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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 71 (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 70 (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 69 (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
Simulation estimation of mixed discrete choice models using randomized and scrambled halton sequences
 Transportation Research Part B
, 2002
"... The use of simulation techniques has been increasing in recent years in the transportation and related fields to accommodate flexible and behaviorally realistic structures for analysis of decision processes. This paper proposes a randomized and scrambled version of the Halton sequence for use in sim ..."
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Cited by 34 (3 self)
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The use of simulation techniques has been increasing in recent years in the transportation and related fields to accommodate flexible and behaviorally realistic structures for analysis of decision processes. This paper proposes a randomized and scrambled version of the Halton sequence for use in simulation estimation of discrete choice models. The scrambling of the Halton sequence is motivated by the rapid deterioration of the standard Halton sequence's coverage of the integration domain in high dimensions of integration. The randomization of the sequence is motivated from a need to statistically compute the simulation variance of model parameters. The resulting hybrid sequence combines the good coverage property of quasiMonte Carlo sequences with the ease of estimating simulation error using traditional Monte Carlo methods. The paper develops an evaluation framework for assessing the performance of the traditional pseudorandom sequence, the standard Halton sequence, and the scrambled Halton sequence. The results of computational experiments indicate that the scrambled Halton sequence performs better than the standard Halton sequence and the traditional pseudorandom sequence for simulation estimation of models with high dimensionality of integration.
Randomized Halton Sequences
 Mathematical and Computer Modelling
, 2000
"... The Halton sequence is a wellknown multidimensional low discrepancy sequence. In this paper, we propose a new method for randomizing the Halton sequence: we randomize the start point of each component of the sequence. This method combines the potential accuracy advantage of Halton sequence in mult ..."
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Cited by 27 (1 self)
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The Halton sequence is a wellknown multidimensional low discrepancy sequence. In this paper, we propose a new method for randomizing the Halton sequence: we randomize the start point of each component of the sequence. This method combines the potential accuracy advantage of Halton sequence in multidimensional integration with the practical error estimation advantage of Monte Carlo methods. Theoretically, using multiple randomized Halton sequences as a variance reduction technique we can obtain an efficiency improvement over standard Monte Carlo under rather general conditions. Numerical results show that randomized Halton sequences have better performance than not only Monte Carlo, but also randomly shifted Halton sequences and (single long) purely deterministic skipped Halton sequence. Key Words: QuasiMonte Carlo methods, low discrepancy sequences, Monte Carlo methods, numerical integration, variance reduction. AMS 1991 Subject Classification: 65C05, 65D30. This work was suppor...
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 24 (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 ...
The Dimension Distribution, and Quadrature Test Functions
"... This paper introduces the dimension distribution for a square integrable function f on [0; 1]^s. The dimension distribution is used to relate several definitions of the effective dimension of a function. Functions of low effective dimension can be easy to integrate numerically. ..."
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Cited by 24 (4 self)
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This paper introduces the dimension distribution for a square integrable function f on [0; 1]^s. The dimension distribution is used to relate several definitions of the effective dimension of a function. Functions of low effective dimension can be easy to integrate numerically.
The Mean Square Discrepancy of Randomized Nets
, 1996
"... this article a formula for the mean square L ..."
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 16 (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...
Fast Generation of Randomized LowDiscrepancy Point Sets
, 2001
"... We introduce two novel techniques for speeding up the generation of digital (t,s)sequences. Based on these results a new algorithm for the construction of Owen's randomly permuted (t,s)sequences is developed and analyzed. An implementation is available at http://www.mcqmc.org/Software.html. ..."
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Cited by 12 (1 self)
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We introduce two novel techniques for speeding up the generation of digital (t,s)sequences. Based on these results a new algorithm for the construction of Owen's randomly permuted (t,s)sequences is developed and analyzed. An implementation is available at http://www.mcqmc.org/Software.html.