Solving sparse rational linear systems

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.

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.

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.

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.

We propose two new conjugate-gradient-type methods for the solution of sparse unsymmetric linear systems. We present a new tridiagonalization process for unsymmetric matrices that is closely related to the Lanczos process. We use orthogonal factorizations of the tridiagonal matrix to derive the new algorithms USYMLQ and USYMQR in the same fashion as SYMMLQ and MINRES [C. C. Paige and M. A. Saunders, “Solution of sparse indefinite systems of linear equations,” SIAM J. Numer. Anal., 12 (1975), pp. 617–629], for symmetric matrices. Some numerical results for the new methods and comparisons with other methods are presented.

We discuss design features of a code which solves sparse unsymmetric systems of linear equations using a frontal method. We consider in particular the user interface, the internal data structures, the pivoting strategy, and the isolation of machine dependencies. We illustrate the performance of our code on a variety of test problems on both the IBM 3033 and the CRAY-1.

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.

The minimum degree algorithm is one of the most widely-used heuristics for reducing the cost of solving large sparse systems of linear equations. It has been studied for nearly half a century and has a rich history of bridging techniques from data structures, graph algorithms, and scientific computing. In this paper, we present a simple but novel combinatorial algorithm for computing an exact minimum degree elimination ordering in $O(nm)$ time, which improves on the best known time complexity of $O(n^3)$ and offers practical improvements for sparse systems with small values of $m$. Our approach leverages a careful amortized analysis, which also allows us to derive output-sensitive bounds for the running time of $O(\min\{m\sqrt{m^+}, \Delta m^+\} \log n)$, where $m^+$ is the number of unique fill edges and original edges that the algorithm encounters and $\Delta$ is the maximum degree of the input graph. Furthermore, we show there cannot exist an exact minimum degree algorithm that runs in $O(nm^{1-\varepsilon})$ time, for any $\varepsilon > 0$, assuming the strong exponential time hypothesis. This fine-grained reduction goes through the orthogonal vectors problem and uses a new low-degree graph construction called $U$-fillers, which act as pathological inputs and cause any minimum degree algorithm to exhibit nearly worst-case performance. With these two results, we nearly characterize the time complexity of computing an exact minimum degree ordering.

Many of the fast methods for factoring integers and computing discrete logarithms require the solution of large sparse linear systems of equations over finite fields. This paper presents the results of implementations of several linear algebra algorithms. It shows that very large sparse systems can be solved efficiently by using combinations of structured Gaussian elimination and the conjugate gradient, Lanczos, and Wiedemann methods.

In this paper, we consider the problem of solving a sparse nonsingular system of linear equations. We show that the structures of the triangular matrices obtained in the $LU$-decomposition of a sparse nonsingular matrix A using Gaussian elimination with partial pivoting are contained in those of the Cholesky factors of $A^T A$, provided that the diagonal elements of A are nonzero. Based on this result, a method for solving sparse linear systems is then described. The main advantage of this method is that the numerical computation can be carried out using a static data structure. Numerical experiments comparing this method with other implementations of Gaussian elimination for solving sparse linear systems are presented and the results indicate that the method proposed in this paper is quite competitive with other approaches.

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.

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.

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.

Introduction. Kron [1] has developed a technique for tearing large, sparse linear systems of algebraic equations into smaller systems, then putting the solutions of these smaller systems together to form the solution of the original problem. This can result in a great savings over solving the whole system in one piece. The trick is to find where to tear so that on the one hand the remaining pieces are small and easily solvable, but on the other hand not so much violence is done to the equations so that the task of putting their solutions back together becomes too difncult. Kron's pro? cedure for solving the pieces and putting their solutions together is based upon representing systems of equations as electric networks. He makes his choice of where to tear from his insight into these networks and the physical processes the equations represent. In the hands of someone such as Kron who has an understanding of the methods and good insight into the prob? lem, this technique has had outstanding success. The widespread use of Kron's technique has perhaps been limited severely for lack of practitioners skilled in this art of determining effective ways of tearing specific problems. Many problems are too large to submit to such insight. It is the purpose of this paper to contribute to the development of procedures which can be performed on a computer for analyzing the structures of the systems of equations themselves as an aid in choosing how to break them up for easier solution. The methods developed here require only that we consider what variables occur in each equation, but not how they appear. Thus, the tech? niques are not limited to linear systems.

A projection method for solving nonsymmetric linear systems on multiprocessors