Arnoldi versus nonsymmetric Lanczos algorithms for solving matrix eigenvalue problems

We present theoretical and numerical comparisons between Arnoldi and nonsymmetric Lanczos procedures for computing eigenvalues of nonsymmetric matrices. In exact arithmetic we prove that any type of eigenvalue convergence behavior obtained using a nonsymmetric Lanczos procedure may also be obtained using an Arnoldi procedure but on a different matrix and with a different starting vector. In exact arithmetic we derive relationships between these types of procedures and normal matrices which suggest some interesting questions regarding the roles of nonnormality and of the choice of starting vectors in any characterizations of the convergence behavior of these procedures. Then, through a set of numerical experiments on a complex Arnoldi and on a complex nonsymmetric Lanczos procedure, we consider the more practical question of the behavior of these procedures when they are applied to the same matrices.This work was supported by NSF grant GER-9450081 while the author was visiting the University of Maryland.     

On Lanczos Based Methods for the Regularization of Discrete Ill-Posed Problems

We study hybrid methods for the solution of linear ill-posed problems. Hybrid methods are based on he Lanczos process, which yields a sequence of small bidiagonal systems approximating the original ill-posed problem. In a second step, some additional regularization, typically the truncated SVD, is used to stabilize the iteration. We investigate two different hybrid methods and interpret these schemes as well-known projection methods, namely least-squares projection and the dual least-squares method. Numerical results are provided to illustrate the potential of these methods. This gives interesting insight in to the behavior of hybrid methods in practice.This revised version was published online in October 2005 with corrections to the Cover Date.     

Family of Projected Descent Methods for Optimization Problems with Simple Bounds

This paper presents a family of projected descent direction algorithms with inexact line search for solving large-scale minimization problems subject to simple bounds on the decision variables. The global convergence of algorithms in this family is ensured by conditions on the descent directions and line search. Whenever a sequence constructed by an algorithm in this family enters a sufficiently small neighborhood of a local minimizer satisfying standard second-order sufficiency conditions, it gets trapped and converges to this local minimizer. Furthermore, in this case, the active constraint set at is identified in a finite number of iterations. This fact is used to ensure that the rate of convergence to a local minimizer, satisfying standard second-order sufficiency conditions, depends only on the behavior of the algorithm in the unconstrained subspace. As a particular example, we present projected versions of the modified Polak–Ribière conjugate gradient method and the limited-memory BFGS quasi-Newton method that retain the convergence properties associated with those algorithms applied to unconstrained problems.     

Superlinear Noninterior One-Step Continuation Method for Monotone LCP in the Absence of Strict Complementarity

We propose a noninterior continuation method for the monotone linear complementarity problem (LCP) by modifying the Burke–Xu framework of the noninterior predictor-corrector path-following method (Refs. 1–2). The new method solves one system of linear equations and carries out only one line search at each iteration. It is shown to converge to the LCP solution globally linearly and locally superlinearly without the assumption of strict complementarity at the solution. Our analysis of the continuation method is based on a broader class of the smooth functions introduced by Chen and Mangasarian (Ref. 3).     

Conjugate gradient predictor corrector method for solving large scale problems

In this paper, we give a new method for solving large scale problems. The basic idea of this method depends on implementing the conjugate gradient as a corrector into a continuation method. We use the Euler method as a predictor. Adaptive steplength control is used during the tracing of the solution curve. We present some of our experimental examples to demonstrate the efficiency of the method.

     

Transpose-free formulations of Lanczos-type methods for nonsymmetric linear systems

We present a transpose-free version of the nonsymmetric scaled Lanczos procedure. It generates the same tridiagonal matrix as the classical algorithm, using two matrix–vector products per iteration without accessing A^T. We apply this algorithm to obtain a transpose-free version of the Quasi-minimal residual method of Freund and Nachtigal [15] (without look-ahead), which requires three matrix–vector products per iteration. We also present a related transpose-free version of the bi-conjugate gradients algorithm. This revised version was published online in June 2006 with corrections to the Cover Date.     

