system of equations. Iteratively solving the sparse linear system yields a sequence of faired (smoothed) meshes as well as faired functions.