arXiv subjects

Charles-Edouard Bréhier

Publications and source records attributed to Charles-Edouard Bréhier.

57 records · Page 4Linked to original sources

arXiv2012

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.

math.NA↗

arXiv2012

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.

math.NA↗

arXiv2012

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.

math.NA↗