关于特征值问题有限元方法研究的一个说法

受林群的微积分哲学公式((相对真理)/(绝对真理)=0.9)的启示,总结了近些年来关于特征值问题有限元方法的研究,并发现其背后同样隐藏着该哲学公式,换句话说,所追求的是特征值问题的有限元数值解和真解的零距离,其实就是追求真解的过程,要经多道(即0.9,0.99,0.999,…),再将比例,即(数值解)/(真解),提到1.     

Interior-Point Gradient Method for Large-Scale Totally Nonnegative Least Squares Problems

We study an interior-point gradient method for solving a class of so-called totally nonnegative least-squares problems. At each iteration, the method decreases the residual norm along a diagonally-scaled negative gradient direction with a special scaling. We establish the global convergence of the method and present some numerical examples to compare the proposed method with a few similar methods including the affine scaling method.This author was supported in part by DOE/LANL Contract 03891-99-23This author was supported in part by NSF Grant DMS-0442065     

A Lanczos-type method for multiple starting vectors

Given a square matrix and single right and left starting vectors, the classical nonsymmetric Lanczos process generates two sequences of biorthogonal basis vectors for the right and left Krylov subspaces induced by the given matrix and vectors. In this paper, we propose a Lanczos-type algorithm that extends the classical Lanczos process for single starting vectors to multiple starting vectors. Given a square matrix and two blocks of right and left starting vectors, the algorithm generates two sequences of biorthogonal basis vectors for the right and left block Krylov subspaces induced by the given data. The algorithm can handle the most general case of right and left starting blocks of arbitrary sizes, while all previously proposed extensions of the Lanczos process are restricted to right and left starting blocks of identical sizes. Other features of our algorithm include a built-in deflation procedure to detect and delete linearly dependent vectors in the block Krylov sequences, and the option to employ look-ahead to remedy the potential breakdowns that may occur in nonsymmetric Lanczos-type methods.

     

Deferred Correction Methods for Initial Value Problems

In this paper we derive high order implicit difference methods for large systems of ODE. The methods are based on the deferred correction principle, yielding accuracy of order p by applying the trapezoidal rule p/2 times in each timestep. Numerical experiments demonstrate the efficiency of the method.This revised version was published online in October 2005 with corrections to the Cover Date.     

L-曲线估计确定正则参数的双网格迭代法

本文考虑对不适定问题离散化得到的大规模不适定线性方程组进行Tiknonov正则化,然后用双网格迭代法求解得到的Tikhonov正则化方程组,并用L-曲线估计法来确定正则参数.试验问题的数值结果表明双网格迭代法求解正则化后的对称正定线性方程组效果很好,且L-曲线估计法确定正则参数计算量很小.     

Tikhonov Regularization Methods for Variational Inequality Problems

Motivated by the work of Facchinei and Kanzow (Ref. 1) on regularization methods for the nonlinear complementarity problem and the work of Ravindran and Gowda (Ref. 2) for the box variational inequality problem, we study regularization methods for the general variational inequality problem. A sufficient condition is given which guarantees that the union of the solution sets of the regularized problems is nonempty and bounded. It is shown that solutions of the regularized problems form a minimizing sequence of the D-gap function under a mild condition.     

Low Regularity Error Analysis for Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

This paper presents error estimates in both an energy norm and the $L^2$-norm for the weak Galerkin (WG) finite element methods for elliptic problems with low regularity solutions. The error analysis for the continuous Galerkin finite element remains same regardless of regularity. A totally different analysis is needed for discontinuous finite element methods if the elliptic regularity is lower than H-1.5. Numerical results confirm the theoretical analysis.     

