Parallel Update Research Articles

Steady-states and kinetics of ordering in bus-route models: connection with the Nagel-Schreckenberg model

A Bus Route Model (BRM) can be defined on a one-dimensional lattice, where buses are represented by “particles” that are driven forward from one site to the next with each site representing a bus stop. We replace the random sequential updating rules in an earlier BRM by parallel updating rules. In order to elucidate the connection between the BRM with parallel updating (BRMPU) and the Nagel-Schreckenberg (NaSch) model, we propose two alternative extensions of the NaSch model with space-/time-dependent hopping rates. Approximating the BRMPU as a generalization of the NaSch model, we calculate analytically the steady-state distribution of the time headways (TH) which are defined as the time intervals between the departures (or arrivals) of two successive particles (i.e., buses) recorded by a detector placed at a fixed site (i.e., bus stop) on the model route. We compare these TH distributions with the corresponding results of our computer simulations of the BRMPU, as well as with the data from the simulation of the two extended NaSch models. We also investigate interesting kinetic properties exhibited by the BRMPU during its time evolution from random initial states towards its steady-states.

Conserved Mass Models and Particle Systems in One Dimension

In this paper we study analytically a simple one-dimensional model of mass transport. We introduce a parameter p that interpolates between continuous-time dynamics (p→0 limit) and discrete parallel update dynamics (p=1). For each p, we study the model with (i) both continuous and discrete masses and (ii) both symmetric and asymmetric transport of masses. In the asymmetric continuous mass model, the two limits p=1 and p→0 reduce respectively to the q-model of force fluctuations in bead packs [S. N. Coppersmith et al., Phys. Rev. E 53:4673 (1996)] and the recently studied asymmetric random average process [J. Krug and J. Garcia, cond-mat/9909034]. We calculate the steady-state mass distribution function P(m) assuming product measure and show that it has an algebraic tail for small m, P(m)∼m −β , where the exponent β depends continuously on p. For the asymmetric case we find β(p)=(1−p)/(2−p) for 0≤p<1 and β(1)=−1, and for the symmetric case, β(p)=(2−p)2/(8−5p+p 2) for all 0≤p≤1. We discuss the conditions under which the product measure ansatz is exact. We also calculate exactly the steady-state mass–mass correlation function and show that while it decouples in the asymmetric model, in the symmetric case it has a nontrivial spatial oscillation with an amplitude decaying exponentially with distance.

Parallelizable Bayesian tomography algorithms with rapid, guaranteed convergence

Bayesian tomographic reconstruction algorithms generally require the efficient optimization of a functional of many variables. In this setting, as well as in many other optimization tasks, functional substitution (FS) has been widely applied to simplify each step of the iterative process. The function to be minimized is replaced locally by an approximation having a more easily manipulated form, e.g., quadratic, but which maintains sufficient similarity to descend the true functional while computing only the substitute. We provide two new applications of FS methods in iterative coordinate descent for Bayesian tomography. The first is a modification of our coordinate descent algorithm with one-dimensional (1-D) Newton-Raphson approximations to an alternative quadratic which allows convergence to be proven easily. In simulations, we find essentially no difference in convergence speed between the two techniques. We also present a new algorithm which exploits the FS method to allow parallel updates of arbitrary sets of pixels using computations similar to iterative coordinate descent. The theoretical potential speed up of parallel implementations is nearly linear with the number of processors if communication costs are neglected.

Associative database of protein sequences.

