Structural properties and decomposition of linear balanced matrices

Claude Berge defines a (0, 1) matrix A to be linear ifA does not contain a 2 × 2 submatrix of all ones.A(0, 1) matrixA is balanced ifA does not contain a square submatrix of odd order with two ones per row and column.The contraction of a rowi of a matrix consists of the removal of rowi and all the columns that have a 1 in the entry corresponding to rowi. In this paper we show that if a linear balanced matrixA does not belong to a subclass of totally unimodular matrices, thenA orA ^T contains a rowi such that the submatrix obtained by contracting rowi has a block-diagonal structure.Partial support from NSF grant DMS 8606188, ECS 8800281 and DDM 8800281.     

Convergence of Non-stationary Parallel Multisplitting Methods for Hermitian Positive Definite Matrices

Non-stationary multisplitting algorithms for the solution of linear systems are studied. Convergence of these algorithms is analyzed when the coefficient matrix of the linear system is hermitian positive definite. Asynchronous versions of these algorithms are considered and their convergence investigated.

     

The efficient evaluation of the hypergeometric function of a matrix argument

We present new algorithms that efficiently approximate the hypergeometric function of a matrix argument through its expansion as a series of Jack functions. Our algorithms exploit the combinatorial properties of the Jack function, and have complexity that is only linear in the size of the matrix.

     

Preconditioning for radial basis functions with domain decomposition methods

In our previous work, an effective preconditioning scheme that is based upon constructing least-squares approximation cardinal basis functions (ACBFs) from linear combinations of the RBF-PDE matrix elements has shown very attractive numerical results. The preconditioner costs O(N²) flops to set up and O(N) storage. The preconditioning technique is sufficiently general that it can be applied to different types of different operators. This was applied to the 2D multiquadric method, with c~1/√N on the Poisson test problem, the preconditioned GMRES converges in tens of iterations. In this paper, we combine the RBF methods and the ACBF preconditioning technique with the domain decomposition method (DDM). We studied different implementations of the ACBF-DDM scheme and provide numerical results for N > 10,000 nodes. We shall demonstrate that the efficiency of the ACBF-DDM scheme improves dramatically as successively finer partitions of the domain are considered.     

Parallel solvers for flow control based on domain decomposition

We sketch a non-overlapping domain decomposition method (DDM) for a linear quadratic optimal control problem governed by the Oseen equations. The DDM is applied to the system of necessary and sufficient optimality conditions. The approach extends balanced Neumann Neumann DDMs from single partial differential equations (PDEs) to the optimization control context, and it extends previous work on balanced Neumann Neumann DDMs for the optimal control of scalar elliptic PDEs to the optimal control of the Oseen equations. This extension requires a careful handling of the incompressibility constraint and resulting compatibility conditions, as well as a careful handling of the advection term. The DDM is used to parallelize the matrix-vector operations for the optimality system, as well as to parallelize the preconditioner. We present two approaches. One tackles the global optimality system directly, the other forms the Schur complement corresponding to variables on the subdomain interfaces. We present numerical experiments which clearly show the potential of the approaches. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

有限元结构分析在MIMD计算机上的并行直接解法

§1.前言 并行计算是近十几年来随着并行计算机发展而发展起来的一门新兴学科,特别是对于多指令流多数据流(MIMD)的并行计算机,由于它是由多台普通计算机甚至是向量计算机相互以一定方式联结起来的新型计算机系统,因此无论是它的运算速度或存贮空     

Convergence of the parallel chaotic waveform relaxation method for stiff systems

In this paper, we establish several algorithms for parallel chaotic waveform relaxation methods for solving linear ordinary differential systems based on some given models. Under some different assumptions on the coefficient matrix A and its multisplittings we obtain corresponding sufficient conditions of convergence of the algorithms. Also a discussion on convergence speed comparison of synchronous and asynchronous algorithms is given.     

