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Article

Monte Carlo Methods and the Koksma-Hlawka Inequality

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The Faculty of Mathematics and Mechanics, St. Petersburg State University, 199034 St. Petersburg, Russia
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Mathematics 2019, 7(8), 725; https://doi.org/10.3390/math7080725
Received: 30 June 2019 / Revised: 5 August 2019 / Accepted: 7 August 2019 / Published: 9 August 2019
(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)
The solution of a wide class of applied problems can be represented as an integral over the trajectories of a random process. The process is usually modeled with the Monte Carlo method and the integral is estimated as the average value of a certain function on the trajectories of this process. Solving this problem with acceptable accuracy usually requires modeling a very large number of trajectories; therefore development of methods to improve the accuracy of such algorithms is extremely important. The paper discusses Monte Carlo method modifications that use some classical results of the theory of cubature formulas (quasi-random methods). A new approach to the derivation of the well known Koksma-Hlawka inequality is pointed out. It is shown that for high ( $s > 5$ ) dimensions of the integral, the asymptotic decrease of the error comparable to the asymptotic behavior of the Monte Carlo method, can be achieved only for a very large number of nodes N. It is shown that a special criterion can serve as a correct characteristic of the error decrease (average order of the error decrease). Using this criterion, it is possible to analyze the error for reasonable values of N and to compare various quasi-random sequences. Several numerical examples are given. Obtained results make it possible to formulate recommendations on the correct use of the quasi-random numbers when calculating integrals over the trajectories of random processes. View Full-Text
MDPI and ACS Style

Ermakov, S.; Leora, S. Monte Carlo Methods and the Koksma-Hlawka Inequality. Mathematics 2019, 7, 725. https://doi.org/10.3390/math7080725

AMA Style

Ermakov S, Leora S. Monte Carlo Methods and the Koksma-Hlawka Inequality. Mathematics. 2019; 7(8):725. https://doi.org/10.3390/math7080725

Chicago/Turabian Style

Ermakov, Sergey, and Svetlana Leora. 2019. "Monte Carlo Methods and the Koksma-Hlawka Inequality" Mathematics 7, no. 8: 725. https://doi.org/10.3390/math7080725

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