Three parallel algorithms for classical molecular dynamics are presented. The first assigns each processor a fixed subset of atoms; the second assigns each a fixed subset of inter-atomic forces to compute; the third assigns each a fixed spatial region. The algorithms are suitable for molecular dynamics models which can be difficult to parallelize efficiently—those with short-range forces where the neighbors of each atom change rapidly. They can be implemented on any distributed-memory parallel machine which allows for message-passing of data between independently executing processors. The algorithms are tested on a standard Lennard-Jones benchmark problem for system sizes ranging from 500 to 100,000,000 atoms on several parallel supercomputers--the nCUBE 2, Intel iPSC/860 and Paragon, and Cray T3D. Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems. For large problems, the spatial algorithm achieves parallel efficiencies of 90% and a 1840-node Intel Paragon performs up to 165 faster than a single Cray C9O processor. Trade-offs between the three algorithms and guidelines for adapting them to more complex molecular dynamics simulations are also discussed.

Fast parallel algorithms for short-range molecular dynamics

Data Mining Practical Machine Learning Tools and Techniques

In this book, Andrew Harvey sets out to provide a unified and comprehensive theory of structural time series models. Unlike the traditional ARIMA models, structural time series models consist explicitly of unobserved components, such as trends and seasonals, which have a direct interpretation. As a result the model selection methodology associated with structural models is much closer to econometric methodology. The link with econometrics is made even closer by the natural way in which the models can be extended to include explanatory variables and to cope with multivariate time series. From the technical point of view, state space models and the Kalman filter play a key role in the statistical treatment of structural time series models. The book includes a detailed treatment of the Kalman filter. This technique was originally developed in control engineering, but is becoming increasingly important in fields such as economics and operations research. This book is concerned primarily with modelling economic and social time series, and with addressing the special problems which the treatment of such series poses. The properties of the models and the methodological techniques used to select them are illustrated with various applications. These range from the modellling of trends and cycles in US macroeconomic time series to to an evaluation of the effects of seat belt legislation in the UK.

Forecasting, Structural Time Series Models and the Kalman Filter

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Much has been written about theory and practice in the law, and the tension between practitioners and theorists. Judges do not cite theoretical articles often; they rarely "apply" theories to particular cases. These arguments are not revisited. Instead the Essay explores the working and interaction of theory and practice, practitioners and theorists. The Essay starts with a story about solving a legal issue using our intellectual tools - theory, practice, and their progenies: experience and "gut." Next the Essay elaborates on the nature of theory, practice, experience and "gut." The third part of the Essay discusses theories that are helpful to practitioners and those that are less helpful. The Essay concludes that practitioners theorize, and theorists practice. They use these intellectual tools differently because the goals and orientations of theorists and practitioners, and the constraints under which they act, differ. Theory, practice, experience and "gut" help us think, remember, decide and create. They complement each other like the two sides of the same coin: distinct but inseparable.

Of Theory and Practice

The basic random variables on which random uncertainties can in a given model depend can be viewed as defining a measure space with respect to which the solution to the mathematical problem can be defined. This measure space is defined on a product measure associated with the collection of basic random variables. This paper clarifies the mathematical structure of this space and its relationship to the underlying spaces associated with each of the random variables. Cases of both dependent and independent basic random variables are addressed. Bases on the product space are developed that can be viewed as generalizations of the standard polynomial chaos approximation. Moreover, two numerical constructions of approximations in this space are presented along with the associated convergence analysis.

/pdf/physical-systems-with-random-uncertainties-chaos-275g1jeiid.pdf

Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure

https://hal-upec-upem.archives-ouvertes.fr/hal-00686293/document

A nonparametric model of random uncertainties for reduced matrix models in structural dynamics

A new approach is presented for analyzing random uncertainties in dynamical systems. This approach consists of modeling random uncertainties by a nonparametric model allowing transient responses of mechanical systems submitted to impulsive loads to be predicted in the context of linear structural dynamics. The information used does not require the description of the local parameters of the mechanical model. The probability model is deduced from the use of the entropy optimization principle, whose available information is constituted of the algebraic properties related to the generalized mass, damping, and stiffness matrices which have to be positive-definite symmetric matrices, and the knowledge of these matrices for the mean reduced matrix model. An explicit construction and representation of the probability model have been obtained and are very well suited to algebraic calculus and to Monte Carlo numerical simulation in order to compute the transient responses of structures submitted to impulsive loads. The fundamental properties related to the convergence of the stochastic solution with respect to the dimension of the random reduced matrix model are analyzed. Finally, an example is presented.

/pdf/maximum-entropy-approach-for-modeling-random-uncertainties-4rw7365rgr.pdf

Maximum entropy approach for modeling random uncertainties in transient elastodynamics.

https://hal-upec-upem.archives-ouvertes.fr/hal-00686187/document

Random matrix theory for modeling uncertainties in computational mechanics

Stochastic Canonical Equation of Multidimensional Nonlinear Dissipative Hamiltonian Dynamical Systems Fundamental Examples of Nonlinear Dynamical Systems and Associated Second Order Equation Brief Review of Probability and Random Variables Probabilistic Tools I. Classical Stochastic Processes Probabilistic Tools II. Mean-Square theory of linear integral transformations and of linear Differential Equations Probabilistic Tools III. Diffusion Processes and Fokker-Planck Equation Probabilistic Tools IV. Stochastic Integrals and Stochastic Differential Equations Stochastic Modelling with Stochastic Differential Equations FKP Equation for the Dissipative Hamiltonian Dynamical Systems Stationary Response of Dissipative Dynamical Systems, Existence and Uniqueness, Explicit Solution of an Invariant Measure Complements for the Normalization Condition, Characteristic Function and Moments of the Invariant Measure Application II. Multidimensional Linear Oscillators Subjected to External and Parametric Random Excitations Application III. Multidimensional Nonlinear Oscillators with Inertial Nonlinearity Subjected to External Random Excitations Application Ill. Multidimensional Nonlinear Oscillators Subjected to External and Parametric Random Excitations Symplectic Change of Variables in the Multidimensional Unsteady FKP Equation.

Christian Soize

Papers

Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure

A nonparametric model of random uncertainties for reduced matrix models in structural dynamics

Maximum entropy approach for modeling random uncertainties in transient elastodynamics.

Random matrix theory for modeling uncertainties in computational mechanics

The Fokker-Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solutions