Multidimensional Monte Carlo Integration Research Articles

Virial equation of state for a granular system.

The equation of state for an ideal gas is simple, which is . In the case of imperfect gases where mutual interactions among the constituents are important, pressure P can be expressed as the series expansion of density n with appropriate coefficients, known as virial coefficients . In this paper, we have obtained the first four virial coefficients for a model interaction potential using multidimensional Monte-Carlo integration and importance sampling methods. Next, we perform molecular dynamics simulations with the same for a many-particle system to obtain P as a function of T and n. We compare our numerical data with the virial equation of state.

Adaptive multidimensional integration: vegas enhanced

We describe a new algorithm, vegas+, for adaptive multidimensional Monte Carlo integration. The new algorithm adds a second adaptive strategy, adaptive stratified sampling, to the adaptive importance sampling that is the basis for its widely used predecessor vegas. Both vegas and vegas+ are effective for integrands with large peaks, but vegas+ can be much more effective for integrands with multiple peaks or other significant structures aligned with diagonals of the integration volume. We give examples where vegas+ is 2–19× more accurate than vegas. We also show how to combine vegas+ with other integrators, such as the widely available miser algorithm, to make new hybrid integrators. For a different kind of hybrid, we show how to use integrand samples, generated using MCMC or other methods, to optimize vegas+ before integrating. We give an example where preconditioned vegas+ is more than 100× as efficient as vegas+ without preconditioning. Finally, we give examples where vegas+ is more than 10× as efficient as MCMC for Bayesian integrals with D=3 and 21 parameters. We explain why vegas+ will often outperform MCMC for small and moderate sized problems.

ZMCintegral: A package for multi-dimensional Monte Carlo integration on multi-GPUs

We have developed a Python package ZMCintegral for multi-dimensional Monte Carlo integration on multiple Graphics Processing Units (GPUs). The package employs a stratified sampling and heuristic tree search algorithm. We have built three versions of this package: one with Tensorflow and other two with Numba, and both support general user defined functions with a user-friendly interface. We have demonstrated that Tensorflow and Numba help inexperienced scientific researchers to parallelize their programs on multiple GPUs with little work. The precision and speed of our package is compared with that of VEGAS for two typical integrands, a 6-dimensional oscillating function and a 9-dimensional Gaussian function. The results show that the speed of ZMCintegral is comparable to that of the VEGAS with a given precision. For heavy calculations, the algorithm can be scaled on distributed clusters of GPUs. Program summaryProgram Title: ZMCintegralProgram Files doi:http://dx.doi.org/10.17632/p7wc7k6mpp.1Licensing provisions: Apache License Version 2.0Programming language: PythonExternal routines/libraries: Tensorflow; Numba; RayNature of problem: Easy to use python package for Multidimensional-multi-GPUs Monte Carlo integrationSolution method: Stratified sampling and heuristic tree search using multiple GPUs on distributed clusters

A stochastic gradient approach for the reliability maximization of passively controlled structures

This paper addresses the reduction of the computational burden of structural reliability maximization of passively controlled buildings subject to transient loading. Such a reduction is accomplished by means of Stochastic Gradient Descent (SGD) algorithms, which replace expensive multi-dimensional Monte Carlo integration by a singleton multi-iteration integration. In order to be able to apply SGD methods, the time-dependent structural reliability evaluation was constructed based on the out-crossing rate approach. Design of single and multi Friction Tuned Mass Dampers (FTMDs) are considered as design examples. The proposed SGD algorithm, the accelerated stochastic gradient descent (ASGD) algorithm, efficiently combines Nesterov acceleration, Polyak–Ruppert averaging and restart techniques. The main result drew from the numerical analysis is that SGD algorithms were able to maximize the structural reliability by providing more accurate results and requiring lower computational cost than well-known optimization methods. Finally, the limitation of the proposed optimization procedure is linked to the validity of the out-crossing rate to approximate the time-dependent structural reliability.

Vectorized algorithm for multidimensional Monte Carlo integration on modern GPU, CPU and MIC architectures

The aim of this paper is to show that the multidimensional Monte Carlo integration can be efficiently implemented on computers with modern multicore CPUs and manycore accelerators including Intel MIC and GPU architectures using a new vectorized version of LCG pseudorandom number generator which requires limited amount of memory. We introduce two new implementations of the algorithm based on directive-based parallel programming standards OpenMP and OpenACC and consider their performance using Hockney–Jesshope theoretical model of vector computations. We also present and discuss the results of experiments performed on dual-processor Intel Xeon E5-2670 computers with Intel Xeon Phi 7120P and NVIDIA K40m.

Using distributed memory parallel computers and GPU clusters for multidimensional Monte Carlo integration

SummaryThe aim of this paper is to show that the multidimensional Monte Carlo integration can be efficiently implemented on various distributed memory parallel computers and clusters of multicore nodes using recently developed parallel versions of linear congruential generator and lagged Fibonacci generator pseudorandom number generators. We show how to accelerate the overall performance by offloading some computations to Graphics Processing Units (GPUs), and we discuss how to transform Message Passing Interface (MPI) + OpenMP programs to MPI + OpenMP + CUDA model. We explain how to utilize multiple cores of CPUs together with multiple GPU accelerators within a single node and how to achieve reasonable load balancing of all computational resources of GPU‐accelerated multicore nodes. We present and discuss the results of experiments performed on the following target architectures: IBM Blue Gene/Q parallel computer, a cluster of Intel Xeon E5‐2660 servers, and a Tesla‐based GPU cluster with Intel Xeon X5650 multicore processors. The results are presented from two points of view: strong scaling and weak scaling. We also compare the performance of all considered architectures. Copyright © 2014 John Wiley & Sons, Ltd.

