An Implicitly Restarted Block Arnoldi Method in a Vector-Wise Fashion

In this paper, we develop an implicitly restarted block Arnoldi algorithm in a vector-wise fashion. The vector-wise construction greatly simplifies both the detection of necessary deflation and the actual deflation itself, so it is preferable to the block-wise construction. The numerical experiment shows that our algorithm is effective.     

Estimates for Piecewise Quadric Approximations of Implicitly Defined Manifolds

Several algorithms have been given for obtaining piecewise quadric approximations of implicitly defined manifolds. This paper is concerned with error estimates for such algorithms.     

Spectral Variants of Krylov Subspace Methods

Krylov iterative methods usually solve an optimization problem, per iteration, to obtain a vector whose components are the step lengths associated with the previous search directions. This vector can be viewed as the solution of a multiparameter optimization problem. In that sense, Krylov methods can be combined with the spectral choice of step length that has recently been developed to accelerate descent methods in optimization. In this work, we discuss different spectral variants of Krylov methods and present encouraging preliminary numerical experiments, with and without preconditioning.     

非线性特征值问题的二次近似方法

本文研究求解非线性特征值问题的数值方法.基于矩阵值函数的二次近似,将非线性特征值问题转化为二次特征值问题,提出了求解非线性特征值问题的逐次二次近似方法,分析了该方法的收敛性.结合求解二次特征值问题的Arnoldi方法和Jacobi-Davidson方法,给出求解非线性特征值问题的一些二次近似方法.数值结果表明本文所给算法是有效的.     

A computational method for large‐scale differential symmetric Stein equation

We propose a numerical method for solving large‐scale differential symmetric Stein equations having low‐rank right constant term. Our approach is based on projection the given problem onto a Krylov subspace then solving the low dimensional matrix problem by using an integration method, and the original problem solution is built by using obtained low‐rank approximate solution. Using the extended block Arnoldi process and backward differentiation formula (BDF), we give statements of the approximate solution and corresponding residual. Some numerical results are given to show the efficiency of the proposed method.     

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.     

A rational Arnoldi process with applications

The rational Arnoldi process is a popular method for the computation of a few eigenvalues of a large non‐Hermitian matrix and for the approximation of matrix functions. The method is particularly attractive when the rational functions that determine the process have only few distinct poles , because then few factorizations of matrices of the form A ? z_jI have to be computed. We discuss recursion relations for orthogonal bases of rational Krylov subspaces determined by rational functions with few distinct poles. These recursion formulas yield a new implementation of the rational Arnoldi process. Applications of the rational Arnoldi process to the approximation of matrix functions as well as to the computation of eigenvalues and pseudospectra of A are described. The new implementation is compared to several available implementations.     

A two-sided Arnoldi algorithm with stopping criterion and MIMO selection procedure

In this paper we introduce a two-sided Arnoldi method for the reduction of high order linear systems and we propose useful extensions, first of all a stopping criterion to find a suitable order for the reduced model and secondly, a selection procedure to significantly improve the performance in the multi-input multi-output (MIMO) case. One application is in micro-electro-mechanical systems (MEMS). We consider a thermo-electric micro thruster model, and a comparison between the commonly used Arnoldi algorithm and the two-sided Arnoldi is performed.     

Skew-symmetric methods for nonsymmetric linear systems with multiple right-hand sides

By transforming nonsymmetric linear systems to the extended skew-symmetric ones, we present the skew-symmetric methods for solving nonsymmetric linear systems with multiple right-hand sides. These methods are based on the block and global Arnoldi algorithm which is formed by implementing orthogonal projections of the initial matrix residual onto a matrix Krylov subspace. The algorithms avoid the tediously long Arnoldi process and highly reduce expensive storage. Numerical experiments show that these algorithms are effective and give better practical performances than global GMRES for solving nonsymmetric linear systems with multiple right-hand sides.     

高阶谱元区域分解算法及其在流动稳定性中的应用

以无时间分裂误差的区域分解Stokes谱元算法为基础构建整体稳定性分析方法.用Jacobian-free的Inexact-Newton-Krylov算法求解不可压缩Navier-Stokes方程的定常解,将Stokes算法的时间推进步作为Newton迭代的预处理,在此基础上采用Arnoldi方法计算大规模特征值问题,对复杂流动进行稳定性分析,该方法能统一处理定常和非定常计算,没有时间分裂误差,无需显式构造Jacobian矩阵,可以减少内存使用,降低计算量,并加速迭代收敛.对有分析解的Kovasznay流动的计算表明,高阶谱元法具有指数收敛的谱精度.对亚临界方腔对称驱动流的各种定常解的计算及其稳定性分析验证了方法的可行性.