An implementable proximal point algorithmic framework for nuclear norm minimization

The nuclear norm minimization problem is to find a matrix with the minimum nuclear norm subject to linear and second order cone constraints. Such a problem often arises from the convex relaxation of a rank minimization problem with noisy data, and arises in many fields of engineering and science. In this paper, we study inexact proximal point algorithms in the primal, dual and primal-dual forms for solving the nuclear norm minimization with linear equality and second order cone constraints. We design efficient implementations of these algorithms and present comprehensive convergence results. In particular, we investigate the performance of our proposed algorithms in which the inner sub-problems are approximately solved by the gradient projection method or the accelerated proximal gradient method. Our numerical results for solving randomly generated matrix completion problems and real matrix completion problems show that our algorithms perform favorably in comparison to several recently proposed state-of-the-art algorithms. Interestingly, our proposed algorithms are connected with other algorithms that have been studied in the literature.     

From polynomial matrices to markov parameters and back: Theory and numerical algorithms

We consider polynomial matrix representations of MIMO linear systems and their connection to Markov parameters. Specifically, we consider polynomial matrix models in an arbitrary operator ρ, and develop theory and numerical algorithms for transforming polynomial matrix models into Markov parameter models, and vice versa. We also provide numerical examples to illustrate the proposed algorithms.     

New techniques for the analysis of linear interval equations

The basic properties of interval matrix multiplication and several well-known solution algorithms for linear interval equations are abstracted by the concept of a sublinear map. The new concept, coupled with a systematic use of Ostrowski's comparison operator (in a form generalized to interval matrices), is used to derive quantitative information about the result of interval Gauss elimination andthe limit of various iterative schemes for the solution of linear interval equations. Moreover, optimality results are prove for (1) the use of the midpoint inverse as a preconditioning matrix, and (2) Gauss-Seidel iteration with componentwise intersection. This extends and improves results by Scheu, Krawczyk, and Alefeld and Herzberger.     

Direct fail-proof triangularization algorithms for AX + XB = C with Error-Free and parallel implementations

Two 0(mn³) inversion-free direct algorithms to compute a solution of the linear system AX +XB = C by triangularizing a Hessenberg matrix are presented. Without any loss of generality the matrix A is assumed upper Hessenberg and the order m of A the order n of B. The algorithms have an in-built consistency check, are capable of pruning redundant rows and converting the resulting matrix into a full row rank matrix, and permit A and —B to be any square matrices with common or distinct eigenvalues. In addition, these algorithms can also solve the homogeneous system AX +XB = 0 (null matrix C). An error-free implementation of the solution X using multiple modulus residue arithmetic as well as a parallelization of the algorithms is discussed.     

On the Linear Stability of Splitting Methods

A comprehensive linear stability analysis of splitting methods is carried out by means of a 2×2 matrix K(x) with polynomial entries (the stability matrix) and the stability polynomial p(x) (the trace of K(x) divided by two). An algorithm is provided for determining the coefficients of all possible time-reversible splitting schemes for a prescribed stability polynomial. It is shown that p(x) carries essentially all the information needed to construct processed splitting methods for numerically approximating the evolution of linear systems. By conveniently selecting the stability polynomial, new integrators with processing for linear equations are built which are orders of magnitude more efficient than other algorithms previously available. This paper is dedicated to Arieh Iserles on the occasion of his 60th anniversary.     

Fast algorithms for hierarchically semiseparable matrices

Semiseparable matrices and many other rank‐structured matrices have been widely used in developing new fast matrix algorithms. In this paper, we generalize the hierarchically semiseparable (HSS) matrix representations and propose some fast algorithms for HSS matrices. We represent HSS matrices in terms of general binary HSS trees and use simplified postordering notation for HSS forms. Fast HSS algorithms including new HSS structure generation and HSS form Cholesky factorization are developed. Moreover, we provide a new linear complexity explicit ULV factorization algorithm for symmetric positive definite HSS matrices with a low‐rank property. The corresponding factors can be used to solve the HSS systems also in linear complexity. Numerical examples demonstrate the efficiency of the algorithms. All these algorithms have nice data locality. They are useful in developing fast‐structured numerical methods for large discretized PDEs (such as elliptic equations), integral equations, eigenvalue problems, etc. Some applications are shown. Copyright © 2009 John Wiley & Sons, Ltd.     