Accurate multidimensional Poisson-disk sampling

We present an accurate and efficient method to generate samples based on a Poisson-disk distribution. This type of distribution, because of its blue noise spectral properties, is useful for image sampling. It is also useful for multidimensional Monte Carlo integration and as part of a procedural object placement function. Our method extends trivially from 2D to 3D or to any higher dimensional space. We demonstrate results for up to four dimensions, which are likely to be the most useful for computer graphics applications. The method is accurate because it generates distributions with the same statistical properties of those generated with the brute-force dart-throwing algorithm, the archetype against which all other Poisson-disk sampling methods are compared. The method is efficient because it employs a spatial subdivision data structure that signals the regions of space where the insertion of new samples is allowed. The method has O ( N log N ) time and space complexity relative to the total number of samples. The method generates maximal distributions in which no further samples can be inserted at the completion of the algorithm. The method is only limited in the number of samples it can generate and the number of dimensions over which it can work by the available physical memory.

Markov chain Monte Carlo algorithms for CDMA and MIMO communication systems

In this paper, we develop novel Bayesian detection methods that are applicable to both synchronous code-division multiple-access and multiple-input multiple-output communication systems. Markov chain Monte Carlo (MCMC) simulation techniques are used to obtain Bayesian estimates (soft information) of the transmitted symbols. Unlike previous reports that widely use statistical inference to estimate a posteriori probability (APP) values, we present alternative statistical methods that are developed by viewing the underlying problem as a multidimensional Monte Carlo integration. We show that this approach leads to results that are similar to those that would be obtained through a proper Rao-Blackwellization technique and thus conclude that our proposed methods are superior to those reported in the literature. We also note that when the channel signal-to-noise ratio is high, MCMC simulator experiences some very slow modes of convergence. Thus accurate estimation of APP values requires simulations of very long Markov chains, which may be infeasible in practice. We propose two solutions to this problem using the theory of importance sampling. Extensive computer simulations show that both solutions improve the system performance greatly. We also compare the proposed MCMC detection algorithms with the sphere decoding and minimum mean square error turbo detectors and show that the MCMC detectors have superior performance.

Distributed Monte Carlo Integration Using Corba and Java

The object-oriented middleware standard, CORBA, is a very useful platform for distributed computing, and in particular for sharing a workload among a collection of possibly polymorphic computers. CORBA has, however, received relatively little attention from the scientific computing community. In this article we demonstrate how CORBA and Java can be used to implement a distributed multi-dimensional Monte Carlo integration algorithm which runs on the internet.

Numerical calculation of channel capacity for wireless infrared multipath channels

The authors present an efficient numerical approach to the channel capacity evaluation of wireless infrared channels, impaired by multipath induced intersymbol interference. The major disadvantage of this technique is that it involves only one-dimensional integration as opposed to previously proposed methods which rely on multidimensional Monte Carlo integration.

Sum of states for non-separable anharmonic oscillators

‘Exact’ vibrational sums of states G v( E) for non-separable anharmonic oscillators in several non-linear triatomics and formaldehyde are obtained by generating all compositions of the relevant quantum numbers from which the correspondmg energy levels are calculated and then counted up. The G v( E) so produced confirm the G v( E) calculated by multidimensional Monte Carlo integration, at less computational effort for the triatomics. The first-order steepest-descent routine in the Morse oscillator representation is shown to be a good smooth-function approximation to the ‘exact’ non-separable anharmonic G v( E) if the molecular frequencies are not too low and the energy and anharmonicity coefficients are not too high.

Cointegration tests in present value relationships: A Bayesian look at the bivariate properties of stock prices and dividends

A Bayesian methodology is used to examine the bivariate time series properties of stock prices and dividends. The hypothesis that stock prices and dividends are cointegrated is emphasized. Exact small sample results are calculated using a specific parameterization for the bivariate model. Some commonly encountered problems in conventional cointegration tests based on asymptotic theory are thereby eliminated. We consider three hypotheses whose nonlinear form entails the need for multidimensional Monte Carlo integration. In addition, three data sets are used, two of which are annual U.S. sets over long time periods and one a post-war Canadian set. The intention is to demonstrate that a more accurate understanding of the value and limitations of cointegration tests for investigating present value relationships is possible with a Bayesian methodology, employed over a variety of data configurations.

Computing variational bounds for flow through random aggregates of spheres

Known formulas for variational bounds on Darcy's constant for slow flow through porous media depend on two-point and three-point spatial correlation functions. Certain bounds due to Prager and Doi depending only a two-point correlation functions have been calculated for the first time for random aggregates of spheres with packing fractions ( η) up to η = 0.64. Three radial distribution functions for hard spheres were tested for η up to 0.49: (1) the uniform distribution or “well-stirred approximation,” (2) the Percus-Yevick approximation, and (3) the semi-empirical distribution of Verlet and Weis. The empirical radial distribution functions of Bennett and Finney were used for packing fractions near the random-close-packing limit ( η RCP ∼ 0.64). An accurate multidimensional Monte Carlo integration method (VEGAS) developed by Lepage was used to compute the required two-point correlation functions. The results show that Doi's bounds are preferred for η < 0.10 while Prager's bounds are preferred for η > 0.10. The “upper bounds” computed using the well-stirred approximation actually become negative (which is physically impossible) as η increases, indicating the very limited value of this approximation. The other two choices of radial distribution function give reasonable results for η up to 0.49. However, these bounds do not decrease with η as fast as expected for large η. It is concluded that variational bounds dependent on three-point correlation functions are required to obtain more accurate bounds on Darcy's constant for large η.