Pipeline implementation of cellular automata for structural design on message-passing multiprocessors

The inherent structure of cellular automata is trivially parallelizable and can directly benefit from massively parallel machines in computationally intensive problems. This paper presents both block synchronous and block pipeline (with asynchronous message passing) parallel implementations of cellular automata on distributed memory (message-passing) architectures. A structural design problem is considered to study the performance of the various cellular automata implementations. The synchronous parallel implementation is a mixture of Jacobi and Gauss–Seidel style iteration, where it becomes more Jacobi like as the number of processors increases. Therefore, it exhibits divergence because of the mathematical characteristics of Jacobi iteration matrix for the structural problem as the number of processors increases. The proposed pipeline implementation preserves convergence by simulating a pure Gauss–Seidel style row-wise iteration. Numerical results for analysis and design of a cantilever plate made of composite material show that the pipeline update scheme is convergent and successfully generates optimal designs.     

Parallel Asynchronous Global Search and the Nested Optimization Scheme

In this paper, a parallel asynchronous information algorithm for solving multidimensional Lipschitz global optimization problems, where times for evaluating the objective function can be different from point to point, is proposed. This method uses the nested optimization scheme and a new parallel asynchronous global optimization method for solving core univariate subproblems generated by the nested scheme. The properties of the scheme related to parallel computations are investigated. Global convergence conditions for the new method and theoretical conditions of speed up, which can be reached by using asynchronous parallelization in comparison with the pure sequential case, are established. Numerical experiments comparing sequential, synchronous, and asynchronous algorithms are also reported.     

Asynchronous multisplitting two-stage iterations for systems of weakly nonlinear equations

For the large sparse systems of weakly nonlinear equations arising in the discretizations of many classical differential and integral equations, this paper presents a class of asynchronous parallel multisplitting two-stage iteration methods for getting their solutions by the high-speed multiprocessor systems. Under suitable assumptions, we study the global convergence properties of these asynchronous multisplitting two-stage iteration methods. Moreover, for this class of new methods, we establish their local convergence theories, and precisely estimate their asymptotic convergence factors under some reasonable assumptions when the involved nonlinear mapping is only assumed to be directionally differentiable. Numerical computations show that our new methods are feasible and efficient for parallely solving the system of weakly nonlinear equations.     

Decomposition Methods for Differentiable Optimization Problems over Cartesian Product Sets

This paper presents a unified analysis of decomposition algorithms for continuously differentiable optimization problems defined on Cartesian products of convex feasible sets. The decomposition algorithms are analyzed using the framework of cost approx imation algorithms. A convergence analysis is made for three decomposition algorithms: a sequential algorithm which extends the classical Gauss-Seidel scheme, a synchronized parallel algorithm which extends the Jacobi method, and a partially asynchronous parallel algorithm. The analysis validates inexact computations in both the subproblem and line search phases, and includes convergence rate results. The range of feasible step lengths within each algorithm is shown to have a direct correspondence to the increasing degree of parallelism and asynchronism, and the resulting usage of more outdated information in the algorithms.     

Modulus-based synchronous two-stage multisplitting iteration methods for linear complementarity problems

In order to solve large sparse linear complementarity problems on parallel multiprocessor systems, we construct modulus-based synchronous two-stage multisplitting iteration methods based on two-stage multisplittings of the system matrices. These iteration methods include the multisplitting relaxation methods such as Jacobi, Gauss–Seidel, SOR and AOR of the modulus type as special cases. We establish the convergence theory of these modulus-based synchronous two-stage multisplitting iteration methods and their relaxed variants when the system matrix is an H _?+?-matrix. Numerical results show that in terms of computing time the modulus-based synchronous two-stage multisplitting relaxation methods are more efficient than the modulus-based synchronous multisplitting relaxation methods in actual implementations.     

A class of asynchronous multisplitting two-stage iterations for large sparse block systems of weakly nonlinear equations

For the block system of weakly nonlinear equations Ax=G(x), where is a large sparse block matrix and is a block nonlinear mapping having certain smoothness properties, we present a class of asynchronous parallel multisplitting block two-stage iteration methods in this paper. These methods are actually the block variants and generalizations of the asynchronous multisplitting two-stage iteration methods studied by Bai and Huang (Journal of Computational and Applied Mathematics 93(1) (1998) 13–33), and they can achieve high parallel efficiency of the multiprocessor system, especially, when there is load imbalance. Under quite general conditions that is a block H-matrix of different types and is a block P-bounded mapping, we establish convergence theories of these asynchronous multisplitting block two-stage iteration methods. Numerical computations show that these new methods are very efficient for solving the block system of weakly nonlinear equations in the asynchronous parallel computing environment.     