We present a new concept that combines data storage and data analysis in genome research, based on an associative network memory. As an illustration, 115 000 conserved regions from over 73 000 published sequences (i.e. from the entire annotated part of the SWISSPROT sequence database) were identified and clustered by a self-organizing network. Similarity and kinship, as well as degree of distance between the conserved protein segments, are visualized as neighborhood relationship on a two-dimensional topographical map. Such a display overcomes the restrictions of linear list processing and allows local and global sequence relationships to be studied visually. Families are memorized as prototype vectors of conserved regions. On a massive parallel machine, clustering and updating of the database take only a few seconds; a rapid analysis of incoming data such as protein sequences or ESTs is carried out on present-day workstations. Access to the database is available at http://www.bioinf.mdc-berlin.de/unter2.html++ + (hanke,lehmann,reich)@mdc-berlin.de; bork@embl-heidelberg.de

Directed percolation universality in asynchronous evolution of spatiotemporal intermittency

We present strong evidence that a coupled-map-lattice model for spatiotemporal intermittency belongs to the universality class of directed percolation when the updating rules are asynchronous, i.e. when only one randomly chosen site is evolved at each time step. In contrast, when the system is subjected to parallel updating, available numerical evidence suggest that it does not belong to this universality class and that it is not even universal. We argue that in the absence of periodic external forcing, the asynchronous rule is the more physical.

A hybrid neural optimization scheme based on parallel updates

A synchronous Hopfield-type neural network model containing units with analog input and binary output, which is suitable for parallel implementation, is examined in the context of solving discrete optimization problems. A hybrid parallel update scheme concerning the stochastic input-output behaviour of each unit is presented. This parallel update scheme maintains the solution quality of the Boltzmann Machine optimizer, which is inherently sequential. Experimental results on the Maximum Independent Set problem demonstrate the benefit of using the proposed optimizer in terms of computation time. Excellent speedup has been obtained through parallel implementation on both shared memory and distributed memory architectures.

The asymmetric exclusion process: Comparison of update procedures

The asymmetric exclusion process (ASEP) has attracted a lot of interest not only because of its many applications, e.g., in the context of the kinetics of biopolymerization and traffic flow theory, but also because it is a paradigmatic model for nonequilibrium systems. Here we study the ASEP for different types of updates, namely random-sequential, sequential, sublattice-parallel, and parallel. In order to compare the effects of the different update procedures on the properties of the stationary state, we use large-scale Monte Carlo simulations and analytical methods, especially the so-called matrix-product Ansatz (MPA). We present in detail the exact solution for the model with sublattice-parallel and sequential updates using the MPA. For the case of parallel update, which is important for applications like traffic flow theory, we determine the phase diagram, the current, and density profiles based on Monte Carlo simulations. We furthermore suggest an MPA for that case and derive the corresponding matrix algebra.

Self-organized criticality in 1D stochastic traffic flow model

Results of computer simulations of a 1D particle hopping model of traffic flow are presented. The model is characterized by parallel update and fully asymmetric stochastic hopping dynamics which allows unbounded series of jumps to empty neighbour sites on the right. The considered case of open boundary conditions can be used to model a “bottleneck” situation in traffic. Evidence for self-organized criticality is found in two aspects: the presence of long-range spatial correlations manifested in the shape of density profiles, and long-time temporal correlations showing up in the low-frequency behaviour of the spectral density of the total particle number and flow. A plausible conjecture is to interpret the observed qualitative changes in these features, as a function of the injection rate and the hopping probability, in terms of a nonequilibrium phase transition between a low-density phase and a maximal current phase. This conjecture is supported by the phase diagram obtained in mean-field approximation.

Island density in homoepitaxial growth: improved Monte Carlo results

We reexamine the density of two dimensional islands in the submonolayer regime of a homoepitaxially growing surface using the coarse grained Monte Carlo simulation with random sequential updating rather than parallel updating. It turns out that the power law dependence of the density of islands on the deposition rate agrees much better with the theoretical prediction than previous data obtained by other methods if random sequential instead of parallel updating is used.

