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.     

Splitting and composition methods for explicit time dependence in separable dynamical systems

We consider splitting methods for the numerical integration of separable non-autonomous differential equations. In recent years, splitting methods have been extensively used as geometric numerical integrators showing excellent performances (both qualitatively and quantitatively) when applied on many problems. They are designed for autonomous separable systems, and a substantial number of methods tailored for different structures of the equations have recently appeared. Splitting methods have also been used for separable non-autonomous problems either by solving each non-autonomous part separately or after each vector field is frozen properly. We show that both procedures correspond to introducing the time as two new coordinates. We generalize these results by considering the time as one or more further coordinates which can be integrated following either of the previous two techniques. We show that the performance as well as the order of the final method can strongly depend on the particular choice. We present a simple analysis which, in many relevant cases, allows one to choose the most appropriate split to retain the high performance the methods show on the autonomous problems. This technique is applied to different problems and its performance is illustrated for several numerical examples.     

A Class of Explicit Exponential General Linear Methods

In this paper, we consider a class of explicit exponential integrators that includes as special cases the explicit exponential Runge–Kutta and exponential Adams–Bashforth methods. The additional freedom in the choice of the numerical schemes allows, in an easy manner, the construction of methods of arbitrarily high order with good stability properties. We provide a convergence analysis for abstract evolution equations in Banach spaces including semilinear parabolic initial-boundary value problems and spatial discretizations thereof. From this analysis, we deduce order conditions which in turn form the basis for the construction of new schemes. Our convergence results are illustrated by numerical examples. AMS subject classification (2000) 65L05, 65L06, 65M12, 65J10     

求解2×2块对称不定线性方程组迭代法的收敛性分析(英文)

本文研究求解系数矩阵为2×2块对称不定矩阵时的线性方程组,提出了一种新的分裂迭代法,并通过研究迭代矩阵的谱半径,详细讨论了新方法的收敛性.最后,我们也讨论了预条件矩阵特征根的几条性质.     

Fourth-Order Splitting Methods for Time-Dependant Differential Equations

This study was suggested by previous work on the simulation of evolution equations with scale-dependent processes,e.g.,wave-propagation or heat-transfer,that are modeled by wave equations or heat equations.Here,we study both parabolic and hyperbolic equations.We focus on ADI (alternating direction implicit) methods and LOD (locally one-dimensional) methods,which are standard splitting methods of lower order,e.g.second-order.Our aim is to develop higher-order ADI methods,which are performed by Richardson extrapolation,Crank-Nicolson methods and higher-order LOD methods,based on locally higher-order methods.We discuss the new theoretical results of the stability and consistency of the ADI methods.The main idea is to apply a higher- order time discretization and combine it with the ADI methods.We also discuss the dis- cretization and splitting methods for first-order and second-order evolution equations. The stability analysis is given for the ADI method for first-order time derivatives and for the LOD (locally one-dimensional) methods for second-order time derivatives.The higher-order methods are unconditionally stable.Some numerical experiments verify our results.     

Runge-Kutta Methods Adapted to Manifolds and Based on Rigid Frames

We consider numerical integration methods for differentiable manifolds as proposed by Crouch and Grossman. The differential system is phrased by means of a system of frame vector fields E ₁, ... , E _n on the manifold. The numerical approximation is obtained by composing flows of certain vector fields in the linear span of E ₁, ... , E _n that are tangent to the differential system at various points. The methods reduce to traditional Runge-Kutta methods if the frame vector fields are chosen as the standard basis of euclidean ⁿ . A complete theory for the order conditions involving ordered rooted trees is developed. Examples of explicit and diagonal implicit methods are presented, along with some numerical results.     

Parallel Interior-Point Method for Linear and Quadratic Programs with Special Structure

This paper concerns the use of iterative solvers in interior-point methods for linear and quadratic programming problems. We state an adaptive termination rule for the inner iterative scheme and we prove the global convergence of the obtained algorithm, exploiting the theory developed for inexact Newton methods. This approach is promising for problems with special structure on parallel computers. We present an application on Cray T3E/256 and SGI Origin 2000/64 arising in stochastic linear programming and robust optimization, where the constraint matrix is block-angular and extremely large.     

Block Projection Methods for Linear Systems

The aim of this paper is to provide a theory of block projection methods for the solution of a system of linear equations with multiple right-hand sides. Our approach allows to obtain recursive algorithms for the implementation of these methods.     

解线性方程组的松弛方法及其应用

线性方程组在科学和工程领域中有着重要的应用,松弛方法是求解线性方程组的有效算法之一.本文在著名的Gauss-Seidel迭代法的基础上,研究了一种有效的松弛方法.理论分析表明,该方法能收敛到线性方程组的唯一解.此外,我们还将该方法应用在鞍点问题和PageRank问题的求解上,并得出了相应的数值结果.结果表明该方法比现有的松弛方法更有效.     

