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Latin Supercube Sampling for Very High Dimensional Simulations
, 1997
"... This paper introduces Latin supercube sampling (LSS) for very high dimensional simulations, such as arise in particle transport, finance and queuing. LSS is developed as a combination of two widely used methods: Latin hypercube sampling (LHS), and QuasiMonte Carlo (QMC). In LSS, the input variables ..."
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Cited by 69 (7 self)
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This paper introduces Latin supercube sampling (LSS) for very high dimensional simulations, such as arise in particle transport, finance and queuing. LSS is developed as a combination of two widely used methods: Latin hypercube sampling (LHS), and QuasiMonte Carlo (QMC). In LSS, the input variables are grouped into subsets, and a lower dimensional QMC method is used within each subset. The QMC points are presented in random order within subsets. QMC methods have been observed to lose effectiveness in high dimensional problems. This paper shows that LSS can extend the benefits of QMC to much higher dimensions, when one can make a good grouping of input variables. Some suggestions for grouping variables are given for the motivating examples. Even a poor grouping can still be expected to do as well as LHS. The paper also extends LHS and LSS to infinite dimensional problems. The paper includes a survey of QMC methods, randomized versions of them (RQMC) and previous methods for extending Q...
Valuation of Mortgage Backed Securities Using Brownian Bridges to Reduce Effective Dimension
, 1997
"... The quasiMonte Carlo method for financial valuation and other integration problems has error bounds of size O((log N) k N \Gamma1 ), or even O((log N) k N \Gamma3=2 ), which suggests significantly better performance than the error size O(N \Gamma1=2 ) for standard Monte Carlo. But in hig ..."
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Cited by 68 (13 self)
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The quasiMonte Carlo method for financial valuation and other integration problems has error bounds of size O((log N) k N \Gamma1 ), or even O((log N) k N \Gamma3=2 ), which suggests significantly better performance than the error size O(N \Gamma1=2 ) for standard Monte Carlo. But in high dimensional problems this benefit might not appear at feasible sample sizes. Substantial improvements from quasiMonte Carlo integration have, however, been reported for problems such as the valuation of mortgagebacked securities, in dimensions as high as 360. We believe that this is due to a lower effective dimension of the integrand in those cases. This paper defines the effective dimension and shows in examples how the effective dimension may be reduced by using a Brownian bridge representation. 1 Introduction Simulation is often the only effective numerical method for the accurate valuation of securities whose value depends on the whole trajectory of interest Mathematics Departmen...
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
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...
On the use of low discrepancy sequences in Monte Carlo methods
 MONTE CARLO METHODS AND APPLICATIONS
, 1996
"... Quasirandom (or low discrepancy) sequences are sequences for which the convergence to the uniform distribution on [0; 1) s occurs rapidly. Such sequences are used in quasiMonte Carlo methods for which the convergence speed, with respect to the N first terms of the sequence, is in O(N \Gamma1 ..."
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Cited by 28 (1 self)
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Quasirandom (or low discrepancy) sequences are sequences for which the convergence to the uniform distribution on [0; 1) s occurs rapidly. Such sequences are used in quasiMonte Carlo methods for which the convergence speed, with respect to the N first terms of the sequence, is in O(N \Gamma1 (ln N) s ), where s is the mathematical dimension of the problem considered. The disadvantage of these methods is that error bounds, even if they exist theoretically, are inefficient in practice. Nevertheless, to take advantage of these methods for what concerns their convergence speed, we use them as a variance reduction technique, which lead to great improvements with respect to standard Monte Carlo methods. We consider in this paper two different approaches which combine Monte Carlo and quasiMonte Carlo methods. The first one can use every low discrepancy sequence and the second one, called Owen's method, uses only Niederreiter sequences. We prove that the first approach has the same...
Efficient Multidimensional Sampling
, 2002
"... Image synthesis often requires the Monte Carlo estimation of integrals. Based on a generalized concept of stratification we present an efficient sampling scheme that consistently outperforms previous techniques. This is achieved by assembling sampling patterns that are stratified in the sense of jit ..."
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Cited by 25 (1 self)
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Image synthesis often requires the Monte Carlo estimation of integrals. Based on a generalized concept of stratification we present an efficient sampling scheme that consistently outperforms previous techniques. This is achieved by assembling sampling patterns that are stratified in the sense of jittered sampling and Nrooks sampling at the same time. The faster convergence and improved antialiasing are demonstrated by numerical experiments.
Scrambling Sobol and NiederreiterXing Points
 Journal of Complexity
, 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. In particular randomized (0; m; s)nets in base b produce unbiased estimates of the integral, have a variance that tend ..."
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Cited by 22 (7 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. In particular randomized (0; m; s)nets in base b produce unbiased estimates of the integral, have a variance that tends to zero faster than 1=n for any square integrable integrand, and have a variance that for finite n is never more than e : = 2:718 times as large as the Monte Carlo variance. Lower bounds than e are known for special cases. Some very important (t; m; s)nets have t ? 0. The widely used Sobol sequences are of this form, as are some recent and very promising nets due to Niederreiter and Xing. Much less is known about randomized versions of these nets, especially in s ? 1 dimensions. This paper shows that scrambled (t; m; s)nets enjoy the same properties as scrambled (0; m; s)nets, except the sampling variance is guaranteed only to be below b t ((b + 1)=(b \Gamma 1)) s times the Monte ...
Monte Carlo Extension Of QuasiMonte Carlo
 Proceedings of the 1998 Winter Simulation Conference
, 1998
"... This paper surveys recent research on using Monte Carlo techniques to improve quasiMonte Carlo techniques. Randomized quasiMonte Carlo methods provide a basis for error estimation. They have, in the special case of scrambled nets, also been observed to improve accuracy. Finally through Latin super ..."
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Cited by 15 (0 self)
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This paper surveys recent research on using Monte Carlo techniques to improve quasiMonte Carlo techniques. Randomized quasiMonte Carlo methods provide a basis for error estimation. They have, in the special case of scrambled nets, also been observed to improve accuracy. Finally through Latin supercube sampling it is possible to use Monte Carlo methods to extend quasiMonte Carlo methods to higher dimensional problems. 1 INTRODUCTION The problem we consider is the estimation of an integral I = Z [0;1] d f(x)dx: (1) Standard manipulations can be applied to express integrals over domains other than the unit cube or with respect to nonuniform measures in the form (1). Similarly, the integrand f in (1) subsumes weighting functions from importance sampling or periodization. We are especially interested in cases where the dimension d is large, and some of the methods considered here apply to the case d = 1. The focus of this article is on ways of combining Monte Carlo and quasiMo...
Computational Investigation of LowDiscrepancy Sequences in . . .
 PROCEEDINGS OF THE SIXTEENTH ANNUAL CONFERENCE ON UNCERTAINTY IN ARTIFICIAL INTELLIGENCE (UAI2000)
, 2000
"... Monte Carlo sampling has become a major vehicle for approximate inference in Bayesian networks. In this paper, we investigate a family of related simulation approaches, known collectively as quasiMonte Carlo methods based on deterministic lowdiscrepancy sequences. We first ..."
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Cited by 12 (2 self)
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Monte Carlo sampling has become a major vehicle for approximate inference in Bayesian networks. In this paper, we investigate a family of related simulation approaches, known collectively as quasiMonte Carlo methods based on deterministic lowdiscrepancy sequences. We first