Coarse-Grained Langevin Approximations and Spatiotemporal Acceleration for Kinetic Monte Carlo Simulations of Diffusion of Interacting Particles

Kinetic Monte Carlo methods provide a powerful computational tool for the simulation of microscopic processes such as the diffusion of interacting particles on a surface, at a detailed atomistic level. However such algorithms are typically computationatly expensive and are restricted to fairly small spatiotemporal scales. One approach towards overcoming this problem was the development of coarse-grained Monte Carlo algorithms. In recent literature, these methods were shown to be capable of efficiently describing much larger length scales while still incorporating information on microscopic interactions and fluctuations. In this paper, a coarse-grained Langevin system of stochastic differential equations as approximations of diffusion of interacting particles is derived, based on these earlier coarse-grained models. The authors demonstrate the asymptotic equivalence of transient and long time behavior of the Langevin approximation and the underlying microscopic process, using asymptotics methods such as large deviations for interacting particles systems, and furthermore, present corresponding numerical simulations, comparing statistical quantities like mean paths, auto correlations and power spectra of the microscopic and the approximating Langevin processes. Finally, it is shown that the Langevin approximations presented here are much more computationally efficient than conventional Kinetic Monte Carlo methods, since in addition to the reduction in the number of spatial degrees of freedom in coarse-grained Monte Carlo methods, the Langevin system of stochastic differential equations allows for multiple particle moves in a single timestep.     

The exponential matrix function and linear differential equations

Monte Carlo optimization has been shown to be useful in solving multivariate optimization problems. In most Monte Carlo simulations, to find the optimal solution of an integer programming problem, the computer gets close to the true optimal solution, but does not find the exact optimal. This can be a problem at times but it can be overcome by multi‐stage Monte Carlo integer programs that find a preliminary nearly optimal solution and use this as a focal point and springboard for finding the true optimal.     

Monte Carlo study of time asymptotics of the polarized radiation intensity

The parameters of time asymptotics of the polarized radiation intensity are estimated. Precision Monte Carlo estimates of these parameters are derived for finite medium layers by iterating the resolvent of the corresponding transfer operator with a given scattering matrix and by evaluating parametric time derivatives. The computations are performed for two versions of the problem: with a Rayleigh scattering matrix and an aerosol scattering matrix. It is shown that the asymptotics of the radiation intensity are affected by polarization, except for the spatially homogeneous problem, for which the results are obtained analytically.     

Dynamic Principal Component Analysis of Multivariate Volatility via Fourier Analysis

A method is proposed to compute a time‐varying correlation matrix between asset prices. The method has a natural geometric interpretation in terms of dynamic principal components analysis. The paper illustrates, via Monte Carlo experiments and data analysis, the potential of the method in computing cross‐correlations; and it describes market integration, introducing the concept of reference asset.     

Hitting time and time change

This paper determines first‐passage time distributions with a twofold emphasis on the dynamics of the state variables and interest rate uncertainty. Underlyings follow two‐dimensional geometric Brownian motions, Ornstein–Uhlenbeck processes or Poisson jump‐diffusion processes, and boundaries are either fixed or indexed on risk‐free bonds. Forward‐neutral changes of numeraire enable one to derive generic valuation expressions, while changing time allows one to determine closed‐form solutions for geometric Brownian motions and moving barriers. In turn, the latter formulas are used to reduce the variance of Monte Carlo simulations in the case of jump‐diffusion processes, by means of the control variate method.     

On file structuring for non-uniform access frequencies

An important property of file structures is their behavior when the underlying distribution of access frequencies is non-uniform. In this note we consider methods for structuring files so that initially unknown but non-uniform access frequencies are exploited in a way that reduces mean search times. As a specific illustration a simple algorithm is applied to sequence search trees and shown to produce structures with mean search times that are significantly less than those produced by an algorithm creating tree structures independent of access frequencies. An analytic result is obtained for this improvement when access frequencies are uniform. Samples from the results of Monte Carlo simulations are used to illustrate this improvement when access frequencies are non-uniform.     

Monte Carlo evaluation of biological variation: Random generation of correlated non-Gaussian model parameters

When modelling the behaviour of horticultural products, demonstrating large sources of biological variation, we often run into the issue of non-Gaussian distributed model parameters. This work presents an algorithm to reproduce such correlated non-Gaussian model parameters for use with Monte Carlo simulations. The algorithm works around the problem of non-Gaussian distributions by transforming the observed non-Gaussian probability distributions using a proposed SKN-distribution function before applying the covariance decomposition algorithm to generate Gaussian random co-varying parameter sets. The proposed SKN-distribution function is based on the standard Gaussian distribution function and can exhibit different degrees of both skewness and kurtosis. This technique is demonstrated using a case study on modelling the ripening of tomato fruit evaluating the propagation of biological variation with time.     

