# NASPDE Conference - Abstracts

### Titles and abstracts

**Xavier Blanc**(University of Paris 6) :

*Some variants of stochastic homogenization*

We present some variants of stochastic homogenization theory for scalar elliptic equations in divergence form. These variants basically consist in defining stochastic coefficients from stochastic deformations (using random diffeormorphisms) of the periodic setting. Some variance reduction, applied to homogenization, will also be presented, their validity being assessed theoretically using the preceding approach. These are joint works with R. Costaouec, F. Legoll, C. Le Bris and P.-L. Lions.

**Charles-Edouard Bréhier**(ENS Cachan Bretagne) :

*Analysis of the Heterogeneous Multiscale Method for a system of SPDEs*

I will present a numerical scheme based on HMM for a system of SPDEs with two-time scales. The error is controlled in a strong and in a weak sense. One of the key elements to ensure the efficiency of the scheme relies on the approximation of the so-called averaged coefficient via realizations of a scheme on the fast equation. I will also discuss this question.

**Zdzislaw Brzezniak**(University of York) :

*Strong and weak solutions to stochastic Landau-Lifshitz and geometric wave equations*

I will speak about the existence of weak solutions (and the existence and uniqueness of strong solutions) to the stochastic geometric wave Landau-Lifshitz equations for multi (and one)-dimensional spatial domains. I will also describe the corresponding Large Deviations principle and it's applications to a ferromagnetic wire.

The talk is based on a joint works with M. Ondrejat, and B. Goldys and T. Jegaraj.

**Julia Charrier**(University of Aix Marseille) :

*Numerical analysis of the advection-diffusion of a solute in porous media with uncertainty*

We consider a probabilistic numerical method to compute the spread, and its derivative, of a solute in a porous medium in the presence of uncertainty. A Monte-Carlo method is used to deal with uncertainty, and the solution of the advection-diffusion equation is approximated thanks to the time discretization of SDEs. Error estimates are established, under some assumptions including the case of random fields of lognormal type with low regularity.

**Fabio Nobile**(EPF Lausanne) :

*Perturbation methods and low rank approximations for the Darcy equation with log-normal permeability*

We consider the Darcy equation to describe the flow in a saturated porous medium. The permeability of the medium is described as a log-normal random field, eventually conditioned to available direct measurements, to account for its relatively large uncertainty and heterogeneity.

We consider perturbation methods based on Taylor expansion of the solution of the PDE around the nominal permeability value. Successive higher order corrections to the statistical moments such as pointwise mean and covariance of the solution can be obtained recursively from the computation of high order correlation functions which, on their turn, solve high dimensional problems. To overcome the curse of dimensionality in computing and storing such high order correlations, we adopt a low-rank format, namely the so called tensor-train (TT) format.

We show that, on the one hand, the Taylor series diverges, so that it only makes sense to compute corrections up to a maximum critical order, beyond which the accuracy of the solution deteriorates instead of improving. On the other hand, we show on some numerical test cases, the effectiveness of the proposed approach in case of a moderately small variance of the log-normal permeability field.

**Andreas Prohl**(University of Tübingen) :

*Strong convergence with rates for discretizations of SPDEs with non-Lipschitz drift*

I discuss the convergence analysis for space-time discretizations of three nonlinear SPDE's: the stochastic Navier-Stokes equation, the stochastic Allen-Cahn equation, and the stochastic mean curvature flow of planar curves of graphs. Depending on the drift operator, optimal rates w.r.t. strong convergence are valid for errors on large subsets, or on the whole sample set.

**Christoph Schwab**(ETH Zürich) :

*Sparse Adaptive Smolyak Quadrature Approach to Bayesian Inverse Problems*

Download abstract

**Aretha Teckentrup**(University of Bath) :

*Multilevel Markov chain Monte Carlo algortihms for uncertainty quantification in subsurface flow*

The quantification of uncertainty in groundwater flow plays a central role in the safety assessment of radioactive waste disposal and of CO2 capture and storage underground. Stochastic modelling of data uncertainties in the rock permeabilities lead to elliptic PDEs with random coefficients. Typical models used for the random coefficients, such as log-normal random fields with exponential covariance, are unbounded and have only limited spatial regularity, making practical computations very expensive and the rigorous numerical analysis challenging.

To overcome the problem of the prohibitively large computational cost of existing Markov chain Monte Carlo (MCMC) methods, we develop and analyse a new multilevel MCMC algorithm, based on a hierarchy of spatial levels/grids. We will demonstrate on a typical model problem the significant gains with respect to conventional MCMC that are possible with this new approach, and provide a full convergence analysis of the new algorithm.

**Gilles Vilmart**(University of Geneva, ENS Rennes and INRIA) : *Weak second order mean-square stable integrators for stiff stochastic differential equations *

We present two families of integrators (implicit and explicit stabilized) for stiff Itô stochastic differential equations which exhibit simultaneously favourable mean-square stability properties and weak second order of accuracy. These constructions inspired the design of a “swiss-knife” integrator for stiff diffusion-advection-reaction problems with noise.

This work in collaboration with Assyr Abdulle (EPF Lausanne) and Konstantinos C. Zygalakis (Univ. Southompton).

**Jochen Voss**(University of Leeds) :

*The Stationary Distribution of Discretised SPDEs*

I will discuss the problem of how discretisation error affects the solution of a stochastic partial differential equation (SPDE). In this talk I will focus on finite element discretisation of SPDEs of reaction-diffusion type (in one space dimension). In this situation it transpires that one can give a surprisingly explicit description of the discretisation error; based on this I will derive a result about speed of convergence of the stationary distribution of the discretised SPDE to the correct stationary distribution.

The talk is based on the following article:

Jochen Voss: The Effect of Finite Element Discretisation on the Stationary Distribution of SPDEs. Communications in Mathematical Sciences, vol. 10, no. 4, pp. 1143–1159, 2012.