Monte-Carlo parallelization algorithm

Parallelization of computer algorithms for multidimensional problems of computational physics is considered. The focus is on efficient data processing by computer systems with a large number of concurrent processors, allowing for data exchange time. The computing time is minimized by a Monte-Carlo technique.Translated from Matematicheskie Modeli Estestvoznaniya, Published by Moscow University, Moscow, 1995, pp. 150–156.     

Parallel algorithms for solving large linear systems

The solution of linear systems continues to play an important role in scientific computing. The problems to be solved often are of very large size, so that solving them requires large computer resources. To solve these problems, at least supercomputers with large shared memory or massive parallel computer systems with distributed memory are needed.

This paper gives a survey of research on parallel implementation of various direct methods to solve dense linear systems. In particular are considered: Gaussian elimination, Gauss-Jordan elimination and a variant due to Huard (1979), and an algorithm due to Enright (1978), designed in relation to solving (stiff) ODEs, such that stepsize and other method parameters can easily be varied.

Some theoretical results are mentioned, including a new result on error analysis of Huard's algorithm. Moreover, practical considerations and results of experiments on supercomputers and on a distributed-memory computer system are presented.     

Computer Algebra Algorithms for Control Related Tests of Implicit Dynamic Systems

This contribution is focused on control related tests for implicit dynamic systems, like accessibility, observability or input to output, input to state linearizability. Since the performance of these tests needs tedious symbolic calculations, computer algebra systems are the ideal tool to cope with this problem. Accessibility and observability are exemplarily used to present a new approach based on Lie groups. It is shown that non accessible or non observable systems admit Lie‐groups acting on their solutions such that distinguished parts remain unchanged. This fact allows us to apply this technique, as well as its realization by computer algebra algorithm, to several fundamental problems in control.     

The multiple pendulum problem via Maple®

The way in which computer algebra systems, such as Maple, have made the study of physical problems of some considerable complexity accessible to mathematicians and scientists with modest computational skills is illustrated by solving the multiple pendulum problem. A solution is obtained for four pendulums with no restriction on the size of the angles through which they move, and for as many as a hundred pendulums if the angles remain sufficiently small. An attractive and informative graphical display of the motion in animated form can be obtained from the solution, and the effect of varying the initial conditions can be investigated visually.     

On a parallel multilevel solver for linear elasticity problems

In this paper an application of the additive multilevel iteration method to parallel solving of large‐scale linear elasticity problems is considered. The results are derived in the framework of the hierarchical basis finite element discretization defined on a tensor product of one‐dimensional grid and a sequence of nested triangulations . The algorithm was tested on a number of model problems, arising from bridge foundation modeling. Parallel performance of the solver is reported for Cray T3E‐600 and Sun ES/4000 computer systems. Copyright © 1999 John Wiley & Sons, Ltd.     

Rotationless formulation for large deformations in flexible multibody dynamics

Especially for specific applications, such as contact problems, computer methods for flexible multibody dynamics that are able to treat large deformation phenomena are important. Classical formalisms for multibody dynamics are based on rigid bodies. Their extension to flexible multibody systems is typically restricted to linear elastic material behavior whereas large deformation phenomena are formulated in the framework of the nonlinear finite element method. In the talk we address computer methods that can handle large deformations in the context of multibody systems. In particular, the link between nonlinear continuum mechanics and multibody systems is facilitated by a specific formulation of rigid body dynamics [1]. It makes possible the incorporation of state-of-the-art computer methods for large deformation problems. In the talk we focus on the treatment of large deformation contact whithin flexible multibody dynamics [2]. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical solution of Volterra integro-differential equations modelling thalamo-cortical systems

Our study concerns thalamo-cortical systems which are modelled by nonlinear systems of Volterra integro-differential equations of convolution type. The thalamo-cortical systems describe a new architecture for a neurocomputer. Such a computer employs principles of human brain. It consists of oscillators which have different frequencies and are weakly connected via a common medium forced by an external input. Since a neurocomputer consists of many interconnected oscillators (referred also as neurons), the thalamo-cortical systems include large numbers of Volterra integro-differential equations. Solving such systems numerically is expensive not only because of their large dimensions but also because of many kernel evaluations which are needed over the whole interval from the initial point, where the initial condition is imposed, up to the present point, where the computations are currently executed. Moreover, the whole computed history of the solution has to be stored in the memory of the computing machine. Therefore, robust and efficient numerical algorithms are needed for computer simulations for the solutions to the thalamocortical systems. In this paper, we illustrate an iteration technique to solve the thalamo-cortical systems. The proposed successive iterates are vector functions of time, which change the original problems into systems of easier and separated equations. Such separated equations can then be solved in parallel computing environments. Results of numerical experiments are presented for large numbers of oscillators. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Output feedback stabilization for a class of nonlinear time-evolution systems

