Overlapping Schwarz Waveform Relaxation for the Heat Equation in N Dimensions

We analyze overlapping Schwarz waveform relaxation for the heat equation in n spatial dimensions. We prove linear convergence of the algorithm on unbounded time intervals and superlinear convergence on bounded time intervals. In both cases the convergence rates are shown to depend on the size of the overlap. The linear convergence result depends also on the number of subdomains because it is limited by the classical steady state result of overlapping Schwarz for elliptic problems. However the superlinear convergence result is independent of the number of subdomains. Thus overlapping Schwarz waveform relaxation does not need a coarse space for robust convergence independent of the number of subdomains, if the algorithm is in the superlinear convergence regime. Numerical experiments confirm our analysis. We also briefly describe how our results can be extended to more general parabolic problems.     

Remarks on the optimal convolution kernel for CSOR waveform relaxation

The convolution SOR waveform relaxation method is a numerical method for solving large-scale systems of ordinary differential equations on parallel computers. It is similar in spirit to the SOR acceleration method for solving linear systems of algebraic equations, but replaces the multiplication with an overrelaxation parameter by a convolution with a time-dependent overrelaxation function. Its convergence depends strongly on the particular choice of this function. In this paper, an analytic expression is presented for the optimal continuous-time convolution kernel and its relation to the optimal kernel for the discrete-time iteration is derived. We investigate whether this analytic expression can be used in actual computations. Also, the validity of the formulae that are currently used to determine the optimal continuous-time and discrete-time kernels is extended towards a larger class of ODE systems.     

波形松弛算法及其在计算流体力学中的应用

本文介绍求解线性常系数微分代数方程组的波形松弛算法, 基于Laplace积分变换得到该算法新的收敛理论. 进一步将波形松弛算法应用于求解非定常Stokes方程, 介绍并讨论了连续时间波形松弛算法CABSOR算法和离散时间波形松弛算法DABSOR算法.     

Overlapping Schwarz waveform relaxation method for the solution of the reaction-diffusion equation

We are interested in solving time dependent problems using domain decomposition methods. In the classical approach, one discretizes first the time dimension and then one solves a sequence of steady problems by a domain decomposition method. In this article, we treat directly the time dependent problem and we study a Schwarz waveform relaxation algorithm for the convection diffusion equation. We study the convergence of the overlapping Schwarz waveform relaxation method for solving the reaction-diffusion equation over multi-overlapped subdomains. Also we will show that the method converges linearly and superlinearly over long and short time intervals, and the convergence depends on the size of overlap. Numerical results are presented from solutions of a specific model problems to demonstrate the convergence, linear and superlinear, and the role of the overlap size.     

Waveform relaxation method for stochastic differential equations with constant delay

This paper extends the waveform relaxation method to stochastic differential equations with constant delay terms, gives sufficient conditions for the mean square convergence of the method. A lot of attention is paid to the rate of convergence of the method. The conditions of the superlinear convergence for a special case, which bases on the special splitting functions, are given. The theory is applied to a one-dimensional model problem and checked against results obtained by numerical experiments.     

Remarks on waveform relaxation method with overlapping splittings

By studying the superlinear convergence of waveform relaxation method on finite time intervals, it has formerly been shown, by using the theory of quasinilpotent operators, that the convergence properties are largely determined by the graph properties of the splitting. In this paper, we show how the directed graphs associated to the decomposition are modified when overlapping splittings are used. In particular, we explain how overlapping should be used in order to best accelerate convergence of the iteration method.     

Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations

We investigate Chebyshev spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations. Waveform relaxation methods allow to replace the system of nonlinear differential equations resulting from the application of spectral collocation methods by a sequence of linear problems which can be effectively integrated in a parallel computing environment by highly stable implicit methods. The effectiveness of this approach is illustrated by numerical experiments on the Hutchinson's equation. The boundedness of waveform relaxation iterations is proved for the Hutchinson's equation. This result is used in the proof of the superlinear convergence of the iterations.     

抛物方程时间周期问题的有限元多格子动力学迭代

本文我们研究线性周期抛物方程的有限元多格子动力学迭代.多格子动力学迭代又称多重网格波形松弛,它是在函数空间中的一种迭代过程.对于由加速技术得到的多格子动力学迭代算子,我们通过计算周期函数的Fourier系数给出了新的谱表达式.从这些有用的表达式出发,我们推导了时间连续和离散格式的迭代收敛条件.数值实验进一步验证了本文的理论结果.     

Waveform Relaxation Methods for Lie-Group Equations

In this paper, we derive and analyse waveform relaxation (WR) methods for solving differential equations evolving on a Lie-group. We present both continuous-time and discrete-time WR methods and study their convergence properties. In the discrete-time case, the novel methods are constructed by combining WR methods with Runge-Kutta-Munthe-Kaas (RK-MK) methods. The obtained methods have both advantages of WR methods and RK-MK methods, which simplify the computation by decoupling strategy and preserve the numerical solution of Lie-group equations on a manifold. Three numerical experiments are given to illustrate the feasibility of the new WR methods.     

WAVEFORM RELAXATION METHODS OF NONLINEAR INTEGRAL-DIFFERENTIAL-ALGEBRAIC EQUATIONS

In this paper we consider continuous-time and discrete-time waveform relaxation meth-ods for general nonlinear integral-differential-algebraic equations. For the continuous-time case we derive the convergence condition of the iterative methods by invoking the spec-tral theory on the resulting iterative operators. By use of the implicit difference forms,namely the backward-differentiation formulae, we also yield the convergence condition of the discrete waveforms. Numerical experiments are provided to illustrate the theoretical work reported here.     