Hamiltonian structure and statistically relevant conserved quantities for the truncated Burgers‐Hopf equation

In dynamical systems with intrinsic chaos, many degrees of freedom, and many conserved quantities, a fundamental issue is the statistical relevance of suitable subsets of these conserved quantities in appropriate regimes. The Galerkin truncation of the Burgers‐Hopf equation has been introduced recently as a prototype model with solutions exhibiting intrinsic stochasticity and a wide range of correlation scaling behavior that can be predicted successfully by simple scaling arguments. Here it is established that the truncated Burgers‐Hopf model is a Hamiltonian system with Hamiltonian given by the integral of the third power. This additional conserved quantity, beyond the energy, has been ignored in previous statistical mechanics studies of this equation. Thus, the question arises of the statistical significance of the Hamiltonian beyond that of the energy. First, an appropriate statistical theory is developed that includes both the energy and Hamiltonian. Then a convergent Monte Carlo algorithm is developed for computing equilibrium statistical distributions. The probability distribution of the Hamiltonian on a microcanonical energy surface is studied through the Monte‐Carlo algorithm and leads to the concept of statistically relevant and irrelevant values for the Hamiltonian. Empirical numerical estimates and simple analysis are combined to demonstrate that the statistically relevant values of the Hamiltonian have vanishingly small measure as the number of degrees of freedom increases with fixed mean energy. The predictions of the theory for relevant and irrelevant values for the Hamiltonian are confirmed through systematic numerical simulations. For statistically relevant values of the Hamiltonian, these simulations show a surprising spectral tilt rather than equipartition of energy. This spectral tilt is predicted and confirmed independently by Monte Carlo simulations based on equilibrium statistical mechanics together with a heuristic formula for the tilt. On the other hand, the theoretically predicted correlation scaling law is satisfied both for statistically relevant and irrelevant values of the Hamiltonian with excellent accuracy. The results established here for the Burgers‐Hopf model are a prototype for similar issues with significant practical importance in much more complex geophysical applications. Several interesting mathematical problems suggested by this study are mentioned in the final section. © 2002 Wiley Periodicals, Inc.     

On the Stability and the Approximation of Branching Distribution Flows,with Applications to Nonlinear Multiple Target Filtering

We analyze the exponential stability properties of a class of measure-valued equations arising in nonlinear multi-target filtering problems. We also prove the uniform convergence properties w.r.t. the time parameter of a rather general class of stochastic filtering algorithms, including sequential Monte Carlo type models and mean field particle interpretation models. We illustrate these results in the context of the Bernoulli and the Probability Hypothesis Density filter, yielding what seems to be the first results of this kind in this subject.     

Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients

In Monte Carlo methods quadrupling the sample size halves the error. In simulations of stochastic partial differential equations (SPDEs), the total work is the sample size times the solution cost of an instance of the partial differential equation. A Multi-level Monte Carlo method is introduced which allows, in certain cases, to reduce the overall work to that of the discretization of one instance of the deterministic PDE. The model problem is an elliptic equation with stochastic coefficients. Multi-level Monte Carlo errors and work estimates are given both for the mean of the solutions and for higher moments. The overall complexity of computing mean fields as well as k-point correlations of the random solution is proved to be of log-linear complexity in the number of unknowns of a single Multi-level solve of the deterministic elliptic problem. Numerical examples complete the theoretical analysis.     

New Monte Carlo scheme for simulating Lagranian particle diffusion with wind shear effects

Problems related to the application of Lagrangian particle methods to air quality diffusion simulations are reviewed. Advantages and shortcomings of these techniques are discussed, and a new Monte Carlo method is proposed for the treatment of cross correlation between wind fluctuation components. Also discussed is how data collected from advanced meteorlogical instrumentation can be manipulated in order to provide the proper input to the proposed numerical modelling techniques.     

Meromorphic symbolic structures for boundary value problems on manifolds with edges

We investigate the ideal of Green and Mellin operators with asymptotics for a manifold with edge‐corner singularities and boundary which belongs to the structure of parametrices of elliptic boundary value problems on a configuration with corners whose base manifolds have edges. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On geometric ergodicity of skewed—SVCHARME models

Markov Chain Monte Carlo is repeatedly used to analyze the properties of intractable distributions in a convenient way. In this paper we derive conditions for geometric ergodicity of a general class of nonparametric stochastic volatility models with skewness driven by the hidden Markov Chain with switching.     

