A Modified Monte-Carlo Quadrature

Haber, Seymour

doi:10.2307/2003589

Cited by 20 publications

(23 citation statements)

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“…In Section 3 we state and prove precise error estimates for the randomized Riemann sum quadrature rule, which are an important ingredient in our error analysis for the randomized Runge-Kutta methods (3) and (4). Randomized quadrature rules are well-known to the literature, see [14,15]. However, this is apparently the first time they are applied in the error analysis of randomized Runge-Kutta methods.…”

Section: Introductionmentioning

confidence: 99%

Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients

Kruse

2017

Computational Methods in Applied Mathematics

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Abstract. This paper contains an error analysis of two randomized explicit Runge-Kutta schemes for ordinary differential equations (ODEs) with timeirregular coefficient functions. In particular, the methods are applicable to ODEs of Carathéodory type, whose coefficient functions are only integrable with respect to the time variable but are not assumed to be continuous. A further field of application are ODEs with coefficient functions that contain weak singularities with respect to the time variable.The main result consists of precise bounds for the discretization error with respect to the L p (Ω; R d )-norm. In addition, convergence rates are also derived in the almost sure sense. An important ingredient in the analysis are corresponding error bounds for the randomized Riemann sum quadrature rule. The theoretical results are illustrated through a few numerical experiments.

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Section: Introductionmentioning

confidence: 99%

Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients

Kruse

2017

Computational Methods in Applied Mathematics

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“…A first approach to obtain a negative sum of covariances in (5) is to stratify the unit hypercube [16]. We partition axis j in k j ≥ 1 equal parts, for j = 1, .…”

Section: Stratificationmentioning

confidence: 99%

“…One can estimate the variance by replicating the scheme m ≥ 2 times, computing the empirical variance in each box, and averaging. If f is continuous and bounded, and all k j are equal to k (so n = k s ), then by using a Taylor expansion in each box one can show [16] that Var[X s,n ] = O(n −1−2/s ). This may provide a significant improvement when s is small, but for large s, the rate is not much better than for MC, and the method quickly becomes impractical because n increases exponentially with k. Nevertheless, it is sometimes effective to apply stratification to just a few important (selected) coordinates.…”

Section: Stratificationmentioning

confidence: 99%

“…We can use RQMC with any of these three estimators. We try a Sobol' net with one extra coordinate (i.e., the Hammersley point set) with n = 2 16 randomized by LMS+DS, the same point set with NUS, and also a randomly shifted lattice rule with a baker's transformation, in two dimensions, with the same n. With this we find that the option price is 1.601 × 10 −4 (the given digits here are significant). Table 4 reports the empirical variances for all the estimators discussed above, with both MC and RQMC.…”

Section: A Financial Option Under a Variance-gamma Processmentioning

confidence: 99%

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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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“…Regarding the Xi as (pairwise) independent random variables uniformly distributed on G 8f Jo is a random variable with mean /; the amount by which it is apt to differ from I is estimated in terms of its standard deviations (Jo). In general (îorf&L 2 …”

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confidence: 99%

A Modified Monte-Carlo Quadrature

Cited by 20 publications

References 0 publications

Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients

Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients

Randomized Quasi-Monte Carlo: An Introduction for Practitioners

A combination of Monte Carlo and classical methods for evaluating multiple integrals

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