Improving an interior-point approach for large block-angular problems by hybrid preconditioners

The computational time required by interior-point methods is often dominated by the solution of linear systems of equations. An efficient specialized interior-point algorithm for primal block-angular problems has been used to solve these systems by combining Cholesky factorizations for the block constraints and a conjugate gradient based on a power series preconditioner for the linking constraints. In some problems this power series preconditioner resulted to be inefficient on the last interior-point iterations, when the systems became ill-conditioned. In this work this approach is combined with a splitting preconditioner based on LU factorization, which works well for the last interior-point iterations. Computational results are provided for three classes of problems: multicommodity flows (oriented and nonoriented), minimum-distance controlled tabular adjustment for statistical data protection, and the minimum congestion problem. The results show that, in most cases, the hybrid preconditioner improves the performance and robustness of the interior-point solver. In particular, for some block-angular problems the solution time is reduced by a factor of 10.     

On the Finite Termination of an Entropy Function Based Non-Interior Continuation Method for Vertical Linear Complementarity Problems

By using a smooth entropy function to approximate the non-smooth max-type function, a vertical linear complementarity problem (VLCP) can be treated as a family of parameterized smooth equations. A Newton-type method with a testing procedure is proposed to solve such a system. We show that under some milder than usual assumptions the proposed algorithm finds an exact solution of VLCP in a finite number of iterations. Some computational results are included to illustrate the potential of this approach. This author’s work was partially supported by the National Natural Science Foundation of China (Grant Nos. 10271002 and 10401038). This author’s work was partially supported by the Scientific Research Foundation of Tianjin University for the Returned Overseas Chinese Scholars and the Scientific Research Foundation of Liu Hui Center for Applied Mathematics, Nankai University-Tianjin University.     

High Order Long-Step Methods for Solving Linear Complementarity Problems

The paper is concerned with methods for solving linear complementarity problems (LCP) that are monotone or at least sufficient in the sense of Cottle, Pang and Venkateswaran (1989). A basic concept of interior-point-methods is the concept of (perhaps weighted) feasible or infeasible interior-point paths. They converge to a solution of the LCP if a natural path parameter, usually the current duality gap, tends to 0.After reviewing some basic analyticity properties of these paths it is shown how these properties can be used to devise also long-step path-following methods (and not only predictor–corrector type methods) for which the duality gap converges Q-superlinearly to 0 with an arbitrarily high order.     

Cascadic Multigrid for Finite Volume Methods for Elliptic Problems

In this paper, some effective cascadic multigrid methods are proposed for solving the large scale symmetric or nonsymmetric algebraic systems arising from the finite volume methods for second order elliptic problems. It is shown that these algorithms are optimal in both accuracy and computational complexity. Numerical experiments are reported to support our theory.     

Fast Iterative Methods for Solving of Boundary Nonlinear Integral Equations with Singularity

A very efficient and fully discrete method for numerical solution of boundary nonlinear integral equation is described. There seems a lack of rigorous numerical analysis because of singular or hypersingular behavior. In this paper, we suggest variants of methods for solving numerical solutions. Moreover, our aim has been to show how the iterations can be effectively and efficiently regularized for solving ill-posed problems by using the preconditioner. We have compared these methods with CPU time and iterations. Finally, some numerical examples show the efficiency of the proposed methods.     

Fourth-Order Symmetric DIRK Methods for Periodic Stiff Problems

New symmetric DIRK methods specially adapted to the numerical integration of first-order stiff ODE systems with periodic solutions are obtained. Our interest is focused on the dispersion (phase errors) of the dominant components in the numerical oscillations when these methods are applied to the homogeneous linear test model. Based on this homogeneous test model we derive the dispersion conditions for symmetric DIRK methods as well as symmetric stability functions with real poles and maximal dispersion order. Two new fourth-order symmetric methods with four and five stages are obtained. One of the methods is fourth-order dispersive whereas the other method is symplectic and sixth-order dispersive. These methods have been applied to a number of test problems (linear as well as nonlinear) and some numerical results are presented to show their efficiency when they are compared with the symplectic DIRK method derived by Sanz-Serna and Abia (SIAM J. Numer. Anal. 28 (1991) 1081–1096).