Iterative solution methods on the Cray YMP/C90

In many scientific computing problems, the overall execution time is dominated by the time to solve very large linear systems. Quite often, the matrices are unsymmetric and ill conditioned, with an irregular sparsity structure reflecting the irregular refinement in the discretization grid. With increasing problem size and problem dimension, direct solvers cannot be used because of the huge memory requirements. The performance of preconditioned iterative solvers is largely dominated by memory-related aspects like size, bandwidth, and indirect addressing speed. This article summarizes the experience of the authors on the relationship between memory aspects and performance in real applications in the domain of very large scale integration (VLSI) device simulation. The authors analyze storage requirements of direct and iterative solvers on a statistical data set, and demonstrate performance variations due to memory-related architectural features on a number of computers ranging from workstations to Cray, NEC, and Fujitsu supercomputers for typical and ill-conditioned linear systems, using different iterative methods and preconditioners. The experiments are done using PILS, a package of iterative linear solvers. PILS implements a large number of iterative methods and preconditioners and allows them to be combined in a flexible way.

SUMMARY Solving the frequency-domain elastic wave equations relies on an efficient linear solver for the large, sparse, indefinite and ill-conditioned linear system derived from the discretization of the elastic wave equation. Direct solvers, which are mostly based on LU decomposition, are efficient for multiple right-hand sides problems, but the memory requirement is huge due to the fill-in effects. On the contrary, iterative solvers fully benefit from the sparsity of the system, but they require problem-specific preconditioners to ensure the convergence because of the ill-conditioning of the system. In this study, we investigate the performance of a robust iterative method named CARP-CG for frequency-domain elastic wave modeling. CARP-CG method turns the original system into a symmetric positive semi-definite system by Kaczmarz row-projections. Such a system can be efficiently solved by the conjugate gradient (CG) method. The row-projections can be seen as a purely algebraic preconditioning technique which is general and is easy to implement. The parallelization is straightforward through a row-block decomposition combined with a component-averaging method. We discretize the 2D frequency-domain elastic wave equation through a 4th order finite difference scheme. Numerical experiments on the Marmousi2 model exhibit a good scalability of CARP-CG. Comparisons between CARP-CG and standard Krylov iterative solvers (GMRES and CGNR) further emphasize the robustness and the fast convergence of CARP-CG method.

SummaryAutomatic result verification is an important tool to guarantee that completely inaccurate results cannot be used for decisions without getting remarked during a numerical computation. Mathematical rigor provided by verified computing allows the computation of an enclosure containing the exact solution of a given problem. Particularly, the computation of linear systems can strongly benefit from this technique in terms of reliability of results. However, in order to compute an enclosure of the exact result of a linear system, more floating‐point operations are necessary, consequently increasing the execution time. In this context, parallelism appears as a good alternative to improve the solver performance. In this paper, we present an approach to solve very large dense linear systems with verified computing on clusters. This approach enabled our parallel solver to compute huge linear systems with point or interval input matrices with dimensions up to 100,000. Numerical experiments show that the new version of our parallel solver introduced in this paper provides good relative speedups and delivers a reliable enclosure of the exact results. Copyright © 2014 John Wiley & Sons, Ltd.

Large sparse linear algebraic systems can be found in a variety of scientific and engineering fields and many scientists strive to solve them in an efficient and robust manner. In this paper, we propose an interpretable neural solver, the Fourier neural solver (FNS), to address them. FNS is based on deep learning and a fast Fourier transform. Because the error between the iterative solution and the ground truth involves a wide range of frequency modes, the FNS combines a stationary iterative method and frequency space correction to eliminate different components of the error. Local Fourier analysis shows that the FNS can pick up on the error components in frequency space that are challenging to eliminate with stationary methods. Numerical experiments on the anisotropic diffusion equation, convection–diffusion equation, and Helmholtz equation show that the FNS is more efficient and more robust than the state-of-the-art neural solver.

Discrete wavelet transform approximation is an established means of approximating dense linear systems arising from discretization of differential and integral equations defined on a one-dimensional domain. For higher dimensional problems, approximation with a sum of Kronecker products has been shown to be effective in reducing storage and computational costs. We have combined these two approaches to enable solution of very large dense linear systems by an iterative technique using a Kronecker product approximation represented in a wavelet basis. Further approximation of the system using only a single Kronecker product provides an effective preconditioner for the system. Here we present our methods and illustrate them with some numerical examples. This technique has the potential for application in a range of areas including computational fluid dynamics, elasticity, lubrication theory and electrostatics.

