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At least 19 recordsLinked to original sources
arXiv1995

A Parallel Processing Algorithm for Computing Short-Range Particle Forces with Inhomogeneous Particle Distributions

We present a computational algorithm for computing short range forces between particles. The algorithm has two distinguishing features. First, it is optimized for multi-processor computers, and will use as many processors as are available. Second, it is optimized for inhomogeneous, dynamic particle distributions; for any distribution the computational load is distributed evenly to all processors, and the communication time is less than 15\% of the total run time. In this talk we present our new algorithm. We developed the program for a grand-challenge problem in cosmology, simulation of the formation of large-scale structure in the universe. This simulation, run on the Thinking Machines Corporation CM-5, uses the particle-particle/particle-mesh (PPPM) \cite{hock_east} algorithm. The particle-particle phase is computed using the algorithm we describe in this paper. We discuss this and other applications.

astro-ph↗
arXiv1993

Braid analysis of (low-dimensional) chaos

Braid theory is used to calcualte the topological entropy of data from the belousov-Zhabotinskii reaction, the results agree well with one-dimensional theory to the order of approximation considered.

chao-dyn↗
arXiv1993

A Dynamical Simulation Facility for Hybrid Systems

This paper establishes a general framework for describing hybrid dynamical systems which is particularly suitable for numerical simulation. In this context, the data structures used to describe the sets and functions which comprise the dynamical system are crucial since they provide the link between a natural mathematical formulation of a problem and the correct application of standard numerical algorithms. We describe a partial implementation of the design methodology and use this simulation tool for a specific control problem in robotics as an illustration of the utility of the approach for practical applications.

chao-dyn↗
arXiv1993

Exhibiting Randomness in Arithmetic using Mathematica and C

In my book "Algorithmic Information Theory" I explain how I constructed a million-character equation that proves that there is randomness in arithmetic. My book only includes a few pages from the monster equation, and omits the software used to construct it. This software has now been rewritten in Mathematica. The Mathematica software for my book, and its input, are here in their entirety. The Mathematica code is remarkably compact, but it sometimes is slow. So one C program plus equipment for automatically generating another is also included in this software package.

chao-dyn↗
arXiv1993

Dependence of extensive chaos on the spatial correlation length (substantial revision)

We consider spatiotemporal chaotic systems for which spatial correlation functions decay substantially over a length scale xi (the spatial correlation length) that is small compared to the system size L. Numerical simulations suggest that such systems generally will be extensive, with the fractal dimension D growing in proportion to the system volume for sufficiently large systems (L >> xi). Intuitively, extensive chaos arises because of spatial disorder. Subsystems that are sufficiently separated in space should be uncorrelated and so contribute to the fractal dimension in proportion to their number. We report here the first numerical calculation that examines quantitatively how one important characterization of extensive chaos---the Lyapunov dimension density---depends on spatial disorder, as measured by the spatial correlation length xi. Surprisingly, we find that a representative extensively chaotic system does not act dynamically as many weakly interacting regions of size xi.

chao-dyn↗
arXiv1994

Chaotic time series Part II: System identification and prediction

This paper is the second in a series of two, and describes the current state of the art in modelling and prediction of chaotic time series. Sampled data from deterministic non-linear systems may look stochastic when analysed with linear methods. However, the deterministic structure may be uncovered and non-linear models constructed that allow improved prediction. We give the background for such methods from a geometrical point of view, and briefly describe the following types of methods: global polynomials, local polynomials, multi layer perceptrons and semi-local methods including radial basis functions. Some illustrative examples from known chaotic systems are presented, emphasising the increase in prediction error with time. We compare some of the algorithms with respect to prediction accuracy and storage requirements, and list applications of these methods to real data from widely different areas.