Incomplete projection algorithms for solving the convex feasibility problem

We present a general scheme for solving the convex feasibility problem and prove its convergence under mild conditions. Unlike previous schemes no exact projections are required. Moreover, we also introduce an acceleration factor, which we denote as the factor, that seems to play a fundamental role to improve the quality of convergence. Numerical tests on systems of linear inequalities randomly generated give impressive results in a multi-processing environment. The speedup is superlinear in some cases. New acceleration techniques are proposed, but no tests are reported here. As a by-product we obtain the rather surprising result that the relaxation factor, usually confined to the interval (0,2), gives better convergence results for values outside this interval.     

Waveform relaxation for reaction-diffusion equations

We report a new waveform relaxation (WR) algorithm for general semi-linear reaction-diffusion equations. The superlinear rate of convergence of the new WR algorithm is proved, and we also show the advantages of the new approach superior to the classical WR algorithms by the estimation on iteration errors. The corresponding discrete WR algorithm for reaction-diffusion equations is presented, and further the parallelism of the discrete WR algorithm is analyzed. Moreover, the new approach is extended to handle the coupled reaction-diffusion equations. Numerical experiments are carried out to verify the effectiveness of the theoretic work.     

An active-set projected trust-region algorithm with limited memory BFGS technique for box-constrained nonsmooth equations

In this article, by means of an active set and limited memory strategy, we propose a trust-region method for box-constrained nonsmooth equations. The global convergence and the superlinear convergence are established under suitable conditions.     

Minimax问题的一个滤子算法

本文提出一个求解不等式约束的Minimax问题的滤子算法,结合序列二次规划方法,并利用滤子以避免罚函数的使用.在适当的条件下,证明了此方法的全局收敛性及超线性收敛性.数值实验表明算法是有效的.     

Anderson accelerated fixed-point iteration for multilinear PageRank

In this paper, we apply the Anderson acceleration technique to the existing relaxation fixed-point iteration for solving the multilinear PageRank. In order to reduce computational cost, we further consider the periodical version of the Anderson acceleration. The convergence of the proposed algorithms is discussed. Numerical experiments on synthetic and real-world datasets are performed to demonstrate the advantages of the proposed algorithms over the relaxation fixed-point iteration and the extrapolated shifted fixed-point method. In particular, we give a strategy for choosing the quasi-optimal parameters of the associated algorithms when they are applied to solve the test problems with different sizes but the same structure.     

Limited-memory BFGS with displacement aggregation

A displacement aggregation strategy is proposed for the curvature pairs stored in a limited-memory BFGS (a.k.a. L-BFGS) method such that the resulting (inverse) Hessian approximations are equal to those that would be derived from a full-memory BFGS method. This means that, if a sufficiently large number of pairs are stored, then an optimization algorithm employing the limited-memory method can achieve the same theoretical convergence properties as when full-memory (inverse) Hessian approximations are stored and employed, such as a local superlinear rate of convergence under assumptions that are common for attaining such guarantees. To the best of our knowledge, this is the first work in which a local superlinear convergence rate guarantee is offered by a quasi-Newton scheme that does not either store all curvature pairs throughout the entire run of the optimization algorithm or store an explicit (inverse) Hessian approximation. Numerical results are presented to show that displacement aggregation within an adaptive L-BFGS scheme can lead to better performance than standard L-BFGS.

     

Overlapping Schwarz waveform relaxation method for the solution of the forward–backward heat equation

In this article we analyzed the convergence of the Schwarz waveform relaxation method for solving the forward–backward heat equation. Numerical results are presented for a specific type of model problem.     

A trust-region method with a conic model for unconstrained optimization

In this paper, we propose and analyze a new conic trust-region algorithm for solving the unconstrained optimization problems. A new strategy is proposed to construct the conic model and the relevant conic trust-region subproblems are solved by an approximate solution method. This approximate solution method is not only easy to implement but also preserves the strong convergence properties of the exact solution methods. Under reasonable conditions, the locally linear and superlinear convergence of the proposed algorithm is established. The numerical experiments show that this algorithm is both feasible and efficient. Copyright © 2008 John Wiley & Sons, Ltd.     

A Truncated Nonmonotone Gauss-Newton Method for Large-Scale Nonlinear Least-Squares Problems

In this paper, a Gauss-Newton method is proposed for the solution of large-scale nonlinear least-squares problems, by introducing a truncation strategy in the method presented in [9]. First, sufficient conditions are established for ensuring the convergence of an iterative method employing a truncation scheme for computing the search direction, as approximate solution of a Gauss-Newton type equation. Then, a specific truncated Gauss-Newton algorithm is described, whose global convergence is ensured under standard assumptions, together with the superlinear convergence rate in the zero-residual case. The results of a computational experimentation on a set of standard test problems are reported.     

Fourier-Laplace analysis of the multigrid waveform relaxation method for hyperbolic equations

The multigrid waveform relaxation (WR) algorithm has been fairly studied and implemented for parabolic equations. It has been found that the performance of the multigrid WR method for a parabolic equation is practically the same as that of multigrid iteration for the associated steady state elliptic equation. However, the properties of the multigrid WR method for hyperbolic problems are relatively unknown. This paper studies the multigrid acceleration to the WR iteration for hyperbolic problems, with a focus on the convergence comparison between the multigrid WR iteration and the multigrid iteration for the corresponding steady state equations. Using a Fourier-Laplace analysis in two case studies, it is found that the multigrid performance on hyperbolic problems no longer shares the close resemblance in convergence factors between the WR iteration for parabolic equations and the iteration for the associated steady state equations.