Graham Horton

Fluid stochastic Petri nets : Theory, applications, and solution techniques

In this paper we introduce a new class of stochastic Petri nets in which one or more places can hold fluid rather than discrete tokens. We define a class of fluid stochastic Petri nets in such a way that the discrete and continuous portions may affect each other. Following this definition we provide equations for their transient and steady-state behavior. We present several examples showing the utility of the construct in communication network modeling and reliability analysis, and discuss important special cases. We then discuss numerical methods for computing the transient behavior of such nets. Finally, some numerical examples are presented.

A space-time multigrid method for parabolic partial differential equations

We consider the solution of parabolic partial differential equations (PDEs). In standard time-stepping techniques multigrid can be used as an iterative solver for the elliptic equations arising at each discrete time step. By contrast, the method presented in this paper treats the whole of the space-time problem simultaneously. Thus the multigrid operations of smoothing and coarse-grid correction are defined on all of the space-time variables of a given grid level. The method is characterized by a coarsening strategy with prolongation and restriction operators which depend at each grid level on the degree of anisotropy of the discretization stencil. Numerical results for the one- and two-dimensional heat equations are presented and are shown to agree closely with predictions from Fourier mode analysis.

A multi-level diffusion method for dynamic load balancing

Abstract We consider the problem of dynamic load balancing for multiprocessors, for which a typical application is a parallel finite element solution method using non-structured grids and adaptive grid refinement. This type of application requires communication between the subproblems which arises from the interdependencies in the data. A load balancing algorithm should ideally not make any assumptions about the physical topology of the parallel machine. Further requirements are that the procedure should be fast and accurate. An new multi-level algorithm is presented for solving the dynamic load balancing problem which has these properties and whose parallel complexity is logarithmic in the number of processors used in the computation.

A multi-level solution algorithm for steady-state Markov chains

A new iterative algorithm, the multi-level algorithm, for the numerical solution of steady state Markov chains is presented. The method utilizes a set of recursively coarsened representations of the original system to achieve accelerated convergence. It is motivated by multigrid methods, which are widely used for fast solution of partial differential equations. Initial results of numerical experiments are reported, showing significant reductions in computation time, often an order of magnitude or more, relative to the Gauss-Seidel and optimal SOR algorithms for a variety of test problems. It is shown how the well-known iterative aggregation-disaggregation algorithm of Takahashi can be interpreted as a special case of the new method.

State space construction and steady-state solution of GSPNs on a shared-memory multiprocessor

A common approach for the quantitative analysis of a generalized stochastic Petri net (GSPN) is to generate its entire state space and then solve the corresponding continuous-time Markov chain (CTMC) numerically. This analysis often suffers from two major problems: the state space explosion and the stiffness of the CTMC. In this paper we present parallel algorithms for shared-memory machines that attempt to alleviate both of these difficulties: the large main memory capacity of a multiprocessor can be utilized and long computation times are reduced by efficient parallelization. The algorithms comprise both CTMC construction and numerical steady-state solution. We give experimental results obtained with a Convex SPP1600 shared-memory multiprocessor that show the behavior of the algorithms and the parallel speedups obtained.

Parallelization of robust multigrid methods: ILU factorization and frequency decomposition method

Using the anisotropic equation as a test problem, the concept of robustness is defined. Two multi-grid methods which are known to have this property are described: the standard multi-grid algorithm with ILU smoothing, and the frequency decomposition method. The parallelization on a MIMD computer is presented, together with results for the speedup obtained. The methods are compared with a standard parallel multi-grid algorithm using a Gaus-Seidel red-black smoother.

An algorithm with polylog parallel complexity for solving parabolic partial differential equations

The standard numerical algorithms for solving parabolic partial differential equations are inherently sequential in the time direction. This paper describes an algorithm for the time-accurate solution of certain classes of parabolic partial differential equations that can be parallelized in both time and space. It has a serial complexity that is proportional to the serial complexities of the best-known algorithms. The algorithm is a variant of the multigrid waveform relaxation method where the scalar ordinary differential equations that make up the kernel of computation are solved using a cyclic reduction-type algorithm. Experimental results obtained on a massively parallel multiprocessor are presented.

Fourier mode analysis of the multigrid waveform relaxation and time-parallel multigrid methods

The advent of parallel computers has led to the development of new solution algorithms for time-dependent partial differential equations. Two recently developed methods, multigrid waveform relaxation and time-parallel multigrid, have been designed to solve parabolic partial differential equations on many time-levels simultaneously. This paper compares the convergence properties of these methods, based on the results of an exponential Fourier mode analysis for a model problem.ZusammenfassungDie Erscheinung von Parallelrechnern hat zur Entwicklung neuer Lösungsverfahren for zeitabhängige partielle Differentialgleichungen geführt. Zwei der in letzter Zeit entwickelten Verfahren — die Mehrgitter-Wellenformrelaxations-Methode und die zeitparallele Mehrgittermethode —haben zum Ziel, die Lösung zu vielen verschiedenen diskreten Zeitpunkten simultan zu berechnen. In dieser Arbeit wird anhand der Ergebnisse einer Fourier-Analyse für ein Modell-problem das Konvergenzverhalten beider Methoden verglichen.

Parallel Shared-Memory State-Space Exploration in Stochastic Modeling

Stochastic modeling forms the basis for analysis in many areas, including biological and economic systems, as well as the performance and reliability modeling of computers and communication networks. One common approach is the state-space-based technique, which, starting from a high-level model, uses depth-first search to generate both a description of every possible state of the model and the dynamics of the transitions between them. However, these state spaces, besides being very irregular in structure, are subject to a combinatorial explosion, and can thus become extremely large. In the interest therefore of utilizing both the large memory capacity and the greater computational performance of modern multiprocessors, we are interested in implementing parallel algorithms for the generation and solution of these problems. In this paper we describe the techniques we use to generate the state space of a stochastic Petri-net model using shared-memory multiprocessors. We describe some of the problems encountered and our solutions, in particular the use of modified B-trees as a data structure for the parallel search process. We present results obtained from experiments on two different shared-memory machines.

A time-parallel multigrid-extrapolation method for parabolic partial differential equations

Abstract We consider the problem of solving unsteady partial differential equations on an MIMD machine. Conventional parallel methods use a data partitioning type approach in which the solution grid at each time-step is divided amongst the available processors. The sequential nature of the time integration is, however, retained. The algorithm presented in this paper makes use of a time-parallel approach, whreby several processors may be employed to solve at several time-steps simultaneously. The time-parallel method enables the inherent parallelism of the extrapolation scheme to be efficiently exploited, allowing a significant increase both in accuracy and in the degree of parallelism. The efficiencies obtained by an implementation on a message-passing multiprocessor demonstrate the suitability of the time-parallel extrapolation method for this type of equation.