A purely algebric approach to solving very large general unstructured dense linear systems, in particular, those that arise in 3D boundary integral applications is suggested. We call this technique the matrix-free approach because it allows one to avoid the necessity of storing the whole coefficient matrix in any form, which provides significant memory and arithmetic savings. We propose to approximate a non-singular coefficient matrix A by a block low-rank matrix A and to use the latter when performing matrix–vector multiplications in iterative solution algorithms. Such approximations are shown to be easily computable, and a reliable a posteriori accuracy estimate of ‖A − A‖2 is derived. We prove that block low-rank approximations are sufficiently accurate for some model cases. However, even in the absence of rigorous proof of the existence of accurate approximations, one can apply the algorithm proposed to compute a block low-rank approximation and then make a decision on its practical suitability. We present numerical examples for the 3D CEM and CFD integral applications, which show that, at least for some industrial applications, the matrix-free approach is robust and cost-effective. © 1997 John Wiley & Sons, Ltd.

The simulation of several physical phenomena, arising from a wide class of scientific fields, requires solving large sparse linear systems effectively. In order to exploit appropriately the available supercomputing infrastructures, designing efficient and scalable solvers for large sparse linear systems is necessary. A hybrid parallel algebraic iterative method based on the Schur complement method and multiprojection techniques that utilizes semiaggregated subdomains is proposed. The graph corresponding to the coefficient matrix of the sparse linear system is partitioned by a graph partitioning algorithm and the Schur complement problem is solved by a preconditioned Krylov subspace method. The preconditioning scheme is based on domain decomposition and the resulting subdomains consist of fine and aggregated components from the inner boundaries. Moreover, two variants for approximating the local Schur complements, derived from the semiaggregation procedure, are proposed in order to improve memory requirements and preprocessing time. The proposed method is suitable for distributed memory systems with multicore nodes and has improved convergence behavior for a large number of workstations. Numerical and comparative results concerning the applicability, the convergence behavior, and the parallel performance of the proposed scheme are given.

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.

The solution of large sparse unsymmetric linear systems is a critical and challenging component of semiconductor device and circuit simulations. The time for a simulation is often dominated by this part. The sparse solver is expected to balance different, and often conflicting requirements. Reliability, a low memory-footprint, and a short solution time are a few of these demands. Currently, no black-box solver exists that can satisfy all criteria. The linear systems from both simulations can be highly ill-conditioned and are, therefore, quite challenging for direct and iterative methods. In this paper, it is shown that algorithms to place large entries on the diagonal using unsymmetric permutations and scalings greatly enhance the reliability of both direct and preconditioned iterative solvers for unsymmetric linear systems arising in semiconductor device and circuit simulations. The numerical experiments indicate that the overall solution strategy is both reliable and cost effective.

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.

The goal of this paper is to develop a parallel algorithm for the direct solution of large sparse linear systems and integrate it into domain decomposition methods. The computational effort for these linear systems, often encountered in numerical simulation of structural mechanics problems by finite element codes, is very significant in terms of run-time and memory requirements.In this paper, a two-level parallelism is exploited. The exploitation of the lower level of parallelism is based on the development of a parallel direct solver with a nested dissection algorithm and to introduce it into the FETI methods. This direct solver has the advantage of handling zero-energy modes in floating structures automatically and properly. The upper level of parallelism is a coarse-grain parallelism between substructures of FETI. Some numerical tests are carried out to evaluate the performance of the direct solver.

A recursive algorithm for generating the transition matrices of multistation multiserver exponential reliable queueing networks

Krylov subspace methods are popular iterative methods to solve large sparse linear systems in the real-world computations due to their cheap memory requirement and computational cost. In this paper, we discuss the solution of singular systems. We will show that the consistency of a singular linear system is not a sufficient condition for a Krylov subspace method to successfully find a solution to the system. The choice of initial guess is a crucial step. If the initial guess is properly chosen, a Krylov method almost surely converges to find a solution from the point of view of probability, otherwise a Krylov subspace method surely diverges. Moreover, our algorithm applied to parallel calculation is discussed in the paper.

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.

Applications involving large sparse nonsymmetric linear systems encourage parallel implementations of robust iterative solution methods, such as GMRES(k). Two parallel versions of GMRES(k) based on different data distributions and using Householder reflections in the orthogonalization phase are analyzed with respect to scalability (their ability to maintain fixed efficiency with an increase in problem size and number of processors). A theoretical algorithm-machine model for scalability of GMRES(k) with fixed k is derived and validated by experiments on three parallel computers, each with different machine characteristics. The analysis for an adaptive version of GMRES(k), in which the restart value k is adapted to the problem, is also presented and scalability results for this case are briefly discussed.