An inertial-like proximal algorithm for equilibrium problems

The paper concerns with an inertial-like algorithm for approximating solutions of equilibrium problems in Hilbert spaces. The algorithm is a combination around the relaxed proximal point method, inertial effect and the Krasnoselski–Mann iteration. The using of the proximal point method with relaxations has allowed us a more flexibility in practical computations. The inertial extrapolation term incorporated in the resulting algorithm is intended to speed up convergence properties. The main convergence result is established under mild conditions imposed on bifunctions and control parameters. Several numerical examples are implemented to support the established convergence result and also to show the computational advantage of our proposed algorithm over other well known algorithms.     

Domains of divergence of the USSOR method applied onp-cyclic matrices

Summary Using a recently derived classical type general functional equation, relating the eigenvalues of a weakly cyclic Jacobi iteration matrix to the eigenvalues of its associated Unsymmetric Successive Overrelaxation (USSOR) iteration matrix, we obtain bounds for the convergence of the USSOR method, when applied to systems with ap-cyclic coefficient matrix.     

影响阻尼牛顿法收敛性的两个重要参数

对阻尼牛顿算法作了适当的改进,证明了新算法的收敛性.基于新算法,运用计算机代数系统Matlab,研究了迭代次数k,参数对(μ,λ)与初值x0三者间的依赖关系,研究了病态问题在新算法下趋于稳定的渐变(瞬变)过程.数值结果表明:(1)阻尼牛顿迭代中,参数对(μ,λ)与迭代次数k间存在特有的非线性关系;(2)适当的参数对(μ,λ)与阻尼因子α的共同作用能够在迭代中大幅度地降低病态问题的Jacobi阵的条件数,使病态问题逐渐趋于稳定,从而改变原问题的收敛性与收敛速度.     

An Adaptive Partial Linearization Method for Optimization Problems on Product Sets

We consider a general class of composite optimization problems where the goal function is the sum of a smooth function and a non-necessary smooth convex separable function associated with some space partition, whereas the feasible set is a Cartesian product concordant to this partition. We suggest an adaptive version of the partial linearization method, which makes selective component-wise steps satisfying some descent condition and utilizes a sequence of control parameters. This technique is destined to reduce the computational expenses per iteration and maintain the basic convergence properties. We also establish its convergence rates and describe some examples of applications. Preliminary results of computations illustrate usefulness of the new method.     

EXPERIMENTAL STUDY OF THE ASYNCHRONOUS MULTISPLITTING RELAXATION METHODS FOR THE LINEAR COMPLEMENTARITY PROBLEMS

We study the numerical behaviours of the relaxed asynchronous multisplitting methods for the linear complementarity problems by solving some typical problems from practical applications on a real multiprocessor system. Numerical results show that the parallel multisplitting relaxation methods always perform much better than the corresponding sequential alternatives, and that the asynchronous multisplitting relaxation methods often outperform their corresponding synchronous counterparts. Moreover, the two-sweep relaxed multisplitting methods have better convergence properties than their corresponding one-sweep relaxed ones in the sense that they have larger convergence domains and faster convergence speeds. Hence, the asynchronous multisplitting unsymmetric relaxation iterations should be the methods of choice for solving the large sparse linear complementarity problems in the parallel computing environments.     

The extrapolated first order method for solving systems with complex eigenvalues

An extrapolated form of the basic first order stationary iterative method for solving linear systems when the associated iteration matrix possesses complex eigenvalues, is investigated. Sufficient (and necessary) conditions are given such that convergence is assured. An analytic determination of good (and sometimes optimum) values of the involved real parameter is presented in terms of certain bounds on the eigenvalues of the iteration matrix. The usefulness of the developed theory is shown through a simple application to the conventional Jacobi method.     

