Approximation of the invariant measure with an Euler scheme for Stochastic PDE's driven by Space-Time White Noise

In this article, we consider a stochastic PDE of parabolic type, driven by a space-time white-noise, and its numerical discretization in time with a semi-implicit Euler scheme. When the nonlinearity is assumed to be bounded, then a dissipativity assumption is satisfied, which ensures that the SDPE admits a unique invariant probability measure, which is ergodic and strongly mixing - with exponential convergence to equilibrium. Considering test functions of class $\mathcal{C}^2$, bounded and with bounded derivatives, we prove that we can approximate this invariant measure using the numerical scheme, with order 1/2 with respect to the time step

Analysis of a HMM time-discretization scheme for a system of Stochastic PDE's

We consider the discretization in time of a system of parabolic stochastic partial differential equations with slow and fast components; the fast equation is driven by an additive space-time white noise. The numerical method is inspired by the Averaging Principle satisfied by this system, and fits to the framework of Heterogeneous Multiscale Methods.The slow and the fast components are approximated with two coupled numerical semi-implicit Euler schemes depending on two different timestep sizes. We derive bounds of the approximation error on the slow component in the strong sense - approximation of trajectories - and in the weak sense - approximation of the laws. The estimates generalize the results of \cite{E-L-V} in the case of infinite dimensional processes

Strong and weak order in averaging for SPDEs

We show an averaging result for a system of stochastic evolution equations of parabolic type with slow and fast time scales. We derive explicit bounds for the approximation error with respect to the small parameter defining the fast time scale. We prove that the slow component of the solution of the system converges towards the solution of the averaged equation with an order of convergence is 1/2 in a strong sense - approximation of trajectories - and 1 in a weak sense - approximation of laws. These orders turn out to be the same as for the SDE case

Large deviations principle for the Adaptive Multilevel Splitting Algorithm in an idealized setting

The Adaptive Multilevel Splitting (AMS) algorithm is a powerful and versatile method for the simulation of rare events. It is based on an interacting (via a mutation-selection procedure) system of replicas, and depends on two integer parameters: n $\in$ N * the size of the system and the number k $\in$ {1, . . . , n -- 1} of the replicas that are eliminated and resampled at each iteration. In an idealized setting, we analyze the performance of this algorithm in terms of a Large Deviations Principle when n goes to infinity, for the estimation of the (small) probability P(X \textgreater{} a) where a is a given threshold and X is real-valued random variable. The proof uses the technique introduced in [BLR15]: in order to study the log-Laplace transform, we rely on an auxiliary functional equation. Such Large Deviations Principle results are potentially useful to study the algorithm beyond the idealized setting, in particular to compute rare transitions probabilities for complex high-dimensional stochastic processes

Kolmogorov Equations and Weak Order Analysis for SPDES with Nonlinear Diffusion Coefficient

We provide new regularity results for the solutions of the Kolmogorov equation associated to a SPDE with nonlinear diffusion coefficients and a Burgers type nonlinearity. This generalizes previous results in the simpler cases of additive or affine noise. The basic tool is a discrete version of a two sided stochastic integral which allows a new formulation for the derivatives of these solutions. We show that this can be used to generalize the weak order analysis performed in [16]. The tools we develop are very general and can be used to study many other examples of applications

Convergence analysis of Adaptive Biasing Potential methods for diffusion processes

This article is concerned with the mathematical analysis of a family of adaptive importance sampling algorithms applied to diffusion processes. These methods, referred to as Adaptive Biasing Potential methods, are designed to efficiently sample the invariant distribution of the diffusion process, thanks to the approximation of the associated free energy function (relative to a reaction coordinate). The bias which is introduced in the dynamics is computed adaptively; it depends on the past of the trajectory of the process through some time-averages. We give a detailed and general construction of such methods. We prove the consistency of the approach (almost sure convergence of well-chosen weighted empirical probability distribution). We justify the efficiency thanks to several qualitative and quantitative additional arguments. To prove these results , we revisit and extend tools from stochastic approximation applied to self-interacting diffusions, in an original context