Ninomiya-Victoir scheme: strong convergence, antithetic version and application to multilevel estimators

TL;DR

This paper establishes the strong convergence order of the Ninomiya-Victoir scheme, proposes a coupled scheme for multilevel Monte Carlo methods, and demonstrates improved efficiency through numerical experiments.

Contribution

It proves strong convergence of the Ninomiya-Victoir scheme, introduces a coupled scheme with Giles-Szpruch for multilevel estimators, and shows efficiency gains in simulations.

Findings

Strong convergence order 1/2 for Ninomiya-Victoir scheme

Coupled scheme improves multilevel Monte Carlo efficiency

Reduced discretization levels due to higher weak convergence order

Abstract

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 complexity $O\left(\epsilon^{-2}\right)$ for the precision $\epsilon$. 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.