Large Sparse Symmetric Matrices Research Articles

Computing Interior Eigenvalues of Large Sparse Symmetric Matrices

In this paper we present new restarted Krylov methods for calculating interior eigenvalues of large sparse symmetric matrices. The proposed methods are compact versions of the Heart iteration which are modified to retain the monotonicity property. Numerical experiments illustrate the usefulness of the proposed approach.

A Robust and Efficient Implementation of LOBPCG

Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is widely used to compute eigenvalues of large sparse symmetric matrices. The algorithm can suffer from numerical instability if it is not implemented with care. This is especially problematic when the number of eigenpairs to be computed is relatively large. In this paper we propose an improved basis selection strategy based on earlier work by Hetmaniuk and Lehoucq as well as a robust convergence criterion which is backward stable to enhance the robustness. We also suggest several algorithmic optimizations that improve performance of practical LOBPCG implementations. Numerical examples confirm that our approach consistently and significantly outperforms previous competing approaches in both stability and speed.

High-performance implementation of Chebyshev filter diagonalization for interior eigenvalue computations

We study Chebyshev filter diagonalization as a tool for the computation of many interior eigenvalues of very large sparse symmetric matrices. In this technique the subspace projection onto the target space of wanted eigenvectors is approximated with filter polynomials obtained from Chebyshev expansions of window functions. After the discussion of the conceptual foundations of Chebyshev filter diagonalization we analyze the impact of the choice of the damping kernel, search space size, and filter polynomial degree on the computational accuracy and effort, before we describe the necessary steps towards a parallel high-performance implementation. Because Chebyshev filter diagonalization avoids the need for matrix inversion it can deal with matrices and problem sizes that are presently not accessible with rational function methods based on direct or iterative linear solvers. To demonstrate the potential of Chebyshev filter diagonalization for large-scale problems of this kind we include as an example the computation of the 102 innermost eigenpairs of a topological insulator matrix with dimension 109 derived from quantum physics applications.

Performance analysis of distributed symmetric sparse matrix vector multiplication algorithm for multi‐core architectures

SummarySparse matrix vector multiply (SpMVM) is an important kernel that frequently arises in high performance computing applications. Due to its low arithmetic intensity, several approaches have been proposed in literature to improve its scalability and efficiency in large scale computations. In this paper, our target systems are high end multi‐core architectures and we use messaging passing interface + open multiprocessing hybrid programming model for parallelism. We analyze the performance of recently proposed implementation of the distributed symmetric SpMVM, originally developed for large sparse symmetric matrices arising in ab initio nuclear structure calculations. We study important features of this implementation and compare with previously reported implementations that do not exploit underlying symmetry. Our SpMVM implementations leverage the hybrid paradigm to efficiently overlap expensive communications with computations. Our main comparison criterion is the ‘CPU core hours’ metric, which is the main measure of resource usage on supercomputers. We analyze the effects of topology‐aware mapping heuristic using simplified network load model. We have tested the different SpMVM implementations on two large clusters with 3D Torus and Dragonfly topology. Our results show that the distributed SpMVM implementation that exploits matrix symmetry and hides communication yields the best value for the ‘CPU core hours’ metric and significantly reduces data movement overheads. Copyright © 2015 John Wiley & Sons, Ltd.

Trace-Penalty Minimization for Large-Scale Eigenspace Computation

In a block algorithm for computing relatively high-dimensional eigenspaces of large sparse symmetric matrices, the Rayleigh-Ritz (RR) procedure often constitutes a major bottleneck. Although dense eigenvalue calculations for subproblems in RR steps can be parallelized to a certain level, their parallel scalability, which is limited by some inherent sequential steps, is lower than dense matrix-matrix multiplications. The primary motivation of this paper is to develop a methodology that reduces the use of the RR procedure in exchange for matrix-matrix multiplications. We propose an unconstrained trace-penalty minimization model and establish its equivalence to the eigenvalue problem. With a suitably chosen penalty parameter, this model possesses far fewer undesirable full-rank stationary points than the classic trace minimization model. More importantly, it enables us to deploy algorithms that makes heavy use of dense matrix-matrix multiplications. Although the proposed algorithm does not necessarily reduce the total number of arithmetic operations, it leverages highly optimized operations on modern high performance computers to achieve parallel scalability. Numerical results based on a preliminary implementation, parallelized using OpenMP, show that our approach is promising.

