A fast processor for 3‐d device simulation using systolic arrays

A projection method for solving nonsymmetric linear systems on multiprocessors

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.

Many applications in science and engineering give rise to large sparse linear systems of equations that need to be solved as efficiently as possible. As the size of the problems of interest increases, it can become necessary to consider exploiting multiprocessors to solve these systems. We report on the design and development of parallel frontal solvers for the numerical solution of large sparse linear systems. Three codes have been developed for the mathematical software library HSL (www.cse.clrc.ac.uk/Activity/HSL). The first is for unsymmetric finite-element problems; the second is for symmetric positive definite finite-element problems; and the third is for highly unsymmetric linear systems such as those that arise in chemical process engineering. In each case, the problem is subdivided into a small number of loosely connected subproblems and a frontal method is then applied to each of the subproblems in parallel. We discuss how our software is designed to achieve the goals of portability, ease of use, efficiency, and flexibility, and illustrate the performance using problems arising from real applications.

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.

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 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.

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.

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, 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.

Many geo-scientific applications involve boundary value problems arising in simulating electrostatic and electromagnetic fields for geophysical prospecting and subsurface imaging of electrical resistivity. Modeling complex geological media with three-dimensional finite-difference grids gives rise to large sparse linear systems of equations. For such systems, we have implemented three common iterative Krylov solution methods on graphics processing units and compared their performance with parallel host-based versions. The benchmarks show that the device efficiency improves with increasing grid sizes. Limitations are currently given by the device memory resources.

Direct methods for solving large sparse linear systems of equations are popular because of their generality and robustness. Their main weakness is that the memory they require increases rapidly with problem size. We discuss the design and development of a new multifrontal solver that aims to circumvent this problem by allowing both the system matrix and its factors to be stored externally. We highlight some of the key features of our new out-of-core package, in particular its use of efficient implementations of dense linear algebra kernels to perform partial factorizations and its memory management system. We present numerical results for some large-scale problems arising from practical applications.

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.

In this chapter, we describe the problem to be solved: elliptic partial differential equation of secondorder with suitable boundary conditions. In particular, we are interested in diffusion problem in 2-d (two dimensions) with variable and possibly discontinuous coefficients. We also describe several discretization methods, including the finite difference, finite volume, and finite element discretization methods. These discretization methods produce large sparse linear systems of equations, whose solution is the subject of this book.

During the past decade Graphics Processing Units (GPU) have been increasingly employed for speeding up compute intensive scientific applications. In this field, the geometric multigrid method (GMG) is one of the most efficient algorithms for solving large sparse linear systems of equations. Herein we analyze the performance of an optimized GPU based implementation of the GMG method on different state-of-the-art NVIDIA GPUs. The GTX Titan Black card, set-up with increased double precision performance leads to the smallest execution time. It is marginally faster than the more recently released GTX Titan X card which has considerably lower double precision performance. Moreover, an energy efficiency analysis reveals that the GTX 660M and the more powerful Titan cards require a similar amount of energy for running the GMG algorithm: the larger execution time is compensated by the lower power consumption.

The polarizable continuum model (PCM) is a well established method for computing solvation effects. Its attractiveness comes from the fact that the solution is modelled as a continuum instead of modelling each atom individually. This allows for more complex simulations. Nevertheless, there are still a couple of challenges to face. One of them being the computational time required at the limit of large systems, or when high accuracy is needed. The wavelet boundary element method can be used to overcome these problems, provided a reliable cavity generator is used. The PCM calculates the molecular free energy in solution as the sum of electrostatic, dispersion-repulsion and the cavitation energy by solving the underlying partial differential equations. Those differential equations can be transformed into integral equations, solely defined on the boundary of the molecular cavity, by applying the integral equation formalism (IEFPCM). The integral equations can then be discretized using the wavelet boundary element method. The resulting sparse system of linear equations can be solved reliably by iterative solvers, leading to high accuracy solutions even for large systems. The challenge of this approach lies in the generation and refinement of the molecular cavities on which the interactions take place. Since wavelets are defined as a tensor product of one-dimensional functions, the wavelet boundary element method needs a discretization into quadrangular patches of the molecular cavity. This feature is missing in common commercial mesh generation tools, which focus on generating triangular meshes. Making use of characteristic functions for defining which points are inside the cavity, a model for generating the molecular surface independently of its exact description is achieved. We present here a way for generating triangular meshes which are subsequently merged into quadrangular ones. We apply geometrical quantities that measure the fitness of the shapes involved and improve the position of individual points iteratively. Thereby, we generate quadrangular meshes with an unprecedented quality, which ultimately can be refined in a hierarchical manner for the use of wavelets. Overall, this method can generate reliable parametrizations on a variety of smooth cavities. With the help of the wavelet boundary element method, the PCM equations can thus be solved with high accuracy also on large systems.