Exact steady states of disordered hopping particle models with parallel and ordered sequential dynamics

A one-dimensional driven lattice gas with disorder in the particle hopping probabilities is considered. It has previously been shown that in the version of the model with random sequential updating, a phase transition occurs from a low density inhomogeneous phase to a high density congested phase. Here the steady states for both parallel (fully synchronous) updating and ordered sequential updating are solved exactly and the phase transition shown to persist in both cases. For parallel dynamics and forward ordered sequential dynamics the phase transition occurs at the same density but for backward ordered sequential dynamics it occurs at a higher density. In both cases the critical density is higher than that for random sequential dynamics. In all the models studied the steady state velocity is related to the fugacity of a Bose system suggesting a principle of minimisation of velocity. A generalisation of the dynamics where the hopping probabilities depend on the number of empty sites in front of the particles, is also solved exactly in the case of parallel updating. The models have natural interpretations as simplistic descriptions of traffic flow. The relation to more sophisticated traffic flow models is discussed.

Partially decoupled Volterra filters: formulation and LMS adaptation

The adaptation of Volterra filters by one particular method-the method of least mean squares (LMS)-while easily implemented, is complicated by the fact that upper hounds for the values of step sizes employed by a parallel update LMS scheme are difficult to obtain. In this paper, we propose a modification of the Volterra filter in which the filter weights of a given order are optimized independently of those weights of higher order. Using this approach, we then solve the minimum mean square error (MMSE) filtering problem as a series of constrained optimization problems, which produce a partially decoupled normal equation for the Volterra filter. From this normal equation, we are able to develop an adaptation routine that uses the principles of partial decoupling that is similar in form to the Volterra LMS algorithm but with important structural differences that allow a straightforward derivation of bounds on the algorithm's step sizes; these bounds can be shown to depend on the respective diagonal blocks of the Volterra autocorrelation matrix. This produces a reliable set of design guidelines that allow more rapid convergence of the lower order weight sets.

Parallel simulations in ferromagnetic spin systems

We study the thermodynamical limits that are obtained from a parallel updating of ferromagnetic spins on a lattice. We investigate the relationships existing between the parallel model and the sequential model. We compare both of their free energy functions. A long-range model for which the mean-field theory is correct is also studied. The last part of the article indicates how parallel updating can be efficiently used in short-range model simulations.

Computing with truly asynchronous threshold logic networks

We present simulation mechanisms by which any network of threshold logic units with either symmetric or asymmetric interunit connections (i.e. a symmetric or asymmetric “Hopfield net”) can be simulated on a network of the same type, but without any a priori constraints on the order of updates of the units. Together with earlier constructions, the results show that the truly asynchronous network model is computationally equivalent to the seemingly more powerful models with either ordered sequential or fully parallel updates.

Discrete State Neural Networks and Energies

In this paper we give under an appropriate theoretical framework a characterization about neural networks (evolving in a binary set of states) which admit an energy. We prove that a neural network, iterated sequentially, admits an energy if and only if the weight matrix verifies two conditions: the diagonal elements are non-negative and the associated incidence graph does not admit non-quasi-symmetric circuits. In this situation the dynamics are robust with respect to a class of small changes of the weight matrix. Further, for the parallel update we prove that a necessary and sufficient condition to admit an energy is that the incidence graph does not contain non-quasi-symmetric circuits. © 1997 Elsevier Science Ltd. All Rights Reserved.

Universality of the chip-firing game

We prove that the parallel updating of the chip-firing game on undirected graphs is universal. To achieve that, we simulate any given two-register machine by chip configurations. As corollaries, we prove that for finite graphs there exists exponential transient time to reach periodic configurations as well as exponential cycles. Also, we prove, for infinite graphs, that the reachability problem is undecidable.

SELF-ORGANIZED CRITICALITY IN 1D TRAFFIC FLOW

Annihilation process of traffic jams is investigated in a one-dimensional traffic flow on a highway. The one-dimensional fully asymmetric exclusion model with open boundaries for parallel update is extended to take into account stochastic transition of cars, where a car moves ahead with transition probability pt. Near pt=1, the system is driven asymptotically into a steady state exhibiting a self-organized criticality. Traffic jams with various lifetimes (or sizes) appear and disappear by colliding with an empty wave. The typical lifetime &lt;m&gt; of traffic jams scales as [Formula: see text], where ∆pt=1−pt. It is shown that the cumulative lifetime distribution Nm(∆pt) satisfies the scaling form [Formula: see text].