Evolving an Improved Algorithm for Envelope Reduction Using a Hyper-Heuristic Approach

Large sparse symmetric matrices are typical characteristics of the linear systems found in various scientific and engineering disciplines, such as fluid mechanics, structural engineering, finite element analysis, and network analysis. In all such systems, the performance of solvers crucially depends on the sum of the distance of each matrix element from the matrix's main diagonal. This quantity is known as the envelope. In order to reduce the envelope of a matrix, its rows and columns should be reordered properly. The problem of minimizing the envelope size is exceptionally hard and known to be NP-complete. A considerable number of methods have been developed for reducing the envelope among which the Sloan algorithm offered a substantial improvement over previous algorithms. In this paper, a hyper-heuristic approach based on genetic programming, which we term genetic hyper-heuristic, is introduced for evolving an enhanced version of the Sloan algorithm. We also present a local search algorithm and integrate it with the new algorithm produced by our hyper-heuristic to further improve the quality of the solutions. The new algorithms are evaluated on a large set of standard benchmarks against six state-of-the-art algorithms from the literature. Our algorithms outperform all the methods under testing by a wide margin.

Parallel Rayleigh Quotient Optimization with FSAI‐Based Preconditioning

The present paper describes a parallel preconditioned algorithm for the solution of partial eigenvalue problems for large sparse symmetric matrices, on parallel computers. Namely, we consider the Deflation‐Accelerated Conjugate Gradient (DACG) algorithm accelerated by factorized‐sparse‐approximate‐inverse‐ (FSAI‐) type preconditioners. We present an enhanced parallel implementation of the FSAI preconditioner and make use of the recently developed Block FSAI‐IC preconditioner, which combines the FSAI and the Block Jacobi‐IC preconditioners. Results onto matrices of large size arising from finite element discretization of geomechanical models reveal that DACG accelerated by these type of preconditioners is competitive with respect to the available public parallel hypre package, especially in the computation of a few of the leftmost eigenpairs. The parallel DACG code accelerated by FSAI is written in MPI‐Fortran 90 language and exhibits good scalability up to one thousand processors.

On Stabilization and Convergence of Clustered Ritz Values in the Lanczos Method

The Lanczos algorithm can be considered as an iterative method for finding a few eigenvalues and eigenvectors of large sparse symmetric matrices. In the present paper we solve a conjecture posed by Zdenek Strakos and Anne Greenbaum in [Open Questions in the Convergence Analysis of the Lanczos Process for the Real Symmetric Eigenvalue Problem, IMA Research Report, 1992] on the clustering of Ritz values, which occurs in finite precision computations. In particular, we prove that the conjecture is valid in most cases and describe the rare case when it is not. The established upper bounds measuring the quality of Ritz approximations imply also that Ritz values cluster only close to an eigenvalue.

Numerical comparison of iterative eigensolvers for large sparse symmetric positive definite matrices

The Jacobi–Davidson (JD) method has been recently proposed for the evaluation of the partial eigenspectrum of large sparse matrices. In this work we report a set of numerical experiments that compare this method with other previously proposed techniques; deflation accelerated conjugate gradient (DACG) and Lanczos (ARPACK), on large sparse symmetric matrices. The results obtained by JD and DACG are benchmarked against those obtained with ARPACK in terms of computational time for the evaluation of s ( s⩽100) leftmost eigenpairs. The comparison is performed on a number of matrices, with dimensions up to a few hundred thousands, arising from the discretization of the diffusion equation by means of finite element, mixed finite element, and finite difference techniques. The results show that DACG and JD are equally more efficient than ARPACK for a small number of eigenpairs, with DACG more performing for small s and particularly for s=1, while ARPACK tends to become competitive when 40 or more eigenpairs are sought. For the largest and worst conditioned matrix DACG proves the only viable alternative.

Design of Generic Direct Sparse Linear System Solver in C++ for Power System Analysis

This article presents the design of the generic linear system solver (LSS) for a class of large sparse symmetric matrices over real and complex numbers. These matrices correspond to one of the following: (1) symmetric positive definite (SPD) matrices, (2) complex Hermitian matrices, or (3) complex matrices with SPD real and imaginary matrices. Such matrices arise in various power system analysis applications like load flow analysis and short circuit analysis. A template facility of C++ is used to write a generic program on float, double, and complex data types. The design of the algorithm guarantees numerical stability and efficient sparsity implementation. A reusable class SET is defined to cater to graph theoretic computations. LSS problems with matrices up to 20,000 nodes have been tested. Another feature of the proposed LSS is the implementation of associative array, which allows subscripting an array with character strings, such as bus names. This helps in making the four power system analysis software user friendly. The proposed LSS reflects an important development towards a truly object-oriented four power system analysis software.

