Solution Of Large-scale Linear Systems Research Articles

The dominant degree for Schur complement of [formula omitted]-strictly diagonally dominant matrix and its applications

We study the dominant degree for Schur complement of S-strictly diagonally dominant matrix and its applications in reducing the order for the solution of large-scale linear systems and the linear complementarity problems. Based on the proposed dominant degree, we obtain an infinite norm bound for the inverse matrix of the Schur complement, and then derive an infinite norm bound for the original matrix. We also give the eigenvalue inclusion set for the Schur complement. In addition, we present a new error bound for the linear complementarity problem with an SB-matrix, which improves the existing results when the dominant degree is tiny. A series of numerical experiments with lots of random matrices are presented to show the efficiency and superiority of our results.

Distributed Solution of Large-Scale Linear Systems via Accelerated Projection-Based Consensus

Solving a large-scale system of linear equations is a key step at the heart of many algorithms in scientific computing, machine learning, and beyond. When the problem dimension is large, computational and/or memory constraints make it desirable, or even necessary, to perform the task in a distributed fashion. In this paper, we consider a common scenario in which a taskmaster intends to solve a large-scale system of linear equations by distributing subsets of the equations among a number of computing machines/cores. We propose a new algorithm called Accelerated Projection-based Consensus , in which at each iteration every machine updates its solution by adding a scaled version of the projection of an error signal onto the nullspace of its system of equations, and the taskmaster conducts an averaging over the solutions with momentum. The convergence behavior of the proposed algorithm is analyzed in detail and analytically shown to compare favorably with the convergence rate of alternative distributed methods, namely distributed gradient descent, distributed versions of Nesterov's accelerated gradient descent and heavy-ball method, the block Cimmino method, and Alternating Direction Method of Multipliers. On randomly chosen linear systems, as well as on real-world data sets, the proposed method offers significant speed-up relative to all the aforementioned methods. Finally, our analysis suggests a novel variation of the distributed heavy-ball method, which employs a particular distributed preconditioning and achieves the same theoretical convergence rate as that in the proposed consensus-based method.

Multi Space Reduced Basis Preconditioners for Large-Scale Parametrized PDEs

In this work we introduce a new two-level preconditioner for the efficient solution of large-scale linear systems arising from the discretization of parametrized PDEs. The proposed preconditioner combines in a multiplicative way a reduced basis solver, which plays the role of coarse component, and a “traditional” fine-grid preconditioner, such as one-level additive Schwarz, block Gauss--Seidel, or block Jacobi preconditioners. The coarse component is built upon a new multi space reduced basis (MSRB) method that we introduce for the first time in this paper, where a reduced basis space is built through the proper orthogonal decomposition algorithm at each step of the iterative method at hand, like the flexible GMRES method. MSRB strategy consists in building reduced basis spaces that are well suited to perform a single iteration, by addressing the error components which have not been treated yet. The Krylov iterations employed to solve the resulting preconditioned system target small tolerances with a very...

Parallel accelerated cyclic reduction preconditioner for three-dimensional elliptic PDEs with variable coefficients

We present a robust and scalable preconditioner for the solution of large-scale linear systems that arise from the discretization of elliptic PDEs amenable to rank compression. The preconditioner is based on hierarchical low-rank approximations and the cyclic reduction method. The setup and application phases of the preconditioner achieve log-linear complexity in memory footprint and number of operations, and numerical experiments exhibit good weak and strong scalability at large processor counts in a distributed memory environment. Numerical experiments with linear systems that feature symmetry and nonsymmetry, definiteness and indefiniteness, constant and variable coefficients demonstrate the preconditioner applicability and robustness. Furthermore, it is possible to control the number of iterations via the accuracy threshold of the hierarchical matrix approximations and their arithmetic operations, and the tuning of the admissibility condition parameter. Together, these parameters allow for optimization of the memory requirements and performance of the preconditioner.