A variety of problems in nonlinear time-evolution systems such as communication networks, computer networks, manufacturing, traffic management, etc., can be modelled as min–max-plus systems in which operations of min, max and addition appear simultaneously. Systems with only maximum (or minimum) constraints can be modelled as max-plus system and handled by max-plus algebra which changes the original nonlinear system in the traditional sense into linear system in this framework. Min-max-plus systems are extensions of max-plus systems and nonlinear even in the max-plus algebra view. Output feedback stabilization for min–max-plus systems with min–max-plus inputs and max-plus outputs is considered in this paper. Max-plus projection representation for the closed-loop system with min–max-plus output feedback is introduced and the formula to calculate the cycle time is presented. Stabilization of reachable systems with at least one observable state and a further result for reachable and observable systems are worked out, during which max-plus output feedbacks are used to stabilize the systems. The method based on the max-plus algebra is constructive in nature.     

Topological design of telecommunication networks-local access design methods

Computer communication networks and telecommunication systems are growing at an explosive rate. Some of the major factors influencing this phenomenal growth rate have been technology driven, deregulation of the telecommunication industry and the breakup of AT&T, product and service introductions and competition, new application areas, price reductions and improved services. Corporations have discovered how to use telecommunication-based systems and computer networks as a strategic competitive weapon. Modern computer networks consist of backbone networks which serve as major highways to transfer large volumes of communication traffic, and local access networks which feed traffic between the backbone network and end user nodes. The design of the local access network is a complex process which builds on many difficult combinatorial optimization problems. This paper surveys many of the problems, presents the state of the art in solving them, and demonstrates a variety of solution procedures. The paper concludes with a list of open problems and areas open for further investigation.This research was partially supported by a Dean's grant for faculty research at the Owen Graduate School of Management, Vanderbilt University, Nashville, TN 37204, USA.     

Grouping of parts and components in flexible manufacturing systems

In this paper, the problem of grouping of parts and components in flexible manufacturing systems is discussed. The actual grouping is done by modelling the problem as an optimal k-decomposition of weighted networks. Algorithms which are suitable for computer implementation and large problems are developed to find an initial solution and for refining this solution. Bounds on algorithm performance are constructed to give an estimate of the quality of the generated solution. A numerical example illustrates these new techniques.     

关于极小极大系统的周期时间配置域

极大极小函数描述的系统常见于计算机科学、控制论、运筹学等领域 .考虑极小极大系统的周期时间能用状态反馈独立配置的区域问题 ,得到了确定系统配置域最重要子域的构造方法 ,并给出了一个例子 .     

Fast transforms for tridiagonal linear equations

In this paper we study the use of the Fourier, Sine and Cosine Transform for solving or preconditioning linear systems, which arise from the discretization of elliptic problems. Recently, R. Chan and T. Chan considered circulant matrices for solving such systems. Instead of using circulant matrices, which are based on the Fourier Transform, we apply the Fourier and the Sine Transform directly. It is shown that tridiagonal matrices arising from the discretization of an onedimensional elliptic PDE are connected with circulant matrices by congruence transformations with the Fourier or the Sine matrix. Therefore, we can solve such linear systems directly, using only Fast Fourier Transforms and the Sherman-Morrison-Woodbury formula. The Fast Fourier Transform is highly parallelizable, and thus such an algorithm is interesting on a parallel computer. Moreover, similar relations hold between block tridiagonal matrices and Block Toeplitz-plus-Hankel matrices of ordern ²×n ² in the 2D case. This can be used to define in some sense natural approximations to the given matrix which lead to preconditioners for solving such linear systems.     

Survey of computer programs for heat transfer analysis

An overview is given of the current capabilities of thirty-three computer programs that are used to solve heat transfer problems. The programs considered range from large general-purpose codes with broad spectrum of capabilities, large user community, and comprehensive user support (e.g., ABAQUS, ANSYS, EAL, MARC, MITAS II, MSC/NASTRAN, and SAMCEF) to the small, special-purpose codes with limited user community such as ANDES, NTEMP, TAC2D, TAC3D, TEPSA and TRUMP. The majority of the programs use either finite elements or finite differences for the spatial discretization. The capabilities of the programs are listed in tabular form followed by a summary of the major features of each program. The information presented herein is based on a questionnaire sent to the developers of each program. This information is preceded by a brief background material needed for effective evaluation and use of computer programs for heat transfer analysis.The present survey is useful in the initial selection of the programs which are most suitable for a particular application. The final selection of the program to be used should, however, be based on a detailed examination of the documentation and the literature about the program.     

