From Massively Parallel Algorithms and Fluctuating Time Horizons to Nonequilibrium Surface Growth

Abstract

We study the asymptotic scaling properties of a massively parallel algorithm for discrete-event simulations where the discrete events are Poisson arrivals. The evolution of the simulated time horizon is analogous to a non-equilibrium surface. Monte Carlo simulations and a coarse-grained approximation indicate that the macroscopic landscape in the steady state is governed by the EdwardsWilkinson Hamiltonian. Since the efficiency of the algorithm corresponds to the density of local minima in the associated surface, our results imply that the algorithm is asymptotically scalable.PACS numbers: 89.80.+h, 02.70.Lq, 68.35.Ct To efficiently utilize modern supercomputers requires massively parallel implementations of dynamic algorithms for various physical, chemical, and biological processes. For many of these there are well-known and routinely used serial Monte Carlo (MC) schemes which are based on the realistic assumption that attempts to update the state of the system form a Poisson process. The parallel implementation of these dynamic MC algorithms belongs to the class of parallel discrete-event simulations, which is one of the most challenging areas in parallel computing [1] and has numerous applications not only in the physical sciences, but also in computer science, queueing theory, and economics. For example, in lattice Ising models the discrete events are spin-flip attempts, while in queueing systems they are job arrivals. Since current special-or multi-purpose parallel computers can have 10 4 − 10 5 processing elements (PE) [2], it is essential to understand and estimate the scaling properties of these algorithms.In this Letter we introduce an approach to investigate the asymptotic scaling properties of an extremely robust parallel scheme [3]. This parallel algorithm is applicable to a wide range of stochastic cellular automata with local dynamics, where the discrete events are Poisson arrivals. Although attempts have been made to estimate its efficiency under some restrictive assumptions [4], the mechanism which ensures the scalability of the algorithm in the "steady state" was never identified. Here we accomplish this by noting that the simulated time horizon is analogous to a growing and fluctuating surface. The local random time increments correspond to the deposition of random amounts of "material" at the local minima of the surface. This correspondence provides a natural ground for cross-disciplinary application of wellknown concepts from non-equilibrium surface growth [5] and driven systems [6] to a certain class of massively parallel computational schemes. To estimate the efficiency of this algorithm one must understand the morphology of the surface associated with the simulated time horizon. In particular, the efficiency of this parallel implementation (the fraction of the non-idling processing elements) exactly corresponds to the density of local minima in the surface model. We show that the steady-state behavior of the macroscopic landscape is governed by the EdwardsWilkinson (EW) Hamiltonian [7],...

Critical behavior of ising models with random long-range (small-world) interactions

“…One type of such graph is the small-world (SW) graph [5]. Such graphs have been used, for example, to improve scalability of parallel computer algorithms [6][7][8][9][10]. Furthermore, the critical behavior of models of materials, such as the Ising model, Heisenberg model, and random-walker models have been studied on SW graphs [11][12][13][14][15][16][17][18][19][20][21].…”

Small-World Synchronized Computing Networks for Scalable Parallel Discrete-Event Simulations

“…Thus, the average progress rate of the simulation approaches a nonzero constant in the asymptotic long-time, large-N limit. For example, for the one-site-per PE basic conservative PDES scheme u ∞ 0.2464 [25,26], due to non-universal short-range correlations between the local slopes [35]. The average width of the virtual time horizon, however, diverges as N →∞ [see (7)], making the measurement phase of the PDES scheme (data collection) not scalable [30].…”

“…(More general PDES schemes, where events to be processed by a PE are initiated (or generated) by the same PE (such as the basic conservative scheme above), are also referred to as self-initiating discrete-event schemes [23,24].) In the original algorithm, the virtual communication topology between PEs mimics the interaction topology of the underlying system [11,12,25]. When "simulating the simulations" based on the above simple "microscopic" rules for the evolution of the time horizon, we implemented periodic boundary conditions, i.e., the PEs are placed on a ring.…”

“…Clearly, the specific value 1/4 in the thermodynamic limit and the prefactor of the 1/N finite-size corrections will differ from those of the actual PDES evolution with its specific "microscopic dynamics". The density of local minima, however, must remain nonzero and it displays universal finite-size effects [25,26,[30][31][32][33][34],…”

“…Here we use a coarse-grained description for the virtual time horizon to perform the scalability analysis [25,26]. It was shown [25] that the virtual time horizon exhibits Kardar-Parisi-Zhang (KPZ)-like [27] kinetic roughening [28] and the steady-state behavior in one dimension is governed by the Edwards-Wilkinson (EW) Hamiltonian [29].…”