Inexact inverse subspace iteration for generalized eigenvalue problems

n this paper, we present an inexact inverse subspace iteration method for computing a few eigenpairs of the generalized eigenvalue problem Ax=λBx. We first formulate a version of inexact inverse subspace iteration in which the approximation from one step is used as an initial approximation for the next step. We then analyze the convergence property, which relates the accuracy in the inner iteration to the convergence rate of the outer iteration. In particular, the linear convergence property of the inverse subspace iteration is preserved. Numerical examples are given to demonstrate the theoretical results.     

An inexact inverse iteration for large sparse eigenvalue problems

In this paper, we propose an inverse inexact iteration method for the computation of the eigenvalue with the smallest modulus and its associated eigenvector for a large sparse matrix. The linear systems of the traditional inverse iteration are solved with accuracy that depends on the eigenvalue with the second smallest modulus and iteration numbers. We prove that this approach preserves the linear convergence of inverse iteration. We also propose two practical formulas for the accuracy bound which are used in actual implementation. © 1997 John Wiley & Sons, Ltd.     

A new subspace iteration method for the algebraic Riccati equation

We consider the numerical solution of the continuous algebraic Riccati equation A*X + XA ? XFX + G = 0, with F = F*,G = G* of low rank and A large and sparse. We develop an algorithm for the low‐rank approximation of X by means of an invariant subspace iteration on a function of the associated Hamiltonian matrix. We show that the sought‐after approximation can be obtained by a low‐rank update, in the style of the well known Alternating Direction Implicit (ADI) iteration for the linear equation, from which the new method inherits many algebraic properties. Moreover, we establish new insightful matrix relations with emerging projection‐type methods, which will help increase our understanding of this latter class of solution strategies. Copyright © 2014 John Wiley & Sons, Ltd.     

A block inverse-free preconditioned Krylov subspace method for symmetric generalized eigenvalue problems

The inverse-free preconditioned Krylov subspace method of Golub and Ye [G.H. Golub, Q. Ye, An inverse free preconditioned Krylov subspace method for symmetric generalized eigenvalue problems, SIAM J. Sci. Comp. 24 (2002) 312-334] is an efficient algorithm for computing a few extreme eigenvalues of the symmetric generalized eigenvalue problem. In this paper, we first present an analysis of the preconditioning strategy based on incomplete factorizations. We then extend the method by developing a block generalization for computing multiple or severely clustered eigenvalues and develop a robust black-box implementation. Numerical examples are given to illustrate the analysis and the efficiency of the block algorithm.     

Restarted generalized Krylov subspace methods for solving large-scale polynomial eigenvalue problems

In this paper, we introduce a generalized Krylov subspace based on a square matrix sequence {A _j } and a vector sequence {u _j }. Next we present a generalized Arnoldi procedure for generating an orthonormal basis of . By applying the projection and the refined technique, we derive a restarted generalized Arnoldi method and a restarted refined generalized Arnoldi method for solving a large-scale polynomial eigenvalue problem (PEP). These two methods are applied to solve the PEP directly. Hence they preserve essential structures and properties of the PEP. Furthermore, restarting reduces the storage requirements. Some theoretical results are presented. Numerical tests report the effectiveness of these methods. Yimin Wei is supported by the National Natural Science Foundation of China and Shanghai Education Committee.     

A geometric theory for preconditioned inverse iteration applied to a subspace

The aim of this paper is to provide a convergence analysis for a preconditioned subspace iteration, which is designated to determine a modest number of the smallest eigenvalues and its corresponding invariant subspace of eigenvectors of a large, symmetric positive definite matrix. The algorithm is built upon a subspace implementation of preconditioned inverse iteration, i.e., the well-known inverse iteration procedure, where the associated system of linear equations is solved approximately by using a preconditioner. This step is followed by a Rayleigh-Ritz projection so that preconditioned inverse iteration is always applied to the Ritz vectors of the actual subspace of approximate eigenvectors. The given theory provides sharp convergence estimates for the Ritz values and is mainly built on arguments exploiting the geometry underlying preconditioned inverse iteration.     