The effect of synchronous firing on the clustering dynamics of social amoebae

A discrete model for computer simulations of the clustering dynamics of social amoebae is presented. This model incorporates the wavelike propagation of extracellular signaling of 3′–5′‐cyclic adenosine monophosphate (cAMP), the sporadic firing of cells at early stage of aggregation, the signal relaying as a response to stimulus, and the inertia and purposeful random walk of the cell movement. It is found that the sporadic firing below the threshold of cAMP concentration plays an important role because it allows time for the cells to form synchronous firing right before the stage of aggregation, and the synchronous firing is critical for the onset of clustering behavior of social amoebae. A Monte‐Carlo simulation was also run which showed the existence of potential equilibriums of mean and variance of aggregation time. The simulation result of this model could well reproduce many phenomena observed by actual experiments. © 2013 Wiley Periodicals, Inc. Complexity 20: 16–26, 2014     

Dual‐weighted‐residual and nonlinear Galerkin methods in the simulation of the aeroelastic response of windturbines

In the simulation of fatigue loading of large wind turbines model reduction and thus reduction of computing time is essential to be able to perform Monte‐Carlo simulations in turbulent wind. We describe the application of two recently proposed methods to increase the accuracy of the reduced model. In most cases only a special functional of the solution is of interest to the engineer. To select the proper basis vectors spanning the subspace of the reduced model according to this functional of interest, the dual‐weighted‐residual method is employed. During the simulation the neglected basis vectors are used to increase the accuracy of the solution based on the idea of the nonlinear and postprocessed Galerkin methods.     

Testing for Correlation Coefficients in Uniform Correlation Longitudinal Mixed Effect Linear Models Based on M-estimation

In this paper, the Fisher scoring method is applied to get M-estimator (robust estimator) in the mixed effects linear model for longitudinal data. The score tests for correlation coefficients in the model with uniform correlation covariance structure based on M-estimator are also studied. Then the properties of test statistics are investigated through Monte Carlo simulations. At last, the methods and properties are illustrated by the grape sugar data example.     

A Path‐Based Method for Simulating Large Deviations and Rare Events in Nonlinear Lightwave Systems

Errors in nonlinear lightwave systems are often associated with rare, noise‐induced, large deviations of the signal. We present a method to determine the most probable manner in which such rare events occur by solving a sequence of constrained optimization problems. These results then guide importance‐sampled Monte Carlo simulations to determine the events' probabilities. The method applies to a general class of intensity‐based optical detectors and to arbitrarily shaped and multiple pulses.     

Circulant Embedding of Approximate Covariances for Inference From Gaussian Data on Large Lattices

Recently proposed computationally efficient Markov chain Monte Carlo (MCMC) and Monte Carlo expectation–maximization (EM) methods for estimating covariance parameters from lattice data rely on successive imputations of values on an embedding lattice that is at least two times larger in each dimension. These methods can be considered exact in some sense, but we demonstrate that using such a large number of imputed values leads to slowly converging Markov chains and EM algorithms. We propose instead the use of a discrete spectral approximation to allow for the implementation of these methods on smaller embedding lattices. While our methods are approximate, our examples indicate that the error introduced by this approximation is small compared to the Monte Carlo errors present in long Markov chains or many iterations of Monte Carlo EM algorithms. Our results are demonstrated in simulation studies, as well as in numerical studies that explore both increasing domain and fixed domain asymptotics. We compare the exact methods to our approximate methods on a large satellite dataset, and show that the approximate methods are also faster to compute, especially when the aliased spectral density is modeled directly. Supplementary materials for this article are available online.     

Random orthogonal matrix simulation

This paper introduces a method for simulating multivariate samples that have exact means, covariances, skewness and kurtosis. We introduce a new class of rectangular orthogonal matrix which is fundamental to the methodology and we call these matrices L matrices. They may be deterministic, parametric or data specific in nature. The target moments determine the L matrix then infinitely many random samples with the same exact moments may be generated by multiplying the L matrix by arbitrary random orthogonal matrices. This methodology is thus termed “ROM simulation”. Considering certain elementary types of random orthogonal matrices we demonstrate that they generate samples with different characteristics. ROM simulation has applications to many problems that are resolved using standard Monte Carlo methods. But no parametric assumptions are required (unless parametric L matrices are used) so there is no sampling error caused by the discrete approximation of a continuous distribution, which is a major source of error in standard Monte Carlo simulations. For illustration, we apply ROM simulation to determine the value-at-risk of a stock portfolio.