chao-dyn↗
arXiv1994

Chaotic time series Part I: Estimation of invariant properies in state space

Certain deterministic non-linear systems may show chaotic behaviour. Time series derived from such systems seem stochastic when analyzed with linear techniques. However, uncovering the deterministic structure is important because it allows for construction of more realistic and better models and thus improved predictive capabilities. This paper describes key features of chaotic systems including strange attractors and Lyapunov exponents. The emphasis is on state space reconstruction techniques that are used to estimate these properties, given scalar observations. Data generated from equations known to display chaotic behaviour are used for illustration. A compilation of applications to real data from widely different fields is given. If chaos is found to be present, one may proceed to build non-linear models, which is the topic of the second paper in this series.

chao-dyn↗
arXiv1993

Some comments on the correlation dimension of $1/f^α$ noise

It has recently been observed that a stochastic (infinite degree of freedom) time series with a $1/f^α$ power spectrum can exhibit a finite correlation dimension, even for arbitrarily large data sets. [A.R. Osborne and A.~Provenzale, {\sl Physica D} {\bf 35}, 357 (1989).] I will discuss the relevance of this observation to the practical estimation of dimension from a time series, and in particular I will argue that a good dimension algorithm need not be trapped by this anomalous fractal scaling. Further, I will analytically treat the case of gaussian \onefas noise, with explicit high and low frequency cutoffs, and derive the scaling of the correlation integral $C(N,r)$ in various regimes of the $(N,r)$ plane. Appears in: {\sl Phys. Lett. A} {\bf 155} (1991) 480--493.

comp-gas↗
arXiv1994

Detecting Nonlinearity in Data with Long Coherence Times

We consider the limitations of two techniques for detecting nonlinearity in time series. The first technique compares the original time series to an ensemble of surrogate time series that are constructed to mimic the linear properties of the original. The second technique compares the forecasting error of linear and nonlinear predictors. Both techniques are found to be problematic when the data has a long coherence time; they tend to indicate nonlinearity even for linear time series. We investigate the causes of these difficulties both analytically and with numerical experiments on ``real'' and computer-generated data. In particular, although we do see some initial evidence for nonlinear structure in the SFI dataset E, we are inclined to dismiss this evidence as an artifact of the long coherence time.

comp-gas↗
arXiv1993

Message-Passing Multi-Cell Molecular Dynamics on the Connection Machine 5

We present a new scalable algorithm for short-range molecular dynamics simulations on distributed memory MIMD multicomputer based on a message-passing multi-cell approach. We have implemented the algorithm on the Connection Machine 5 (CM-5) and demonstrate that meso-scale molecular dynamics with more than $10^8$ particles is now possible on massively parallel MIMD computers. Typical runs show single particle update-times of $0.15 μs$ in 2 dimensions (2D) and approximately $1 μs$ in 3 dimensions (3D) on a 1024 node CM-5 without vector units, corresponding to more than 1.8 GFlops overall performance. We also present a scaling equation which agrees well with actually observed timings.

comp-gas↗
arXiv1993

The Lattice Boltzmann Equation Method for the Simulation of Compressible Fluid Flow

We systematically derived hydrodynamic equations and transport coefficients for a class of multi-speed lattice Boltzmann models in D dimensions, using the multi-scale technique. The constitutive relation of physical fluid is recovered by a modified equilibrium distribution in Maxwell-Boltzmann type. With the use of the rest particles and the particle reservoir, we were able to add one degree of freedom into the sound speed of the modeled fluid. When the sound speed is tuned small enough, the compressible region of fluid flow can be reached. An example 2-D model is presented, together with the numerical verification for its transport coefficients.

comp-gas↗
arXiv1993

Wavelets and Fast Numerical Algorithms

Wavelet based algorithms in numerical analysis are similar to other transform methods in that vectors and operators are expanded into a basis and the computations take place in this new system of coordinates. However, due to the recursive definition of wavelets, their controllable localization in both space and wave number (time and frequency) domains, and the vanishing moments property, wavelet based algorithms exhibit new and important properties. For example, the multiresolution structure of the wavelet expansions brings about an efficient organization of transformations on a given scale and of interactions between different neighbouring scales. Moreover, wide classes of operators which naively would require a full (dense) matrix for their numerical description, have sparse representations in wavelet bases. For these operators sparse representations lead to fast numerical algorithms, and thus address a critical numerical issue. We note that wavelet based algorithms provide a systematic generalization of the Fast Multipole Method (FMM) and its descendents. These topics will be the subject of the lecture. Starting from the notion of multiresolution analysis, we will consider the so-called non-standard form (which achieves decoupling among the scales) and the associated fast numerical algorithms. Examples of non-standard forms of several basic operators (e.g. derivatives) will be computed explicitly.

