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.     

一种新的局部时间积分的区域分解波形松弛算法

提出一种新的区域分解波形松弛算法, 使得可以在不同的子域采用不同的时间步长来并行求解线性抛物方程的初边值问题. 与传统的区域分解波形松弛算法相比, 该算法可以通过预条件子来加快收敛速度, 并且对内存的需求大大降低. 给出了局部时间步长一种具体的实现方法, 证明了离散解的存在唯一性, 并在时间连续水平分析了预条件系统. 数值实验显示了新算法的有效性.     

Space-time domain decomposition for parabolic problems

We analyze a space-time domain decomposition iteration, for a model advection diffusion equation in one and two dimensions. The discretization of this iteration is the block red-black variant of the waveform relaxation method, and our analysis provides new convergence results for this scheme. The asymptotic convergence rate is super-linear, and it is governed by the diffusion of the error across the overlap between subdomains. Hence, it depends on both the size of this overlap and the diffusion coefficient in the equation. However it is independent of the number of subdomains, provided the size of the overlap remains fixed. The convergence rate for the heat equation in a large time window is initially linear and it deteriorates as the number of subdomains increases. The duration of the transient linear regime is proportional to the length of the time window. For advection dominated problems, the convergence rate is initially linear and it improves as the the ratio of advection to diffusion increases. Moreover, it is independent of the size of the time window and of the number of subdomains. Numerical calculations illustrate our analysis.     

Overlapping Schwarz waveform relaxation method for the solution of the convection–diffusion equation

In this article we study the convergence of the overlapping Schwarz wave form relaxation method for solving the convection–diffusion equation over multi-overlapped subdomains. It is shown that the method converges linearly and superlinearly over long and short time intervals, and the convergence depends on the size of the overlap. Numerical results are presented from solving specific types of model problems to demonstrate the convergence and the role of the size of the overlap. Copyright © 2007 John Wiley & Sons, Ltd.     

An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-diffusion problems

We consider a scalar advection-diffusion problem and a recently proposed discontinuous Galerkin approximation, which employs discontinuous finite element spaces and suitable bilinear forms containing interface terms that ensure consistency. For the corresponding sparse, nonsymmetric linear system, we propose and study an additive, two-level overlapping Schwarz preconditioner, consisting of a coarse problem on a coarse triangulation and local solvers associated to a family of subdomains. This is a generalization of the corresponding overlapping method for approximations on continuous finite element spaces. Related to the lack of continuity of our approximation spaces, some interesting new features arise in our generalization, which have no analog in the conforming case. We prove an upper bound for the number of iterations obtained by using this preconditioner with GMRES, which is independent of the number of degrees of freedom of the original problem and the number of subdomains. The performance of the method is illustrated by several numerical experiments for different test problems using linear finite elements in two dimensions.

     

Optimization of Schwarz waveform relaxation over short time windows

Schwarz waveform relaxation algorithms (SWR) are naturally parallel solvers for evolution partial differential equations. They are based on a decomposition of the spatial domain into subdomains, and a partition of the time interval of interest into time windows. On each time window, an iteration, during which subproblems are solved in space-time subdomains, is then used to obtain better and better approximations of the overall solution. The information exchange between subdomains in space-time is performed through classical or optimized transmission conditions (TCs). We analyze in this paper the optimization problem when the time windows are short. We use as our model problem the optimized SWR algorithm with Robin TCs applied to the heat equation. After a general convergence analysis using energy estimates, we prove that in one spatial dimension, the optimized Robin parameter scales like the inverse of the length of the time window, which is fundamentally different from the known scaling on general bounded time windows, which is like the inverse of the square root of the time window length. We illustrate our analysis with a numerical experiment.     

Neumann–Neumann waveform relaxation algorithm in multiple subdomains for hyperbolic problems in 1D and 2D

We present a Waveform Relaxation (WR) version of the Neumann–Neumann algorithm for the wave equation in space‐time. The method is based on a nonoverlapping spatial domain decomposition, and the iteration involves subdomain solves in space‐time with corresponding interface conditions, followed by a correction step. Using a Fourier‐Laplace transform argument, for a particular relaxation parameter, we prove convergence of the algorithm in a finite number of steps for the finite time intervals. The number of steps depends on the size of the subdomains and the time window length on which the algorithm is employed. We illustrate the performance of the algorithm with numerical results, followed by a comparison with classical and optimized Schwarz WR methods. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 514–530, 2017     

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.     

