Asynchronous Methods in Scientific and Mathematical Computing

Kalyan S. Perumalla

The following is a small selection of publications that illustrate the wide range of areas in which asynchronous methods are being applied:

References

[1]        H. Karimabadi, J. Driscoll, Y. Omelchenko, and N. Omidi, "A New Asynchronous Methodology for Modeling of Physical Systems: Breaking the Curse of Courant Condition," Journal of Computational Physics, vol. 205, 2005.

[2]        A. Lew, J. E. Marsden, M. Ortiz, and M. West, "Asynchronous Variational Integrators," Archives for Rational Mechanics and Analysis, vol. 167, pp. 85-146, 2002.

[3]        Y. Tang, K. S. Perumalla, R. M. Fujimoto, H. Karimabadi, J. Driscoll, and Y. Omelchenko, "Optimistic Simulations of Physical Systems using Reverse Computation," SIMULATION: Transactions of The Society for Modeling and Simulation International, vol. 82, pp. 61-73, 2006.

[4]        A. H. Buss and P. J. Sanchez, "Simple Movement and Detection in Discrete Event Simulation," presented at Winter Simulation Conference, Orlando, FL, 2005.

[5]        J. Nutaro, "Parallel Discrete Event Simulation with Application to Continuous Systems," in Department of Electrical and Computer Engineering, vol. Ph.D. Tucson, AZ: University of Arizona, 2003, pp. 182.

[6]        B. Lubachevsky, "Several Unsolved Problems in Large-Scale Discrete Event Simulations," presented at Workshop on Parallel and Distributed Simulation, 1993.

[7]        S. Miller and S. Luding, "Event-driven Molecular Dynamics in Parallel," Journal of Computational Physics, vol. 193, pp. 306-316, 2004.

[8]        J. M. Esposito and V. Kumar, "An Asynchronous Integration and Event Detection Algorithm for Simulating Mult-Agent Hybrid Systems," ACM Transactioins on Modeling and Computer Simulation, vol. 14, pp. 363-388, 2004.

 

Abstracts

The abstracts of the preceding references are reproduced here (from open online sources) as an at-a-glance snapshot for reader convenience.

·         Sequential discrete event simulation approaches to electrostatic codes [1]

Abstract Computer simulation of many important complex physical systems has reached a plateau because most conventional techniques are ill equipped to deal with the multi-scale nature of such systems. The traditional technique to simulate physical systems modeled by partial differential equations consists of breaking the simulation domain into a spatial grid and then advancing the state of the system synchronously at regular discrete time intervals. This so-called time-driven (or time-stepped) simulation (TDS) has inherent inefficiencies such as the time step restriction imposed by a global CFL (Courant–Friedrichs–Levy) condition. There is ongoing research on introducing local time refinement (local versus global CFL) within the time-stepped methodology. Here, we propose an entirely different (asynchronous) simulation methodology which uses the spatial grid but the time advance is based on a discrete event-driven (as opposed to time-driven) approach. This new technique immediately offers several major advantages over TDS. First, it allows each part of the simulation, that is the individual cells in case of fluid simulations and individual particles within a cell in case of particle simulations, to evolve based on their own physically determined time scales. Second, unlike in the TDS where the system is updated faithfully in time based on a pre-selected user specified time step, here the role of time step is replaced by a threshold for local state change. In our technique, individual parts of the global simulation state are updated on a “need-to-be-done-only” basis. Individual parts of the simulation domain set their own time scales for change, which may vary in space as well as during the run. In a particle-in-cell (PIC) simulation, DES enables a self-adjusting temporal mesh for each simulation entity down to assigning an individual time step to each particle. In this paper, we illustrate this new technique via the example of a spacecraft charging in a neutral plasma due to injection of a charged beam particle from its surface and compare its performance with the traditional techniques. We find that even in one-dimension, the new DES technology can be more than 300 times faster than the traditional TDS. Aside from sheer performance advantages, the real power of this technique is in its inherent ability to adapt to the spatial inhomogeneity of the problem. This enables building intelligent algorithms where interaction of simulation entities (e.g., cells, particles) follow elementary rules set by the underlying physics laws. Finally, our extensions of this technique to other problems such as the solution of diffusion equation and electromagnetic codes are briefly discussed.

·         Asynchronous update approach to numerical integration, applied to material science [2]

Abstract We describe a new class of asynchronous variational integrators (AVI) for nonlinear elastodynamics. The AVIs are distinguished by the following attributes: (i) The algorithms permit the selection of independent time steps in each element, and the local time steps need not bear an integral relation to each other; (ii) the algorithms derive from a spacetime form of a discrete version of Hamilton's variational principle. As a consequence of this variational structure, the algorithms conserve local momenta and a local discrete multisymplectic structure exactly. To guide the development of the discretizations, a spacetime multisymplectic formulation of elastodynamics is presented. The variational principle used incorporates both configuration and spacetime reference variations. This allows a unified treatment of all the conservation properties of the system. A discrete version of reference configuration is also considered, providing a natural definition of a discrete energy. The possibilities for discrete energy conservation are evaluated. Numerical tests reveal that, even when local energy balance is not enforced exactly, the global and local energy behavior of the AVIs is quite remarkable, a property which can probably be traced to the symplectic nature of the algorithm.

