Section: New Results

Analysis of probabilistic numerical methods

Particles approximation of mean-field SDEs

O. Bencheikh and Benjamin Jourdain have proved that the weak error between a stochastic differential equation with nonlinearity in the sense of McKean given by moments and its approximation by the Euler discretization with time-step $h$ of a system of $N$ interacting particles is $𝒪(N^{-1}+h)$. Numerical experiments confirm this behaviour and show that it extends to more general mean-field interaction.

Approximation of Markov processes

V. Bally worked on general approximation schemes in total variation distance for diffusion processes in collaboration with his former Phd student Clément Rey [52] This work includes high order schemes as Victoir-Ninomya for example. Further development in this direction is under study in collaboration with A. Alfonsi. Moreover, in collaboration with his former Phd student V. Rabiet and with D. Goreac (University Paris Est Marne la Vallée), V. Bally is studying approximations schemes for Piecewise Deterministic Markov Processes (see [17], [51]). In this framework the goal is to replace small jumps by a Brownien component - such a procedure is popular for "usual" jump equations, but the estimate of the error in the case of PDMP's is much more delicate. A significant example is the Bolzmann equation [28].

High order approximation for diffusion processes

A. Alfonsi and V. Bally are working on a generic method to achieve any weak order of convergence for approximating SDEs.

Adaptive MCMC methods

The Self-Healing Umbrella Sampling (SHUS) algorithm is an adaptive biasing algorithm which has been proposed in order to efficiently sample a multimodal probability measure.

In [21], G. Fort, B. Jourdain, T. Lelièvre and G. Stoltz extend previous works [68], [66], [67] and study a larger class of algorithms where the target distribution is biased using only a fraction of the free energy and which includes a discrete version of well-tempered metadynamics.