Inexact Operator Splitting Methods with Selfadaptive Strategy for Variational Inequality Problems

The Peaceman-Rachford and Douglas-Rachford operator splitting methods are advantageous for solving variational inequality problems, since they attack the original problems via solving a sequence of systems of smooth equations, which are much easier to solve than the variational inequalities. However, solving the subproblems exactly may be prohibitively difficult or even impossible. In this paper, we propose an inexact operator splitting method, where the subproblems are solved approximately with some relative error tolerance. Another contribution is that we adjust the scalar parameter automatically at each iteration and the adjustment parameter can be a positive constant, which makes the methods more practical and efficient. We prove the convergence of the method and present some preliminary computational results, showing that the proposed method is promising. This work was supported by the NSFC grant 10501024.     

Stabilization of Interior-Point Methods for Linear Programming

The paper studies numerical stability problems arising in the application of interior-point methods to primal degenerate linear programs. A stabilization procedure based on Gaussian elimination is proposed and it is shown that it stabilizes all path following methods, original and modified Dikin's method, Karmarkar's method, etc.     

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.     

Block Descent Methods and Hybrid Procedures for Linear Systems

In this paper, we define and study several types of block descent methods for the simultaneous solution of a system of linear equations with several right hand sides. Then, improved block EN methods will be proposed. Finally, block hybrid and minimal residual smoothing procedures will be considered.     

General Linear Methods for Stiff Differential Equations

A new type of general linear method is constructed which combines A-stability or L-stability with ease of implementation. The method is structured in such a manner that its stability region is identical with that of a Runge-Kutta method, using a restriction known as inherent RK stability.This revised version was published online in October 2005 with corrections to the Cover Date.     

Acceleration Scheme for Parallel Projected Aggregation Methods for Solving Large Linear Systems

The Projected Aggregation methods generate the new point x ^k+1 as the projection of x ^k onto an aggregate hyperplane usually arising from linear combinations of the hyperplanes defined by the blocks. The aim of this paper is to improve the speed of convergence of a particular kind of them by projecting the directions given by the blocks onto the aggregate hyperplane defined in the last iteration. For that purpose we apply the scheme introduced in A new method for solving large sparse systems of linear equations using row projections [11], for a given block projection algorithm, to some new methods here introduced whose main features are related to the fact that the projections do not need to be accurately computed. Adaptative splitting schemes are applied which take into account the structure and conditioning of the matrix. It is proved that these new highly parallel algorithms improve the original convergence rate and present numerical results which show their computational efficiency.     

Energy and Quadratic Invariants Preserving Methods for Hamiltonian Systems with Holonomic Constraints

We introduce a new class of parametrized structure--preserving partitioned Runge-Kutta ($\alpha$-PRK) methods for Hamiltonian systems with holonomic constraints. The methods are symplectic for any fixed scalar parameter $\alpha$, and are reduced to the usual symplectic PRK methods like Shake-Rattle method or PRK schemes based on Lobatto IIIA-IIIB pairs when $\alpha=0$. We provide a new variational formulation for symplectic PRK schemes and use it to prove that the $\alpha$-PRK methods can preserve the quadratic invariants for Hamiltonian systems subject to holonomic constraints. Meanwhile, for any given consistent initial values $(p_{0}, q_0)$ and small step size $h>0$, it is proved that there exists $\alpha^*=\alpha(h, p_0, q_0)$ such that the Hamiltonian energy can also be exactly preserved at each step. Based on this, we propose some energy and quadratic invariants preserving $\alpha$-PRK methods. These $\alpha$-PRK methods are shown to have the same convergence rate as the usual PRK methods and perform very well in various numerical experiments.     

Symmetric Projection Methods for Differential Equations on Manifolds

Projection methods are a standard approach for the numerical solution of differential equations on manifolds. It is known that geometric properties (such as symplecticity or reversibility) are usually destroyed by such a discretization, even when the basic method is symplectic or symmetric. In this article, we introduce a new kind of projection methods, which allows us to recover the time-reversibility, an important property for long-time integrations.     

Stokes方程的投影／方向分裂谱方法

我们提出和分析了一种求解Stokes方程的数值方法．新方法基于空间上的Legendre谱离散,时间上则采用投影／方向分裂格式．更确切地说,时间离散的出发点是旋度形式的压力校正投影法,在此基础上进一步应用方向分裂法,把速度和压力方程分裂为一系列一维的椭圆型子问题．然后生成的这些一维子问题用Legendre谱方法进行空间离散．另外,我们证明了全离散格式的稳定性．一些数值实验验证了收敛性和方法的有效性．     

松弛算子分裂法线性收敛性分析的新框架

提出了一种新的分析框架来研究松弛算子分裂法的线性收敛性,可以将这种框架看成是经典的Krasnosel''-Mann迭代和Banach-Picard收缩的扩展形式.随后,将提出的这个框架应用于分析广义邻近点算法和松弛向前向后分裂算法的线性收敛性,其过程十分简洁和直接.