Cellular Automata for Elementary Image Enhancement

We study various cellular automata as algorithms for elementary image enhancement, which refers to methods used to improve features of an image without previous information about them that can be implemented by straightforward techniques. Cellular automata appear as natural tools for image processing due to their local nature and simple parallel computer implementation. For this reason various cellular automata algorithms for sharpening and smoothing are presented and studied in this context. Their dynamical behavior is characterized for sequential and parallel updating by associating to them strictly decreasing functionals with the dynamics of the automata. Since tight bounds for the transient time usually cannot be obtained from these operators, a numerical study is needed to analyze their typical performance and effects. For this purpose we compare them with the classical methods for real two dimensional images in terms of convergence rate, effects, and stability in front of noise. The cellular automata methods studied present very fast convergence to fixed points, noise stability, and improvements on real images, which are features that allow us to propose them as a first level elementary image enhancement.

Effect of car acceleration on traffic flow in 1D stochastic CA model

The one-dimensional (1D) asymmetric simple-exclusion model with parallel update (the deterministic cellular automaton (CA) 184) is extended to take into account acceleration of cars. Velocity and acceleration are assigned to each particle (car). The acceleration of a car is determined by the dynamical equation of motion. The stochastic cellular automaton model mimics the traffic flow on a highway. In congested traffic flow, the properties of traffic jams are studied by computer simulation. The dependence of the propagating velocity of traffic jams upon car acceleration is shown. Also, a simple mean-field theory is given for the congested traffic flow. It is found that the propagating velocity of a traffic jam obtained from simulation agrees with the analytical result for not small values of car acceleration.

Creation and annihilation of traffic jams in a stochastic asymmetric exclusion model with open boundaries: a computer simulation

The creation and annihilation of traffic jams are studied by a computer simulation. The one-dimensional (1D) fully-asymmetric exclusion model with open boundaries for parallel update is extended to take into account stochastic transition of particles (cars) where a particle moves ahead with transition probability pt if the forward nearest neighbour is not occupied. Near pt=1, the system is derived asymptotically into a steady state exhibiting a self-organized criticality. In the self-organized critical state, a traffic jam (start-stop wave) and an empty wave are created at the same time when a car stops temporarily. The traffic jam disappears by colliding with the empty wave. The coalescence process between traffic jams and empty waves is described by the ballistic annihilation process with pair creation. The resulting problem near pt=1 is consistent with the ballistic process in the context of 1D crystal growth studied by Krug and Spohn (1988). The typical lifetime of start-stop waves scales as approximately= Delta pt-0.54+or-0.04 where Delta pt=1-pt. It is shown that the cumulative distribution Nm( Delta pt) of lifetimes satisfies the scaling form Nm( Delta pt) approximately= Delta pt1.1f(m Delta pt0.54). Also, the typical interval between consecutive traffic jams scales as approximately= Delta pt-0.5+or-0.04. The cumulative interval distribution Ns( Delta pt) of traffic jams satisfies the scaling form Ns( Delta pt) approximately= Delta pt0.50g(s Delta pt0.50). For pt<1, no scaling holds.

Analysis of synfire chains

The biological implications of synfire-chain neural networks are explored by studying two idealized models. First a network model with binary firing neurons and parallel updating is proposed. This model can be solved exactly in the thermodynamic limit using the mean-field theory. An explicit equation for the capacity is obtained. In the second model, the synchrony of the pulse of activity along a synfire chain is investigated in the context of simple integrate-and-fire neurons. It is found that under natural assumptions a near-synchronous wave of activity can propagate stably along a synfire chain. The relevance of this result for real systems is discussed.