Ordering symmetric sparse matrices for small profile and wavefront

The ordering of large sparse symmetric matrices for small profile and wavefront or for small bandwidth is important for the efficiency of frontal and variable-band solvers. In this paper, we look at the computation of pseudoperipheral nodes and compare the effectiveness of using an algorithm based on level-set structures with using the spectral method as the basis of the Reverse Cuthill–McKee algorithm for bandwidth reduction. We also consider a number of ways of improving the performance and efficiency of Sloan's algorithm for profile and wavefront reduction, including the use of different weights, the use of supervariables, and implementing the priority queue as a binary heap. We also examine the use of the spectral ordering in combination with Sloan's algorithm. The design of software to implement the reverse Cuthill–McKee algorithm and a modified Sloan's algorithm is discussed. Extensive numerical experiments that justify our choice of algorithm are reported on. Copyright © 1999 John Wiley & Sons, Ltd.

Implementation of a variable block Davidson method with deflation for solving large sparse eigenproblems

The Davidson method is a preconditioned eigenvalue technique aimed at computing a few of the extreme (i.e., leftmost or rightmost) eigenpairs of large sparse symmetric matrices. This paper describes a software package which implements a deflated and variable-block version of the Davidson method. Information on how to use the software is provided. Guidelines for its upgrading or for its incorporation into existing packages are also included. Various experiments are performed on an SGI Power Challenge and comparisons with ARPACK are reported.

Nested Iterations for Symmetric Eigenproblems

The evaluation of the leftmost eigenspectrum of large sparse symmetric matrices is of great interest in many physical and engineering applications. Recently an efficient iterative method called DACG (deflation-accelerated conjugate gradient) based on the conjugate gradient optimization of successive deflated Rayleigh quotients, was developed. In the present paper the authors study the possibility of increasing the efficiency of this scheme by means of a nested iteration (NI) technique that acts on nested finite element grids and calculates an improved initial guess to be used by the DACG procedure. The new NI-DACG method is tested on two sample problems of large size, which make use of nested regular and irregular finite element grids. The results obtained in the calculation of the 40 leftmost eigenpairs for both problems show that the computational efficiency of DACG is increased by a factor of five for the regular mesh and two for the irregular one. They also show the limitation of simple (low-order) interpolation schemes for the intergrid transfer in improving the initial eigenvector estimates, and call for more research in this area.

Generalizations of Davidson’s Method for Computing Eigenvalues of Sparse Symmetric Matrices

This paper analyzes Davidson’s method for computing a few eigenpairs of large sparse symmetric matrices. An explanation is given for why Davidson’s method often performs well but occasionally performs very badly. Davidson’s method is then generalized to a method which offers a powerful way of applying preconditioning techniques developed for solving systems of linear equations to solving eigenvalue problems.

The block conjugate gradient algorithm and related methods

The development of the Lanczos algorithm for finding eigenvalues of large sparse symmetric matrices was followed by that of block forms of the algorithm. In this paper, similar extensions are carried out for a relative of the Lanczos method, the conjugate gradient algorithm. The resulting block algorithms are useful for simultaneously solving multiple linear systems or for solving a single linear system in which the matrix has several separated eigenvalues or is not easily accessed on a computer. We develop a block biconjugate gradient algorithm for general matrices, and develop block conjugate gradient, minimum residual, and minimum error algorithms for symmetric semidefinite matrices. Bounds on the rate of convergence of the block conjugate gradient algorithm are presented, and issues related to computational implementation are discussed. Variants of the block conjugate gradient algorithm applicable to symmetric indefinite matrices are also developed.

How to make the Lanczos algorithm converge slowly

The Paige style Lanczos algorithm is an iterative method for finding a few eigenvalues of large sparse symmetric matrices. Some beautiful relationships among the elements of the eigenvectors of a symmetric tridiagonal matrix are used to derive a perverse starting vector which delays convergence as long as possible. Why such slow convergence is never seen in practice is also examined.

Iterative eigenvalue algorithms based on convergent splittings

Application of direct iterations, based on convergent splittings, to the eigenvalue problem of large sparse symmetric matrices is discussed. A general convergence proof is given, and it is shown how parameters should be chosen to give the best rate of convergence. As special examples are considered, SOR iteration and iteration based on the use of a fast direct Poisson solver. Numerical tests are reported.