Inverse subspace problems with applications

SUMMARY Given a square matrix A, the inverse subspace problem is concerned with determining a closest matrix to A with a prescribed invariant subspace. When A is Hermitian, the closest matrix may be required to be Hermitian. We measure distance in the Frobenius norm and discuss applications to Krylov subspace methods for the solution of large-scale linear systems of equations and eigenvalue problems as well as to the construction of blurring matrices. Extensions that allow the matrix A to be rectangular and applications to Lanczos bidiagonalization, as well as to the recently proposed subspace-restricted SVD method for the solution of linear discrete ill-posed problems, also are considered.Copyright © 2013 John Wiley & Sons, Ltd.

On the Easy Use of Scientific Computing Services for Large Scale Linear Algebra and Parallel Decision Making with the P-Grade Portal

Scientific research is becoming increasingly dependent on the large-scale analysis of data using distributed computing infrastructures (Grid, cloud, GPU, etc.). Scientific computing (Petitet et al. 1999) aims at constructing mathematical models and numerical solution techniques for solving problems arising in science and engineering. In this paper, we describe the services of an integrated portal based on the P-Grade (Parallel Grid Run-time and Application Development Environment) portal ( http://www.p-grade.hu ) that enables the solution of large-scale linear systems of equations using direct solvers, makes easier the use of parallel block iterative algorithm and provides an interface for parallel decision making algorithms. The ultimate goal is to develop a single sign on integrated multi-service environment providing an easy access to different kind of mathematical calculations and algorithms to be performed on hybrid distributed computing infrastructures combining the benefits of large clusters, Grid or cloud, when needed.

The BPS preconditioner on Beowulf cluster

This work presents the implementation on a Linux Cluster of a parallel preconditioner for the solution of the linear system resulting from the finite element discretization of a 2D second order elliptic boundary value problem. The numerical method, proposed by Bramble, Pasciak and Schatz, is developed using Domain Decomposition techniques, which are based on the splitting of the computational domain into subregionsof smaller size, enforcing suitable compatibility conditions. The Fortran code is implemented using PETSc: a suite of data structures and routines devoted to the scientific parallel computing and based on the MPI standard for all message-passing communications. The main interest of the paper is to present an efficient and portable code for the solution of large-scale linear systems and to investigate how the architecturalaspects of the cluster influence the performance of the considered algorithm. We provide an analysis of the execution times as well as of the scalability, using as test case the classical Poisson equation with Dirichlet boundary conditions.

Simultaneous solution of large-scale linear systems and eigenvalue problems with a parallel GMRES method

A method for simultaneous solution of large and sparse linearized equation sets and the corresponding eigenvalue problems is presented. Such problems arise from the discretization and the solution of nonlinear problems with the finite element method and Newton iteration. The method is based on a parallel version of the preconditioned GMRES( m ) by deflation. The parallel code exploits the architecture of the computational clusters using the MPI (Message Passing Interface). The convergence rate, the parallel speedup and the memory requirements of the proposed method are reported and evaluated.

Greedy Coarsening Strategies for Nonsymmetric Problems

The solution of large-scale linear systems in computational science and engineering requires efficient solvers and preconditioners. Often, the most effective such techniques are those based on multilevel splittings of the problem. In this paper, we consider the problem of partitioning both symmetric and nonsymmetric matrices based solely on algebraic criteria. A new algorithm is proposed that combines attractive features of two previous techniques proposed by the authors. It offers rigorous guarantees of certain properties of the partitioning, yet is naturally compatible with the threshold based dropping known to be effective for incomplete factorizations. Numerical results show that the new partitioning scheme leads to improved results for a variety of problems. The effects of further matrix reordering within the fine-scale block are also considered.

Cokriging—a computer program

Cokriging is a process wherein several variables can be jointly estimated on the basis of intervariable and spatial structure information. Presented herein is the program COKRIG, for punctual cokriging, a program in a simple form to demonstrate the utility of cokriging. Equation solution follows a modification of a method developed for the solution of large-scale linear systems. Several example problems show that, at least for earthquake data, the inclusion of intervariable information results in a more accurate BLUE (best linear unbiased estimator).

Iterative Solution of Large-Scale Linear Systems

Iterative Solution of Large-Scale Linear Systems