comp-gas↗
arXiv1993

Identification of Continuous-Time Dynamical Systems: Neural Network Based Algorithms and Parallel Implementation

Time-delay mappings constructed using neural networks have proven successful in performing nonlinear system identification; however, because of their discrete nature, their use in bifurcation analysis of continuous-time systems is limited. This shortcoming can be avoided by embedding the neural networks in a training algorithm that mimics a numerical integrator. Both explicit and implicit integrators can be used. The former case is based on repeated evaluations of the network in a feedforward implementation; the latter relies on a recurrent network implementation. Here the algorithms and their implementation on parallel machines (SIMD and MIMD architectures) are discussed.

comp-gas↗
arXiv1993

Convergence of Convective-Diffusive Lattice Boltzmann Methods

Lattice Boltzmann methods are numerical schemes derived as a kinetic approximation of an underlying lattice gas. A numerical convergence theory for nonlinear convective-diffusive lattice Boltzmann methods is established. Convergence, consistency, and stability are defined through truncated Hilbert expansions. In this setting it is shown that consistency and stability imply convergence. Monotone lattice Boltzmann methods are defined and shown to be stable, hence convergent when consistent. Examples of diffusive and convective-diffusive lattice Boltzmann methods that are both consistent and monotone are presented.

comp-gas↗
arXiv1993

Stability Analysis of Lattice Boltzmann Methods

The lattice Boltzmann equation describes the evolution of the velocity distribution function on a lattice in a manner that macroscopic fluid dynamical behavior is recovered. Although the equation is a derivative of lattice gas automata, it may be interpreted as a Lagrangian finite-difference method for the numerical simulation of the discrete-velocity Boltzmann equation that makes use of a BGK collision operator. As a result, it is not surprising that numerical instability of lattice Boltzmann methods have been frequently encountered by researchers. We present an analysis of the stability of perturbations of the particle populations linearized about equilibrium values corresponding to a constant-density uniform mean flow. The linear stability depends on the following parameters: the distribution of the mass at a site between the different discrete speeds, the BGK relaxation time, the mean velocity, and the wavenumber of the perturbations. This parameter space is too large to compute the complete stability characteristics. We report some stability results for a

comp-gas↗
arXiv1993

Numerical Solution of the Schroedinger Equation using a Quantum Lattice Boltzmann Equation}

The quantum Lattice Boltzmann equation (QLBe), a new variant of the lattice Boltzmann equation, specifically designed to describe non relativistic quantum motion, is validated for the case of a free-particle in (1+1) space-time dimensions. The proper parameter regime under which the method needs to be operated in order to reproduce faithful non-relativistic quantum motion is also discussed.

comp-gas↗
arXiv1993

Accuracy of Discrete-Velocity BGK Models for the Simulation of the Incompressible Navier-Stokes Equations

Two discretizations of a 9-velocity Boltzmann equation with a BGK collision operator are studied. A Chapman-Enskog expansion of the PDE system predicts that the macroscopic behavior corresponds to the incompressible Navier-Stokes equations with additional terms of order Mach number squared. We introduce a fourth-order scheme and compare results with those of the commonly used lattice Boltzmann discretization and with finite-difference schemes applied to the incompressible Navier-Stokes equations in primitive-variable form. We numerically demonstrate convergence of the BGK schemes to the incompressible Navier-Stokes equations and quantify the errors associated with compressibility and discretization effects. When compressibility error is smaller than discretization error, convergence in both grid spacing and time step is shown to be second-order for the LB method and is confirmed to be fourth-order for the fourth-order BGK solver. However, when the compressibility error is simultaneously reduced as the grid is refined, the LB method behaves as a first-order scheme in time.

comp-gas↗