A block Arnoldi-type contour integral spectral projection method for solving generalized eigenvalue problems

For generalized eigenvalue problems, we consider computing all eigenvalues located in a certain region and their corresponding eigenvectors. Recently, contour integral spectral projection methods have been proposed for solving such problems. In this study, from the analysis of the relationship between the contour integral spectral projection and the Krylov subspace, we conclude that the Rayleigh–Ritz-type of the contour integral spectral projection method is mathematically equivalent to the Arnoldi method with the projected vectors obtained from the contour integration. By this Arnoldi-based interpretation, we then propose a block Arnoldi-type contour integral spectral projection method for solving the eigenvalue problem.     

Krylov subspace methods with deflation and balancing preconditioners for least squares problems

For solving least squares problems, the CGLS method is a typical method in the point of view of iterative methods. When the least squares problems are ill-conditioned, the convergence behavior of the CGLS method will present a deteriorated result. We expect to select other iterative Krylov subspace methods to overcome the disadvantage of CGLS. Here the GMRES method is a suitable algorithm for the reason that it is derived from the minimal residual norm approach, which coincides with least squares problems. Ken Hayami proposed BAGMRES for solving least squares problems in [\emph{GMRES Methods for Least Squares Problems, SIAM J. Matrix Anal. Appl., 31(2010)}, pp.2400-2430]. The deflation and balancing preconditioners can optimize the convergence rate through modulating spectral distribution. Hence, in this paper we utilize preconditioned iterative Krylov subspace methods with deflation and balancing preconditioners in order to solve ill-conditioned least squares problems. Numerical experiments show that the methods proposed in this paper are better than the CGLS method.     

Fast enclosure for all eigenvalues in generalized eigenvalue problems

A fast method for enclosing all eigenvalues in generalized eigenvalue problems Ax=λBx is proposed. Firstly a theorem for enclosing all eigenvalues, which is applicable even if A is not Hermitian and/or B is not Hermitian positive definite, is presented. Next a theorem for accelerating the enclosure is presented. The proposed method is established based on these theorems. Numerical examples show the performance and property of the proposed method. As an application of the proposed method, an efficient method for enclosing all eigenvalues in polynomial eigenvalue problems is also sketched.     

Tuned preconditioners for inexact two‐sided inverse and Rayleigh quotient iteration

Convergence results are provided for inexact two‐sided inverse and Rayleigh quotient iteration, which extend the previously established results to the generalized non‐Hermitian eigenproblem and inexact solves with a decreasing solve tolerance. Moreover, the simultaneous solution of the forward and adjoint problem arising in two‐sided methods is considered, and the successful tuning strategy for preconditioners is extended to two‐sided methods, creating a novel way of preconditioning two‐sided algorithms. Furthermore, it is shown that inexact two‐sided Rayleigh quotient iteration and the inexact two‐sided Jacobi‐Davidson method (without subspace expansion) applied to the generalized preconditioned eigenvalue problem are equivalent when a certain number of steps of a Petrov–Galerkin–Krylov method is used and when this specific tuning strategy is applied. Copyright © 2014 John Wiley & Sons, Ltd.     

Convergence of a generalized Newton and an inexact generalized Newton algorithms for solving nonlinear equations with nondifferentiable terms

In this paper, we consider two versions of the Newton-type method for solving a nonlinear equations with nondifferentiable terms, which uses as iteration matrices, any matrix from B-differential of semismooth terms. Local and global convergence theorems for the generalized Newton and inexact generalized Newton method are proved. Linear convergence of the algorithms is obtained under very mild assumptions. The superlinear convergence holds under some conditions imposed on both terms of equation. Some numerical results indicate that both algorithms works quite well in practice.      

