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

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

Asymptotic error distribution for the Ninomiya-Victoir scheme in the commutative case

In a previous work, we proved strong convergence with order $1$ of the Ninomiya-Victoir scheme $X^{NV}$ with time step $T/N$ to the solution $X$ of the limiting SDE when the Brownian vector fields commute. In this paper, we prove that the normalized error process $N \left(X - X^{NV}\right)$ converges to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields and the drift vector field. This result ensures that the strong convergence rate is actually $1$ when the Brownian vector fields commute, but at least one of them does not commute with the drift vector field. When all the vector fields commute the limit vanishes. Our result is consistent with the fact that the Ninomiya-Victoir scheme solves the SDE in this case.Comment: arXiv admin note: text overlap with arXiv:1601.0526

A Radio-fingerprinting-based Vehicle Classification System for Intelligent Traffic Control in Smart Cities

The measurement and provision of precise and upto-date traffic-related key performance indicators is a key element and crucial factor for intelligent traffic controls systems in upcoming smart cities. The street network is considered as a highly-dynamic Cyber Physical System (CPS) where measured information forms the foundation for dynamic control methods aiming to optimize the overall system state. Apart from global system parameters like traffic flow and density, specific data such as velocity of individual vehicles as well as vehicle type information can be leveraged for highly sophisticated traffic control methods like dynamic type-specific lane assignments. Consequently, solutions for acquiring these kinds of information are required and have to comply with strict requirements ranging from accuracy over cost-efficiency to privacy preservation. In this paper, we present a system for classifying vehicles based on their radio-fingerprint. In contrast to other approaches, the proposed system is able to provide real-time capable and precise vehicle classification as well as cost-efficient installation and maintenance, privacy preservation and weather independence. The system performance in terms of accuracy and resource-efficiency is evaluated in the field using comprehensive measurements. Using a machine learning based approach, the resulting success ratio for classifying cars and trucks is above 99%