Obtaining O(N - 2+∈) Convergence for Lattice Quadrature Rules

Hickernell, Fred J.

doi:10.1007/978-3-642-56046-0_18

Cited by 68 publications

(98 citation statements)

References 17 publications

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“…But even in these cases, there can be significant variance reductions compared with standard MC, and sometimes compared with classical RQMC as well. Our paper also provides novel empirical evidence of the effectiveness of applying the baker's transformation over a randomly shifted lattice rule, an idea that was studied theoretically by Hickernell (2002).…”

Section: Resultsmentioning

confidence: 97%

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A Randomized Quasi-Monte Carlo Simulation Method for Markov Chains

L’Ecuyer

Lécot

Tuffin

2008

Operations Research

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Abstract. We introduce and study a randomized quasi-Monte Carlo method for estimating the state distribution at each step of a Markov chain. The number of steps in the chain can be random and unbounded. The method simulates n copies of the chain in parallel, using a (d + 1)-dimensional highly-uniform point set of cardinality n, randomized independently at each step, where d is the number of uniform random numbers required at each transition of the Markov chain. This technique is effective in particular to obtain a low-variance unbiased estimator of the expected total cost up to some random stopping time, when state-dependent costs are paid at each step. It is generally more effective when the state space has a natural order related to the cost function.We provide numerical illustrations where the variance reduction with respect to standard Monte Carlo is substantial. The variance can be reduced by factors of several thousands in some cases. We prove bounds on the convergence rate of the worst-case error and variance for special situations. In line with what is typically observed in randomized quasi-Monte Carlo contexts, our empirical results indicate much better convergence than what these bounds guarantee.

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Section: Resultsmentioning

confidence: 97%

“…This transformation has a "locally antithetic" effect and provably improves the convergence rate when integrating a smooth function with a randomly shifted lattice rule (Hickernell, 2002). For the parameters, we take n equal to the largest prime number smaller than 2 k for k = 10, 12, .…”

Section: Numerical Illustrationsmentioning

confidence: 99%

A Randomized Quasi-Monte Carlo Simulation Method for Markov Chains

L’Ecuyer

Lécot

Tuffin

2008

Operations Research

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“…Consider the change of variable v = ϕ(u) = 2u if u ≤ 1/2 and 1 − 2u if u > 1/2. It is known as the baker transformation [20]: it stretches the points by a factor of 2, from [0, 1) to [0, 2), and folds back the segment [1, 2) to [1, 0). Its impact on the RQMC estimator (4) is exactly equivalent to compressing the graph of f horizontally by a factor of 1/2, and then making a mirror copy on the interval [1/2, 1).…”

Section: Lattice Rulesmentioning

confidence: 99%

“…SS does not improve the variance rate very much, BSS does better, and PCA much better. For PCA, ν ≈ 1.9 and the variance is reduced by a factor of over two millions for n = 2 20 . Similar improvements were obtained in [35] with lattice rules constructed by Lattice Builder [36].…”

Section: A Financial Option Under a Variance-gamma Processmentioning

confidence: 99%

Randomized Quasi-Monte Carlo: An Introduction for Practitioners

L’Ecuyer

2018

Springer Proceedings in Mathematics &Amp; Statistics

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We survey basic ideas and results on randomized quasi-Monte Carlo (RQMC) methods, discuss their practical aspects, and give numerical illustrations. RQMC can improve accuracy compared with standard Monte Carlo (MC) when estimating an integral interpreted as a mathematical expectation. RQMC estimators are unbiased and their variance converges at a faster rate (under certain conditions) than MC estimators, as a function of the sample size. Variants of RQMC also work for the simulation of Markov chains, for function approximation and optimization, for solving partial differential equations, etc. In this introductory survey, we look at how RQMC point sets and sequences are constructed, how we measure their uniformity, why they can work for high-dimensional integrals, and how can they work when simulating Markov chains over a large number of steps.

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“…Here we give the results for (a) a (d + 1)-dimensional Korobov lattice rule with its first coordinate skipped, randomized by a random shift modulo 1 followed by a baker's transformation [1] (denoted ArrayKorobov ) and (b) the first n points of a Sobol sequence randomized by a left (upper triangular) matrix scrambling followed by a random digital shift [6,9] (denoted Array-Sobol ). For Array-Korobov, the multiplier a of the twodimensional Korobov lattice rule was selected so that n is prime, a is a primitive element modulo n (this requirement is actually not needed), and a/n is close to the golden ratio.…”

Section: An M/m/1 Queue With D =mentioning

confidence: 99%

Randomized Quasi-Monte Carlo Simulation of Markov Chains with an Ordered State Space

L’Ecuyer¹,

Lécot²,

Tuffin³

Monte Carlo and Quasi-Monte Carlo Methods 2004

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Summary. We study a randomized quasi-Monte Carlo method for estimating the state distribution at each step of a Markov chain with totally ordered (discrete or continuous) state space. The number of steps in the chain can be random and unbounded. The method can be used in particular to get a low-variance unbiased estimator of the expected total cost up to some random stopping time, when statedependent costs are paid at each step. We provide numerical illustrations where the variance reduction with respect to standard Monte Carlo is substantial.

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Obtaining O(N - 2+∈) Convergence for Lattice Quadrature Rules

Cited by 68 publications

References 17 publications

A Randomized Quasi-Monte Carlo Simulation Method for Markov Chains

A Randomized Quasi-Monte Carlo Simulation Method for Markov Chains

Randomized Quasi-Monte Carlo: An Introduction for Practitioners

Randomized Quasi-Monte Carlo Simulation of Markov Chains with an Ordered State Space

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