A FEAST algorithm with oblique projection for generalized eigenvalue problems

The contour integral‐based eigensolvers are the recent efforts for computing the eigenvalues inside a given region in the complex plane. The best‐known members are the Sakurai–Sugiura method, its stable version CIRR, and the FEAST algorithm. An attractive computational advantage of these methods is that they are easily parallelizable. The FEAST algorithm was developed for the generalized Hermitian eigenvalue problems. It is stable and accurate. However, it may fail when applied to non‐Hermitian problems. Recently, a dual subspace FEAST algorithm was proposed to extend the FEAST algorithm to non‐Hermitian problems. In this paper, we instead use the oblique projection technique to extend FEAST to the non‐Hermitian problems. Our approach can be summarized as follows: (a) construct a particular contour integral to form a search subspace containing the desired eigenspace and (b) use the oblique projection technique to extract desired eigenpairs with appropriately chosen test subspace. The related mathematical framework is established. Comparing to the dual subspace FEAST algorithm, we can save the computational cost roughly by a half if only the eigenvalues or the eigenvalues together with their right eigenvectors are needed. We also address some implementation issues such as how to choose a suitable starting matrix and design‐efficient stopping criteria. Numerical experiments are provided to illustrate that our method is stable and efficient.     

关于广义鞍点问题的HSS迭代方法的收缩因子(英文)

白中治等提出了解非埃尔米特正定线性方程组的埃尔米特和反埃尔米特分裂(HSS)迭代方法(Bai Z Z,Golub G H,Ng M K.Hermitian and skew-Hermitian splitting methodsfor non-Hermitian positive definite linear systems.SIAM J.Matrix Anal.Appl.,2003,24:603-626).本文精确地估计了用HSS迭代方法求解广义鞍点问题时在加权2-范数和2-范数下的收缩因子.在实际的计算中,正是这些收缩因子而不是迭代矩阵的谱半径,本质上控制着HSS迭代方法的实际收敛速度.根据文中的分析,求解广义鞍点问题的HSS迭代方法的收缩因子在加权2-范数下等于1,在2-范数下它会大于等于1,而在某种适当选取的范数之下,它则会小于1.最后,用数值算例说明了理论结果的正确性.     

一类含有p-拉普拉斯算子的二阶边值问题的单调迭代正解（英文）

研究了一类含有p-拉普拉斯算子的微分方程积分边值问题.运用迭代技巧,给出了这一类边值问题的单调正解,值得感兴趣的是微分方程中的非线性项含有一阶导数.     

Accelerated Landweber iteration with convex penalty for linear inverse problems in Banach spaces

In recent years, Landweber iteration has been extended to solve linear inverse problems in Banach spaces by incorporating non-smooth convex penalty functionals to capture features of solutions. This method is known to be slowly convergent. However, because it is simple to implement, it still receives a lot of attention. By making use of the subspace optimization technique, we propose an accelerated version of Landweber iteration with non-smooth convex penalty which significantly speeds up the method. Numerical simulations are given to test the efficiency.     

A policy iteration algorithm for fixed point problems with nonexpansive operators

The aim of this paper is to solve the fixed point problems:

On Minty variational principle for nonsmooth vector optimization problems with generalized approximate convexity

In this paper, we consider a nonsmooth vector optimization problem involving locally Lipschitz generalized approximate convex functions and find some relations between approximate convexity and generalized approximate convexity. We establish relationships between vector variational inequalities and nonsmooth vector optimization problem using the generalized approximate convexity as a tool.     

Gap functions for a system of generalized vector quasi-equilibrium problems with set-valued mappings

In this paper, some gap functions for three classes of a system of generalized vector quasi-equilibrium problems with set-valued mappings (for short, SGVQEP) are investigated by virtue of the nonlinear scalarization function of Chen, Yang and Yu. Three examples are then provided to demonstrate these gap functions. Also, some gap functions for three classes of generalized finite dimensional vector equilibrium problems (GFVEP) are derived without using the nonlinear scalarization function method. Furthermore, a set-valued function is obtained as a gap function for one of (GFVEP) under certain assumptions.