A class of discrete time generalized Riccati equations

In this paper, a class of discrete-time backward non-linear equations defined on some ordered Hilbert spaces of symmetric matrices is considered. The problem of the existence of some global solutions is investigated. The class of considered discrete-time non-linear equations contains, as special cases, a great number of difference Riccati equations both from the deterministic and the stochastic framework. The results proved in the paper provide the sets of necessary and sufficient conditions that guarantee the existence of some special solutions of the considered equations as: the maximal solution, the stabilizing solution and the minimal positive semi-definite solution. These conditions are expressed in terms of the feasibility of some suitable systems of linear matrix inequalities (LMI). One shows that in the case of the equations with periodic coefficients to verify the conditions that guarantee the existence of the maximal or the stabilizing solution, we have to check the solvability of some systems of LMI with a finite number of inequations. The proofs are based on some suitable properties of discrete-time linear equations defined by the positive operators on some ordered Hilbert spaces chosen adequately. The results derived in this paper provide useful conditions that guarantee the existence of the maximal solution or the stabilizing solution for different classes of difference matrix Riccati equations involved in many problems of robust control both in the deterministic and the stochastic framework. The proofs are deterministic and are accessible to the readers less familiarized with the stochastic reasonings.     

Square-root variable-metric methods for minimization

Variable-metric methods are presented which do not need an accurate one-dimensional search and eliminate roundoff error problems which can occur in updating the metric for large-dimension systems. The methods are based on updating the square root of the metric, so that a positive-definite metric always results. The disadvantage of intentionally relaxing the accuracy of the one-dimensional search is that the number of iterations (and hence, gradient evaluations) increases. For problems involving a large number of variables, the square-root method is presented in a triangular form to reduce the amount of computation. Also, for usual optimization problems, the square-root procedure can be carried out entirely in terms of the metric, eliminating storage and computer time associated with computations of the square root of the metric.     

一类高维生物数学模型小扰动下的Hopf分支

讨论一类六维生物数学模型,在［１,２］的基础上,进一步考虑生物种群之间的相互作用,引进扰动项,研究了中心流形及Ｈｏｐｆ分支问题．利用计算机代数符号软件计算了中心流形,得到了产生Ｈｏｐｆ分支的判定条件,并给出了数值算例,所用机器证明的思想方法可广泛运用于类似复杂生物数学模型的研究．     

Dynamics of controlled hybrid systems of aerial cable-ways

The dynamics of the hybrid systems of aerial cable-ways is investigated. The eigenvalue problems are considered for such hybrid systems with different assumptions. An overview of different methods for eigenvalue problems is given. In the research, the method of normal fundamental systems is applied, which turns out to be very effective for the considered problems. Changes in the dynamical characteristics of the systems depending on the controlled parameter are studied.     

Modeling of Galactic Wind Formation from Supernovae Using High-Performance Computations

Fundamental processes in the dynamics of the interstellar medium, namely, galactic wind, i.e., ejections of interstellar matter from central regions of galaxies, which are presumably caused by the formation of supernovae, are mathematically modeled in detail on high-performance parallel computer systems. The mathematical simulation is based on a kinetically consistent gasdynamic approach developed for such class of problems in astrophysics. A kinetically consistent algorithm is well adapted to the architecture of high-performance computer systems with massive parallelism, so that complex large-scale astrophysical phenomena can be efficiently studied with a high resolution. The approach, method, and algorithms are described, and numerical results are presented.

     

The Impact of Operational Research on Agriculture

The paper shows that agriculture is one of the United Kingdom's largest industries. It would therefore be expected that O.R. could have made a significant contribution to decision making. But achievements in practice have been disappointingly small. The industry comprises of a large number of small individual businesses which do not permit specialisation in management functions. Consequently, technical advice and much R and D is provided from public funds. O.R. applications for agriculture have mainly been developed by Universities, Colleges, State Advisory Services and QUANGOS.The paper discusses some techniques used in agriculture—linear programming, dynamic programming and simulation—and outlines some problems encountered with these. Other techniques have had limited uptake and application. Reasons for the disappointing impact of O.R. are discussed as a set of problems-those specific to farmers and their systems; those specific to computer use; problems in recruiting and training O.R. specialists and problems in communication.