Discretized Multisplitting AOR Waveform Relaxation Algorithms for Initial Value Problem of Systems of ODEs

1.IntroductionInthedevelopmentofnewelectricalcircuits,thesimulationofthebehaviourofthecircuithasbecomeanessentialtoolforelectricalengineers.Fromthelayoutofthecircuitanonlinearsystemofordinarydifferentialequationsisgeneratedwhichdescribesthedynamicalbehaviourofthecircuit.Inthesimulationofverylargescaleintegrated(VLSI)circuitsthedimensionofthesystemofODEscanbecomeverylarge.Moreoversincethesystemisstiff,solvingthesesystemsisaverycomputionallyintensivetaskandtheuseofsupercomputersbecomesin-evit…     

Overlapping Schwarz preconditioners for spectral element methods in nonstandard domains and heterogeneous media

Overlapping Schwarz preconditioners are constructed and numerically studied for Gauss-Lobatto-Legendre (GLL) spectral element discretizations of heterogeneous elliptic problems on nonstandard domains defined by Gordon-Hall transfinite mappings. The results of several test problems in the plane show that the proposed preconditioners retain the good convergence properties of overlapping Schwarz preconditioners for standard affine GLL spectral elements, i.e. their convergence rate is independent of the number of subdomains, of the spectral degree in the case of generous overlap and of the discontinuity jumps in the coefficients of the elliptic operator, while in the case of small overlap, the convergence rate depends on the inverse of the overlap size.     

On the geometric convergence of optimized Schwarz methods with applications to elliptic problems

The Schwarz method can be used for the iterative solution of elliptic boundary value problems on a large domain Ω. One subdivides Ω into smaller, more manageable, subdomains and solves the differential equation in these subdomains using appropriate boundary conditions. Optimized Schwarz Methods use Robin conditions on the artificial interfaces for information exchange at each iteration, and for which one can optimize the Robin parameters. While the convergence theory of classical Schwarz methods (with Dirichlet conditions on the artificial interface) is well understood, the overlapping Optimized Schwarz Methods still lack a complete theory. In this paper, an abstract Hilbert space version of the Optimized Schwarz Method (OSM) is presented, together with an analysis of conditions for its geometric convergence. It is also shown that if the overlap is relatively uniform, these convergence conditions are met for Optimized Schwarz Methods for two-dimensional elliptic problems, for any positive Robin parameter. In the discrete setting, we obtain that the convergence factor ρ(h) varies like a polylogarithm of h. Numerical experiments show that the methods work well and that the convergence factor does not appear to depend on h.     

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.     

Piecewise homotopy analysis method and convergence analysis for formally well-posed initial value problems

In this paper, we apply the Schwarz waveform relaxation (SWR) method to the one-dimensional Schrödinger equation with a general linear or a nonlinear potential. We propose a new algorithm for the Schrödinger equation with time-independent linear potential, which is robust and scalable up to 500 subdomains. It reduces significantly computation time compared with the classical algorithms. Concerning the case of time-dependent linear potential or the nonlinear potential, we use a preprocessed linear operator for the zero potential case as a preconditioner which leads to a preconditioned algorithm. This ensures high scalability. In addition, some newly constructed absorbing boundary conditions are used as the transmission conditions and compared numerically.     

Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems,part I: Algorithms and numerical results

Summary We describe sequential and parallel algorithms based on the Schwarz alternating method for the solution of mixed finite element discretizations of elliptic problems using the Raviart-Thomas finite element spaces. These lead to symmetric indefinite linear systems and the algorithms have some similarities with the traditional block Gauss-Seidel or block Jacobi methods with overlapping blocks. The indefiniteness requires special treatment. The sub-blocks used in the algorithm correspond to problems on a coarse grid and some overlapping subdomains and is based on a similar partition used in an algorithm of Dryja and Widlund for standard elliptic problems. If there is sufficient overlap between the subdomains, the algorithm converges with a rate independent of the mesh size, the number of subdomains and discontinuities of the coefficients. Extensions of the above algorithms to the case of local grid refinement is also described. Convergence theory for these algorithms will be presented in a subsequent paper.This work was supported in part by the National Science Foundation under Grant NSF-CCR-8903003, while the author was a graduate student at New York University, and in part by the Army Research Office under Grant DAAL 03-91-G-0150, while the author was a Visiting Assistant Researcher at UCLA     

