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38
QuasiRandom Sequences and Their Discrepancies
 SIAM J. Sci. Comput
, 1994
"... Quasirandom (also called low discrepancy) sequences are a deterministic alternative to random sequences for use in Monte Carlo methods, such as integration and particle simulations of transport processes. The error in uniformity for such a sequence of N points in the sdimensional unit cube is meas ..."
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Cited by 75 (6 self)
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Quasirandom (also called low discrepancy) sequences are a deterministic alternative to random sequences for use in Monte Carlo methods, such as integration and particle simulations of transport processes. The error in uniformity for such a sequence of N points in the sdimensional unit cube is measured by its discrepancy, which is of size (log N) s N \Gamma1 for large N , as opposed to discrepancy of size (log log N) 1=2 N \Gamma1=2 for a random sequence (i.e. for almost any randomlychosen sequence). Several types of discrepancy, one of which is new, are defined and analyzed. A critical discussion of the theoretical bounds on these discrepancies is presented. Computations of discrepancy are presented for a wide choice of dimension s, number of points N and different quasirandom sequences. In particular for moderate or large s, there is an intermediate regime in which the discrepancy of a quasirandom sequence is almost exactly the same as that of a randomly chosen sequence...
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...
Recent Advances In Randomized QuasiMonte Carlo Methods
"... We survey some of the recent developments on quasiMonte Carlo (QMC) methods, which, in their basic form, are a deterministic counterpart to the Monte Carlo (MC) method. Our main focus is the applicability of these methods to practical problems that involve the estimation of a highdimensional inte ..."
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Cited by 59 (12 self)
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We survey some of the recent developments on quasiMonte Carlo (QMC) methods, which, in their basic form, are a deterministic counterpart to the Monte Carlo (MC) method. Our main focus is the applicability of these methods to practical problems that involve the estimation of a highdimensional integral. We review several QMC constructions and dierent randomizations that have been proposed to provide unbiased estimators and for error estimation. Randomizing QMC methods allows us to view them as variance reduction techniques. New and old results on this topic are used to explain how these methods can improve over the MC method in practice. We also discuss how this methodology can be coupled with clever transformations of the integrand in order to reduce the variance further. Additional topics included in this survey are the description of gures of merit used to measure the quality of the constructions underlying these methods, and other related techniques for multidimensional integration. 1 2 1.
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 30 (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...
Faster Evaluation of Multidimensional Integrals
 Computers in Physics
, 1997
"... In a recent paper Keister proposed two quadrature rules as alternatives to Monte Carlo for certain multidimensional integrals and reported his test results. In earlier work we had shown that the quasiMonte Carlo method with generalized Faure points is very effective for a variety of high dimensiona ..."
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Cited by 22 (1 self)
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In a recent paper Keister proposed two quadrature rules as alternatives to Monte Carlo for certain multidimensional integrals and reported his test results. In earlier work we had shown that the quasiMonte Carlo method with generalized Faure points is very effective for a variety of high dimensional integrals occuring in mathematical finance. In this paper we report test results of this method on Keister's examples of dimension 9 and 25, and also for examples of dimension 60, 80 and 100. For the 25 dimensional integral we achieved accuracy of 10 \Gamma2 with less than 500 points while the two methods tested by Keister used more than 220; 000 points. In all of our tests, for n sample points we obtained an empirical convergence rate proportional to n \Gamma1 rather than the n \Gamma1=2 of Monte Carlo. 1 1 Introduction Keister [1] points out that multidimensional integrals arise frequently in many branches of physics. He rules out product rules of onedimensional methods becaus...
QuasiMonte Carlo Node Sets from Linear Congruential Generators
 and QuasiMonte Carlo Methods
, 1998
"... . In this paper we present a new approach to finding good lattice points. We employ the spectral test, a wellknown figure of merit for uniform random number generators. This concept leads to an assessment of lattice points g that is closely related to the classical BabenkoZaremba quantity ae(g; N ..."
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Cited by 17 (9 self)
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. In this paper we present a new approach to finding good lattice points. We employ the spectral test, a wellknown figure of merit for uniform random number generators. This concept leads to an assessment of lattice points g that is closely related to the classical BabenkoZaremba quantity ae(g; N ). The associated lattice rules are good uniformly over a whole range of dimensions. Our numerical examples suggest that this simple approach leads to quasiMonte Carlo node sets that perform very well in comparison to the best available (t; m; s)nets. 1 Introduction There is no contradiction in the title of this paper. We show how to employ concepts that belong to the field of random number generation to obtain excellent node sets for quasiMonte Carlo integration in high dimensions. Our approach uses linear congruential generators ("LCGs") and the spectral test to find good lattice points ("GLPs") of the Korobov type. LCGs are a classical method for generating uniform random numbers, see...
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.
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
The asymptotic efficiency of randomized nets for quadrature
 Math. Comp
, 1999
"... Abstract. An L2type discrepancy arises in the average and worstcase error analyses for multidimensional quadrature rules. This discrepancy is uniquely defined by K(x, y), which serves as the covariance kernel for the space of random functions in the averagecase analysis and a reproducing kernel ..."
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Cited by 7 (3 self)
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Abstract. An L2type discrepancy arises in the average and worstcase error analyses for multidimensional quadrature rules. This discrepancy is uniquely defined by K(x, y), which serves as the covariance kernel for the space of random functions in the averagecase analysis and a reproducing kernel for the space of functions in the worstcase analysis. This article investigates the asymptotic order of the root mean square discrepancy for randomized (0,m,s)nets in base b. For moderately smooth K(x, y) the discrepancy is O(N −1 [log(N)] (s−1)/2), and for K(x, y) with greater smoothness the discrepancy is O(N −3/2 [log(N)] (s−1)/2), where N = b m is the number of points in the net. Numerical experiments indicate that the (t, m, s)nets of Faure, Niederreiter and Sobol ′ do not necessarily attain the higher order of decay for sufficiently smooth kernels. However, Niederreiter nets may attain the higher order for kernels corresponding to spaces of periodic functions. 1.