A fast minimal residual algorithm for shifted unitary matrices

A new iterative scheme is described for the solution of large linear systems of equations with a matrix of the form A = ρU + ζI, where ρ and ζ are constants, U is a unitary matrix and I is the identity matrix. We show that for such matrices a Krylov subspace basis can be generated by recursion formulas with few terms. This leads to a minimal residual algorithm that requires little storage and makes it possible to determine each iterate with fairly little arithmetic work. This algorithm provides a model for iterative methods for non-Hermitian linear systems of equations, in a similar way to the conjugate gradient and conjugate residual algorithms. Our iterative scheme illustrates that results by Faber and Manteuffel [3,4] on the existence of conjugate gradient algorithms with short recurrence relations, and related results by Joubert and Young [13], can be extended.     

Algorithms for solving matrix polynomial equations of special form

In this paper we consider a series of algorithms for calculating radicals of matrix polynomial equations. A particular aspect of this problem arise in author’s work, concerning parameter identification of linear dynamic stochastic system. Special attention is given to searching the solution of an equation in a neighbourhood of some initial approximation. The offered approaches and algorithms allow us to receive fast and quite exact solution. We give some recommendations for application of given algorithms.     

Solving confluent Vandermonde systems of Hermite type

In this paper the authors develop two algorithms to solve systems of linear equations where the coefficients form a confluent Vandermonde matrix of Hermite type, or its transpose. These algorithms reduce the given system to upper triangular form by means of elementary matrix transformations. Recursive formulas to obtain the upper triangular form in an economical way are derived. Applications and numerical results are included.Algol-60 programs are appended.     

Reguläre Zerlegungen und Berechnung des Spektralradius nichtnegativer Matrizen

Summary On the basis of a Rayleigh Quotient Iteration method in [10] and a Maximal Quotient Iteration method in [5, 8] two algorithms for solving special eigenvalue problems are developed. The characteristic properties of these methods lie in the application of iterative linear methods to solving systems of linear equations. The convergence properties are investigated. We apply the algorithms to the computation of the spectralradius of a nonnegative irreducible matrix.

     

Cyclic and simultaneous iterative methods to matrix equations of the form A i X B i = F i

This paper deals with a general type of linear matrix equation problem. It presents new iterative algorithms to solve the matrix equations of the form A _i X?B _i = F _i . These algorithms are based on the incremental subgradient and the parallel subgradient methods. The convergence region of these algorithms are larger than other existing iterative algorithms. Finally, some experimental results are presented to show the efficiency of the proposed algorithms.     

Matrix computations and polynomial root-finding with preprocessing

Solution of homogeneous linear systems of equations is a basic operation of matrix computations. The customary algorithms rely on pivoting, orthogonalization and SVD, but we employ randomized preprocessing instead. This enables us to accelerate the solution dramatically, both in terms of the estimated arithmetic cost and the observed CPU time. The approach is effective in the cases of both general and structured input matrices and we extend it and its computational advantages to the solution of nonhomogeneous linear systems of equations, matrix eigen-solving, the solution of polynomial and secular equations, and approximation of a matrix by a nearby matrix that has a smaller rank or a fixed structure (e.g., of the Toeplitz or Hankel type). Our analysis and extensive experiments show the power of the presented algorithms.     

不等式约束优化一个新型可行QP-free算法

本文对非线性不等式约束优化问题提出了一个新的可行 QP-free 算法. 新算法保存了现有算法的优点, 并具有以下特性: (1) 算法每次迭代只需求解三个具有相同系数矩阵的线性方程组, 计算量小; (2) 可行下降方向只需通过求解一个线性方程组即可获得, 克服了以往分别求解两个线性方程组获得下降方向和可行方向, 然后再做凸组合的困难;(3) 迭代点均为可行点, 并不要求是严格内点; (4) 算法中采用了试探性线搜索,可以进一步减少计算量; (5) 算法中参数很少,数值试验表明算法具有较好的数值效果和较强的稳定性.