Convergence analysis of waveform relaxation methods for neutral differential-functional systems

In this paper, the problems of convergence and superlinear convergence of continuous-time waveform relaxation method applied to Volterra type systems of neutral functional-differential equations are discussed. Under a Lipschitz condition with time- and delay-dependent right-hand side imposed on the so-called splitting function, more suitable conditions about convergence and superlinear convergence of continuous-time WR method are obtained. We also investigate the initial interval acceleration strategy for the practical implementation of the continuous-time waveform relaxation method, i.e., discrete-time waveform relaxation method. It is shown by numerical results that this strategy is efficacious and has the essential acceleration effect for the whole computation process.     

A domain decomposition preconditioner for steady groundwater flow in porous media

In this paper an algorithm is presented based on the additive Schwarz method for steady groundwater flow in a porous medium. The subproblems in the algorithm correspond to the problem on a coarse grid and some overlapping subdomains. It will be shown that the rate of convergence is independent of the mesh parameters and discontinuities of the coefficients, and depends on the overlap ratio.     

A parameter-uniform Schwarz method for a coupled system of reaction–diffusion equations

We consider an arbitrarily sized coupled system of one-dimensional reaction–diffusion problems that are singularly perturbed in nature. We describe an algorithm that uses a discrete Schwarz method on three overlapping subdomains, extending the method in [H. MacMullen, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, A second-order parameter-uniform overlapping Schwarz method for reaction-diffusion problems with boundary layers, J. Comput. Appl. Math. 130 (2001) 231–244] to a coupled system. On each subdomain we use a standard finite difference operator on a uniform mesh. We prove that when appropriate subdomains are used the method produces ε-uniform results. Furthermore we improve upon the analysis of the above-mentioned reference to show that, for small ε, just one iteration is required to achieve the expected accuracy.     

Parametric Method for Global Optimization

This paper considers constrained and unconstrained parametric global optimization problems in a real Hilbert space. We assume that the gradient of the cost functional is Lipschitz continuous but not smooth. A suitable choice of parameters implies the linear or superlinear (supergeometric) convergence of the iterative method. From the numerical experiments, we conclude that our algorithm is faster than other existing algorithms for continuous but nonsmooth problems, when applied to unconstrained global optimization problems. However, because we solve 2n + 1 subproblems for a large number n of independent variables, our algorithm is somewhat slower than other algorithms, when applied to constrained global optimization.This work was partially supported by the NATO Outreach Fellowship - Mathematics 219.33.We thank Professor Hans D. Mittelmann, Arizona State University, for cooperation and support.     

Quadratic convergence of potential-reduction methods for degenerate problems

Global and local convergence properties of a primal-dual interior-point pure potential-reduction algorithm for linear programming problems is analyzed. This algorithm is a primal-dual variant of the Iri-Imai method and uses modified Newton search directions to minimize the Tanabe-Todd-Ye (TTY) potential function. A polynomial time complexity for the method is demonstrated. Furthermore, this method is shown to have a unique accumulation point even for degenerate problems and to have Q-quadratic convergence to this point by an appropriate choice of the step-sizes. This is, to the best of our knowledge, the first superlinear convergence result on degenerate linear programs for primal-dual interior-point algorithms that do not follow the central path. Received: February 12, 1998 / Accepted: March 3, 2000?Published online January 17, 2001     

无界区域上基于自然边界归化的一种重叠型区域分解法及其离散化

1．引言由于科学技术的迅猛发展，人们遇到许多大规模科学和工程计算问题．随着并行计算机的出现和应用，并行技术越来越得到人们的重视和研究．区域分解法成为并行计算和处理这类问题的主要方法之一．但是，对于无界区域上的椭圆边值问题，因进行区域分解后至少有一个区域仍为无界区域，故仅应用通常的区域分解算法求解是不够的．由于边界归化是处理无界区域问题的有效手段，通常采用边界元和有限元耦合的方法求解此类问题IZ，6。8。121．或片什适当的人工边界并在此边界上加近似边界条件，再在有限区域应用有限元方法求解【人习．近年来…