Quasi-Monte Carlo simulation of differential equations

Chouraqui Aïcha (), Lécot Christian () and Djebbar Bachir ()
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Chouraqui Aïcha: Faculté des Sciences, Université de Tlemcen, B.P. 119, 13000Tlemcen, Algeria
Lécot Christian: Université Savoie Mont Blanc, CNRS, LAMA, 73000Chambéry, France
Djebbar Bachir: Faculté de Mathématiques et d’Informatique, Université des Sciences et de la Technologie d’Oran,B.P. 1505, 31000Oran, Algeria

Monte Carlo Methods and Applications, 2017, vol. 23, issue 4, 265-275

Abstract: We are interested in the numerical solution of the ordinary differential equation y′⁢(t)=f⁢(t,y⁢(t)){y^{\prime}(t)=f(t,y(t))} when f is smooth in y but lacks regularity in t. We describe a family of methods akin to the Runge–Kutta family. It involves Monte Carlo simulation of integrals. We focus on third-order schemes which use random samples in dimension three. We give error bounds in terms of the step size and the discrepancy of the set used for the Monte Carlo approximations. We solve a model problem in which f undergoes rapid time variations. It is shown for this example that, by using a quasi-random point set in place of pseudo-random samples, we are able to obtain reduced errors.

Keywords: Monte Carlo methods; quasi-Monte Carlo methods; Runge–Kutta methods; ordinary differential equations; discrepancy (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1515/mcma-2017-0114

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