Results 1  10
of
21
Computer Experiments
, 1996
"... 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, a ..."
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Cited by 67 (5 self)
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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
Hypercube Sampling and the Propagation of Uncertainty in Analyses of Complex Systems
, 2002
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Monte Carlo Variance of Scrambled Net Quadrature
 SIAM J. Numer. Anal
, 1997
"... . 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 (w ..."
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Cited by 28 (1 self)
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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 lowdimensional 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...
Fast Numerical Methods for Stochastic Computations: A Review
, 2009
"... This paper presents a review of the current stateoftheart of numerical methods for stochastic computations. The focus is on efficient highorder methods suitable for practical applications, with a particular emphasis on those based on generalized polynomial chaos (gPC) methodology. The framework ..."
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Cited by 15 (1 self)
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This paper presents a review of the current stateoftheart of numerical methods for stochastic computations. The focus is on efficient highorder 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 multidimensional random spaces.
MULTIPROCESS PARALLEL ANTITHETIC COUPLING FOR BACKWARD AND FORWARD Markov Chain Monte Carlo
, 2005
"... Antithetic coupling is a general stratification strategy for reducing Monte Carlo variance without increasing the simulation size. The use of the antithetic principle in the Monte Carlo literature typically employs two strata via antithetic quantile coupling. We demonstrate here that further stratif ..."
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Cited by 14 (6 self)
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Antithetic coupling is a general stratification strategy for reducing Monte Carlo variance without increasing the simulation size. The use of the antithetic principle in the Monte Carlo literature typically employs two strata via antithetic quantile coupling. We demonstrate here that further stratification, obtained by using k>2(e.g.,k = 3–10) antithetically coupled variates, can offer substantial additional gain in Monte Carlo efficiency, in terms of both variance and bias. The reason for reduced bias is that antithetically coupled chains can provide a more dispersed search of the state space than multiple independent chains. The emerging area of perfect simulation provides a perfect setting for implementing the kprocess parallel antithetic coupling for MCMC because, without antithetic coupling, this class of methods delivers genuine independent draws. Furthermore, antithetic backward coupling provides a very convenient theoretical tool for investigating antithetic forward coupling. However, the generation of k>2 antithetic variates that are negatively associated, that is, they preserve negative correlation under monotone
On rates of convergence for stochastic optimization problems under nonI.I.D. sampling
, 2006
"... 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 ..."
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Cited by 11 (1 self)
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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 quasiMonte 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 twostage stochastic programs from the literature to illustrate the discussed ideas.
Nested Latin Hypercube Design
 Biometrika
, 2009
"... We propose an approach to constructing nested Latin hypercube designs. Such designs are useful for conducting multiple computer experiments with different levels of accuracy. A nested Latin hypercube design with two layers is defined to be a special Latin hypercube design that contains a smaller Lat ..."
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Cited by 5 (4 self)
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We propose an approach to constructing nested Latin hypercube designs. Such designs are useful for conducting multiple computer experiments with different levels of accuracy. A nested Latin hypercube design with two layers is defined to be a special Latin hypercube design that contains a smaller Latin hypercube design as a subset. Our method is easy to implement and can accommodate any number of factors. We also extend this method to construct nested Latin hypercube designs with more than two layers. Illustrative examples are given. Some statistical properties of the constructed designs are derived.
Antithetic Coupling for Perfect Sampling
 In Proceedings of the 2000 ISBA conference
, 2001
"... 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 coupli ..."
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Cited by 3 (0 self)
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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 nonpositively correlated after any (componentwise) 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 "burnin" 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...
Stochastic Markovian modeling of electrophysiology of ion channels: Reconstruction of standard deviations in macroscopic currents
, 2007
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Model Independent Parametric Decision Making
, 2004
"... Accurate knowledge of the effect of parameter uncertainty on process design and operation is essential for optimal and feasible operation of a process plant. Existing approaches dealing with uncertainty in the design and process operations level assume the existence of a well defined model to repre ..."
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Cited by 1 (1 self)
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Accurate knowledge of the effect of parameter uncertainty on process design and operation is essential for optimal and feasible operation of a process plant. Existing approaches dealing with uncertainty in the design and process operations level assume the existence of a well defined model to represent process behavior and in almost all cases convexity of the involved equations. However, most of the realistic case studies cannot be described by well characterised models. Thus, a new approach is presented in this paper based on the idea of High Dimensional Model Reduction technique which utilize a reduced number of model runs to build an uncertainty propagation model that expresses process feasibility. Building on this idea a systematic iterative procedure is developed for design under uncertainty with a unique characteristic of providing parametric expression of the optimal objective with respect to uncertain parameters. The proposed approach treats the system as a black box since it does not rely on the nature of the mathematical model of the process, as is illustrated through a number of examples.