International Conference on Monte Carlo techniques
Closing conference of thematic cycle

Paris July 5-8th 2016 
Campus les cordeliers
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Ninomiya-Victoir scheme: strong convergence, antithetic version and application to multilevel estimators
Anis Al Gerbi  1@  , Benjamin Jourdain  1@  , Emmanuelle Clément  2@  
1 : Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique  (CERMICS)  -  Website
Université Paris Est (UPE), École des Ponts ParisTech (ENPC)
6 et 8 avenue Blaise Pascal Cité Descartes - Champs sur Marne 77455 Marne la Vallée Cedex 2 -  France
2 : Laboratoire d'Analyse et de Mathématiques Appliquées  (LAMA)  -  Website
Fédération de Recherche Bézout, CNRS : UMR8050, Université Paris-Est Créteil Val-de-Marne (UPEC), Université Paris-Est Marne-la-Vallée (UPEMLV)
Université de Paris-Est - Marne-la-Vallée, Cité Descartes, Bâtiment Copernic, 5 bd Descartes, 77454 Marne-la-Vallée Cedex 2, Lab Anal & Math Appl, Equipe Anal & Math Appl -  France

In this paper, we are interested in the strong convergence properties of the Ninomiya-Victoir scheme which is known to exhibit weak convergence with order 2. We prove strong convergence with order 1/2. This study is aimed at analysing the use of this scheme either at each level or only at the finest level of a multilevel Monte Carlo estimator: indeed, the variance of a multilevel Monte Carlo estimator is related to the strong error between the two schemes used on the coarse and fine grids at each level. Recently, Giles and Szpruch proposed a scheme permitting to construct a multilevel Monte Carlo estimator achieving the optimal complexityO(ϵ2) for the precision ϵ. In the same spirit, we propose a modified Ninomiya-Victoir scheme, which may be strongly coupled with order 1 to the Giles-Szpruch scheme at the finest level of a multilevel Monte Carlo estimator. Numerical experiments show that this choice improves the efficiency, since the order 2 of weak convergence of the Ninomiya-Victoir scheme permits to reduce the number of discretization levels.



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