A Tool for Custom Construction of QMC and RQMC Point Sets

L’Ecuyer, Pierre; Marion, Pierre; Godin, Maxime; Puchhammer, Florian

doi:10.1007/978-3-030-98319-2_3

Cited by 13 publications

(7 citation statements)

References 54 publications

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“…However, many applications exhibit a structure where the uniformity of a subset of dimensions greatly impacts the integration error. This has already been observed in the work of Joe and Kuo [3] and L'Ecuyer et al [6]. As the dimension of the subsets often is much smaller than the global dimension, the 𝑡-value of the subset of dimensions theoretically can be smaller than the global 𝑡-value.…”

Section: Overlapping Constraintssupporting

confidence: 53%

Generator Matrices by Solving Integer Linear Programs

Paulin¹,

Cœurjolly²,

Bonneel³

et al. 2023

Preprint

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In quasi-Monte Carlo methods, generating high-dimensional low discrepancy sequences by generator matrices is a popular and efficient approach. Historically, constructing or finding such generator matrices has been a hard problem. In particular, it is challenging to take advantage of the intrinsic structure of a given numerical problem to design samplers of low discrepancy in certain subsets of dimensions. To address this issue, we devise a greedy algorithm allowing us to translate desired net properties into linear constraints on the generator matrix entries. Solving the resulting integer linear program yields generator matrices that satisfy the desired net properties. We demonstrate that our method finds generator matrices in challenging settings, offering low discrepancy sequences beyond the limitations of classic constructions.

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Section: Overlapping Constraintssupporting

confidence: 53%

Generator Matrices by Solving Integer Linear Programs

Paulin¹,

Cœurjolly²,

Bonneel³

et al. 2023

Preprint

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“…, n − 1. In our experiments, the vector a a a was selected using LatNet Builder [20,19] with the P α criterion with α = 2, product weights, and a weight of γ j = 2/(2 + j) for coordinate j. See [19] for the details.…”

Section: Point Sets and Randomization Methodsmentioning

confidence: 99%

“…As a special case, if f is assumed to have bounded variation in the sense of Hardy and Krause, the rate is O(n −1 (ln n) s ). Explicit constructions of these point sets and sequences are also available, mainly in the form of lattice point sets and digital nets and sequences [5,19,4]. One limitation, however, is that QMC fails to provide practical (easily computable) error bounds, so it is difficult to assess its accuracy [39,26].…”

Section: Introductionmentioning

confidence: 99%

Confidence Intervals for Randomized Quasi-Monte Carlo Estimators

L’Ecuyer,

Nakayama,

Owen

et al. 2023

2023 Winter Simulation Conference (WSC)

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Randomized Quasi-Monte Carlo (RQMC) methods provide unbiased estimators whose variance often converges at a faster rate than standard Monte Carlo as a function of the sample size. However, computing valid confidence intervals is challenging because the observations from a single randomization are dependent and the central limit theorem does not ordinarily apply. The natural solution is to replicate the RQMC process independently a small number of times to estimate the variance and use a standard confidence interval based on the normal or Student distribution. In this paper we investigate bootstrap methods for getting nonparametic confidence intervals for the mean using a modest number of replicates. Our main conclusion is that intervals based on the Student t distribution are more reliable than even the bootstrap t method on the integration problems arising from RQMC.In our examples, the individual RQMC estimates often had extreme kurtosis but always had mild skewness. That situation is very favorable for the standard confidence intervals. Simulation experiments cannot cover all the cases of interest, so we add some caveats in our conclusions.

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“…Randomized quasi-Monte Carlo (RQMC) is an alternative sampling approach which under favorable conditions can improve this convergence rate of the variance to O(𝑛 −𝛼+𝜖 ) for any 𝜖 > 0, for some constant 𝛼 that can often reach 2, and even larger values in special situations (Owen, 1997b;L'Ecuyer and Lemieux, 2002;L'Ecuyer, 2009L'Ecuyer, , 2018L'Ecuyer et al, 2020). Quasi-Monte Carlo (QMC) replaces the 𝑛 independent vectors of uniform random numbers that drive the simulations by 𝑛 deterministic vectors with a sufficient number of coordinates to simulate the system and which cover the space (the unit hypercube) more evenly than typical independent random points (Niederreiter, 1992;Dick and Pillichshammer, 2010).…”

Section: Introductionmentioning

confidence: 99%

Variance Reduction with Array-RQMC for Tau-Leaping Simulation of Stochastic Biological and Chemical Reaction Networks

2021

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We explore the use of Array-RQMC, a randomized quasi-Monte Carlo method designed for the simulation of Markov chains, to reduce the variance when simulating stochastic biological or chemical reaction networks with 𝜏-leaping. The task is to estimate the expectation of a function of molecule copy numbers at a given future time 𝑇 by the sample average over 𝑛 sample paths, and the goal is to reduce the variance of this sample-average estimator. We find that when the method is properly applied, variance reductions by factors in the thousands can be obtained. These factors are much larger than those observed previously by other authors who tried RQMC methods for the same examples. Array-RQMC simulates an array of realizations of the Markov chain and requires a sorting function to reorder these chains according to their states, after each step. The choice of sorting function is a key ingredient for the efficiency of the method, although in our experiments, Array-RQMC was never worse than ordinary Monte Carlo, regardless of the sorting method. The expected number of reactions of each type per step also has an impact on the efficiency gain.

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A Tool for Custom Construction of QMC and RQMC Point Sets

Cited by 13 publications

References 54 publications

Generator Matrices by Solving Integer Linear Programs

Generator Matrices by Solving Integer Linear Programs

Confidence Intervals for Randomized Quasi-Monte Carlo Estimators

Variance Reduction with Array-RQMC for Tau-Leaping Simulation of Stochastic Biological and Chemical Reaction Networks

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