·         Parallel simulation of physical systems [3]

Abstract Efficient computer simulation of complex physical phenomena has long been challenging due to their multiphysics and multiscale nature. In contrast to traditional time-stepped execution methods, the authors describe an approach using optimistic parallel discrete event simulation (PDES) and reverse computation techniques to execute plasma physics codes. They show that reverse computation-based optimistic parallel execution can significantly reduce the execution time of an example plasma simulation without requiring a significant amount of additional memory compared to conservative execution techniques. The authors describe an application-level reverse computation technique that is efficient and suitable for complex scientific simulations.

·         Discrete event approach to coverage crossing detection [4]

Abstract Many scenarios involving simulation require modeling movement and sensing. Traditionally, this has been done in a time-stepped manner, often because of a mistaken belief that using a pure discrete event approach is infeasible. This paper discusses how simple motion (linear, uniform, two-dimensional) and simple sensing can be modeled with a pure Discrete Event approach. We demonstrate that this approach is not only feasible, it is often more desirable from several standpoints.

·         Discrete event system specification for asynchronous simulation of continuous and/or hybrid systems [5]

Abstract Recent advances in discrete event modeling of continuous systems have emphasized the need for high performance simulation engines. This need is particularly acute when discrete event methods are applied to the numerical solution of partial differential equations. Accurate approximations can require thousands, or even millions, of cells. The corresponding requirements for memory and computing power can readily exceed what is available on a single processor computer. Discrete event simulations are characterized by asynchronous and irregular, random, or data dependent behavior. This makes parallel algorithm design particularly challenging. Known parallel discrete event simulation algorithms have been developed in terms of event and process oriented world views. In contrast to this, the Discrete Event System Specification (DEVS) forms the foundation of research into discrete event approximations of continuous systems. While event and process oriented models can be expressed in terms of the DEVS modeling formalism, there are DEVS models that do not seem to have an equivalent representation in the event or process oriented world views. This leaves open the question of how existing parallel discrete event simulation algorithms must be adapted in order to simulate DEVS models. In this dissertation, discrete simulation algorithms are built up from the basic definition of a discrete event system. The parallel algorithms that are developed through this approach are shown to operate correctly. To conclude this study, these algorithms are applied to producing numerical solutions of a hyperbolic conservation law (Sod's shock tube problem) and the wave equation.

·         Optimal packing problems with application to physical systems and communication networks [6]

Abstract Several mathematical and algorithmic problems that have arisen in discrete event simulations of large systems are described. The simulated systems belong to the areas of computational physics, queueing networks, and econometric models.

·         Event-driven molecular dynamics [7]

Abstract Although event-driven algorithms have been shown to be far more efficient than time-driven methods such as conventional molecular dynamics, they have not become as popular. The main obstacle seems to be the difficulty of parallelizing event-driven molecular dynamics. Several basic ideas have been discussed in recent years, but to our knowledge no complete implementation has been published yet. In this paper we present a parallel event-driven algorithm including dynamic load-balancing, which can be easily implemented on any computer architecture. To simplify matters our explanations refer to a basic multi-particle system of hard spheres, but can be extended easily to a wide variety of possible models.

·         Hybrid system simulation [8]

Abstract A simulation algorithm is presented for multi-agent hybrid systems---systems consisting of many sets of nonsmooth differential equations---such as systems involving multiple rigid bodies, vehicles, or airplanes. The differential equations are partitioned into coupled subsystems, called "agents"; and the conditions which trigger the discontinuities in the derivatives, called "events", may depend on the global state vector. Such systems normally require significant computational resources to simulate because a global time step is used to ensure the discontinuity is properly handled. When the number of systems is large, forcing all system to be simulated at the same rate creates a computational bottleneck, dramatically decreasing efficiency. By using a control systems approach for selecting integration step sizes, we avoid using a global time step. Each subsystem can be simulated asynchronously when the state is away from the event. As the state approaches the event, the simulation is able to synchronize each of the local time clocks in such a way that the discontinuities are properly handled without the need for "roll back". The algorithm's operation and utility is demonstrated on an example problem inspired by autonomous highway vehicles. Using a combination of stochastic modelling and numerical experiments we show that the algorithm requires significantly less computation time when compared with traditional simulation techniques for such problems, and scales more favorably with problem size.