Block and asynchronous two-stage methods for mildly nonlinear systems

Block parallel iterative methods for the solution of mildly nonlinear systems of equations of the form are studied. Two-stage methods, where the solution of each block is approximated by an inner iteration, are treated. Both synchronous and asynchronous versions are analyzed, and both pointwise and blockwise convergence theorems provided. The case where there are overlapping blocks is also considered. The analysis of the asynchronous method when applied to linear systems includes cases not treated before in the literature. Received June 5, 1997 / Revised version received December 29, 1997     

矩阵多分裂迭代法的收敛条件

本文给出了求解非奇异线性方程组的矩阵多分裂并行迭代法的一些新的收敛结果.当系数矩阵单调和多分裂序列为弱正则分裂时,得到了几个与已有的收敛准则等价的条件,并且证明了异步迭代法在较弱条件下的收敛性.对于同步迭代,给出了与异步迭代不同且较为宽松的收敛条件.     

Asynchronous multisplitting iteration with different weighting schemes

The multisplitting iteration method was presented by O’Leary and White [5] for solving large sparse linear systems on parallel multiprocessor system. In this paper, we further set up an asynchronous variant for the multisplitting iteration method with different weighting schemes studied by White [8]. Moreover, we establish a general convergence criterion for asynchronous iteration framework, and then prove the convergence of the new asynchronous multisplitting iteration method with different weighting schemes by making use of this general criterion.     

Multisplitting Iteration Schemes for Solving a Class of Nonlinear Complementarity Problems

We consider several synchronous and asynchronous multisplitting iteration schemes for solving aclass of nonlinear complementarity problems with the system matrix being an H-matrix.We establish theconvergence theorems for the schemes.The numerical experiments show that the schemes are efficient forsolving the class of nonlinear complementarity problems.     

Perturbed fixed point iterative methods based on pattern structure

We introduce a new idea of algorithmic structure, called assigning algorithm, using a finite collection of a subclass of strictly quasi‐nonexpansive operators. This new algorithm allows the iteration vectors to take steps on a pattern which is based on a connected directed acyclic graph. The sequential, simultaneous, and string‐averaging methods for solving convex feasibility problems are the special cases of the new algorithm which may be used to reduce idle time of processors in parallel implementations. We give a convergence analysis for such algorithmic structure with perturbation. Also, we extend some existence results of the split common fixed point problem based on the new algorithm. The performance of the new algorithm is illustrated with numerical examples from computed tomography.     

Modulus‐based synchronous multisplitting iteration methods for linear complementarity problems

By an equivalent reformulation of the linear complementarity problem into a system of fixed‐point equations, we construct modulus‐based synchronous multisplitting iteration methods based on multiple splittings of the system matrix. These iteration methods are suitable to high‐speed parallel multiprocessor systems and include the multisplitting relaxation methods such as Jacobi, Gauss–Seidel, successive overrelaxation, and accelerated overrelaxation of the modulus type as special cases. We establish the convergence theory of these modulus‐based synchronous multisplitting iteration methods and their relaxed variants when the system matrix is an H ₊ ‐matrix. Numerical results show that these new iteration methods can achieve high parallel computational efficiency in actual implementations. Copyright © 2012 John Wiley & Sons, Ltd.     

Q-learning and policy iteration algorithms for stochastic shortest path problems

We consider the stochastic shortest path problem, a classical finite-state Markovian decision problem with a termination state, and we propose new convergent Q-learning algorithms that combine elements of policy iteration and classical Q-learning/value iteration. These algorithms are related to the ones introduced by the authors for discounted problems in Bertsekas and Yu (Math. Oper. Res. 37(1):66-94, 2012). The main difference from the standard policy iteration approach is in the policy evaluation phase: instead of solving a linear system of equations, our algorithm solves an optimal stopping problem inexactly with a finite number of value iterations. The main advantage over the standard Q-learning approach is lower overhead: most iterations do not require a minimization over all controls, in the spirit of modified policy iteration. We prove the convergence of asynchronous deterministic and stochastic lookup table implementations of our method for undiscounted, total cost stochastic shortest path problems. These implementations overcome some of the traditional convergence difficulties of asynchronous modified policy iteration, and provide policy iteration-like alternative Q-learning schemes with as reliable convergence as classical Q-learning. We also discuss methods that use basis function approximations of Q-factors and we give an associated error bound.