Introduction Deterministic computer simulations of physical phenomena are becoming widely used in science and engineering. Computers are used to describe the flow of air over an airplane wing, combustion of gasses in a flame, behavior of a metal structure under stress, safety of a nuclear reactor, and so on. Some of the most widely used computer models, and the ones that lead us to work in this area, arise in the design of the semiconductors used in the computers themselves. A process simulator starts with a data structure representing an unprocessed piece of silicon and simulates the steps such as oxidation, etching and ion injection that produce a semiconductor device such as a transistor. A device simulator takes a description of such a device and simulates the flow of current through it under varying conditions to determine properties of the device such as its switching speed and the critical voltage at which it switches. A circuit simulator takes a list of devices and the

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. Hybrids of equidistribution and Monte Carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter. This paper studies the variance of one such hybrid, scrambled nets, by applying a multidimensional multiresolution (wavelet) analysis to the integrand. The integrand is assumed to be measurable and square integrable but not necessarily of bounded variation. In simple Monte Carlo, every nonconstant term of the multiresolution contributes to the variance of the estimated integral. For scrambled nets, certain low-dimensional and coarse terms do not contribute to the variance. For any integrand in L 2 , the sampling variance tends to zero faster under scrambled net quadrature than under Monte Carlo sampling, as the number of function evaluations n tends to infinity. Some finite n results bound the variance under scrambled net quadrature by a small constant multiple of the Monte Carlo variance, uniformly ove...

In this paper we discuss the issue of solving stochastic optimization problems by means of sample average approximations. Our focus is on rates of convergence of estimators of optimal solutions and optimal values with respect to the sample size. This is a well studied problem in case the samples are independent and identically distributed (i.e., when standard Monte Carlo is used); here, we study the case where that assumption is dropped. Broadly speaking, our results show that, under appropriate assumptions, the rates of convergence for pointwise estimators under a sampling scheme carry over to the optimization case, in the sense that convergence of approximating optimal solutions and optimal values to their true counterparts has the same rates as in pointwise estimation. Our motivation for the study arises from two types of sampling methods that have been widely used in the Statistics literature. One is Latin Hypercube Sampling (LHS), a stratified sampling method originally proposed in the seventies by McKay, Beckman, and Conover (1979). The other is the class of quasi-Monte Carlo (QMC) methods, which have become popular especially after the work of Niederreiter (1992). The advantage of such methods is that they typically yield pointwise estimators which not only have lower variance than standard Monte Carlo but also possess better rates of convergence. Thus, it is important to study the use of these techniques in samplingbased optimization. The novelty of our work arises from the fact that, while there has been some work on the use of variance reduction techniques and QMC methods in stochastic optimization, none of the existing work — to the best of our knowledge — has provided a theoretical study on the effect of these techniques on rates of convergence for the optimization problem. We present numerical results for some two-stage stochastic programs from the literature to illustrate the discussed ideas. Key words: Stochastic optimization, two-stage stochastic programming with recourse,

This paper presents a review of the current state-of-the-art of numerical methods for stochastic computations. The focus is on efficient high-order methods suitable for practical applications, with a particular emphasis on those based on generalized polynomial chaos (gPC) methodology. The framework of gPC is reviewed, along with its Galerkin and collocation approaches for solving stochastic equations. Properties of these methods are summarized by using results from literature. This paper also attempts to present the gPC based methods in a unified framework based on an extension of the classical spectral methods into multi-dimensional random spaces.

This paper reports some initial investigations of the use of antithetic variates in perfect sampling. A simple random walk example is presented to illustrate the key ingredients of antithetic coupling for perfect sampling as well as its potential benefit. A key step in implementing antithetic coupling is to generate random variates that are negatively associated, a stronger condition than negative correlation as it requires that the variates remain non-positively correlated after any (component-wise) monotone transformations have been applied. For , this step is typically trivial (e.g., by taking and 3 , where ) and it constitutes much of the common use of antithetic variates in Monte Carlo simulation. Our emphasis is on because we have observed some general gains in going beyond the commonly used pair of antithetic variates. We discuss several ways of generating negatively associated random variates for arbitrary , and our comparison generally favors Iterative Latin Hypercube Sampling. Keywords: ANTITHETIC VARIABLES, COUPLING FROM THE PAST, EXACT SAMPLING, ITERATIVE LATIN HYPERCUBE SAMPLING, NEGATIVE ASSOCIATION, RANDOM WALK. 1. PERFECT SAMPLING Exploring a probability distribution using MCMC methods is now a routine practice in Bayesian statistics. The main idea is to run a Markov chain whose stationary distribution is . After an initial "burn-in" period, the frequency with which the chain moves within the state space can be used to approximate the target distribution. An important practical issue is to determine how long we need to run the chain in order to achieve acceptable accuracy in this approximation. As discussed in Wilson (2000), among all methods that are currently available, the best one, not surprisin...

Abstract — We consider the problem of approximating an arbitrary multi–dimensional probability density function by means of a Dirac mixture density. Instead of an optimal solution based on minimizing a global distance measure between the true density and its approximation, a fast suboptimal anytime procedure is proposed, which is based on sequentially partitioning the state space and component placement by local optimization. The proposed procedure adaptively covers the entire state space with a gradually increasing resolution. It can be efficiently implemented by means of a pre–allocated tree structure in a straightforward manner. The resulting computational complexity is linear in the number of components and linear in the number of dimensions. This allows a large number of components to be handled, which is especially useful in high–dimensional state spaces.

[1] This paper introduces a ‘‘refractivity from clutter’ ’ (RFC) approach with an inversion method based on a pregenerated database. The RFC method exploits the information contained in the radar sea clutter return to estimate the refractive index profile. Whereas initial efforts are based on algorithms giving a good accuracy involving high computational needs, the present method is based on a learning machine algorithm in order to obtain a real-time system. This paper shows the feasibility of a RFC technique based on the least squares support vector machine inversion method by comparing it to a genetic algorithm on simulated and noise-free data, at 1 and 5 GHz. These data are simulated in the presence of ideal trilinear surface-based ducts. The learning machine is based on a pregenerated database computed using Latin hypercube sampling to improve the efficiency of the learning. The results show that little accuracy is lost compared to a genetic algorithm approach. The computational time of a genetic algorithm is very high, whereas the learning machine approach is real time. The advantage of a real-time RFC system is that it could work on several azimuths in near real time.

Bootstrap principle is briefly reviewed. Hall’s (1989) antithetic variates method for bootstrap is discussed and extended to more than two antithetic resampling processes. We illustrate the theory with a simulation study. The numerical results show that increasing the number of antithetic resampling processes produces significant smaller variances of the bootstrap estimator over the paired case.

The Multiple-Try Metropolis is a recent extension of the Metropolis algorithm in which the next state of the chain is selected among a set of proposals. We propose a modification of the Multiple-Try Metropolis algorithm which allows for the use of correlated proposals, particularly antithetic and stratified proposals. The method is particularly useful for random walk Metropolis in high dimensional spaces and can be used easily when the proposal distribution is Gaussian. We explore the use of quasi Monte Carlo (QMC) methods to generate highly stratified samples. A series of examples is presented to evaluate the potential of the method.