# Timetable (SPDW04)

## Stochastic Partial Differential Equations (SPDEs) : Approximation, Asymtotics and Computation

Monday 28th June 2010 to Friday 2nd July 2010

 09:00 to 09:55 Registration 09:55 to 10:00 Welcome from David Wallace (INI Director) 10:00 to 10:50 Accelerated numerical schemes for deterministic and stochastic partial differential equations We present some recent joint results with Nicolai Krylov on accelerated numerical schemes for some classes of deterministic and stochastic PDEs. INI 1 10:50 to 11:30 Morning Coffee 11:30 to 12:20 Particle approximations for strong solutions of linear SPDEs with multiplicative noise Two classes of particle approximation for strong solutions of linear SPDEs with multiplicative noise are presented. The first is a Monte-Carlo type method and the second is based on the recent Kusuoka-Lyons-Victoir approach to approximate solution of SDEs. The work is motivated by and applied to nonlinear filtering. INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:10 to 15:00 J Mattingly ([Duke])SPDE scaling limits of an Markov chain Montecarlo algorithm I will discuss how a simple random walk metropolis algorithm converges to an SPDE as the dimension of the sample space goes to infinity. I will discuss how this the limiting SPDE gives insight into how one should tune the algorithm to obtain an asymptotically optimal mixing rate. This is joint work with Andrew Stuart and Natesh Pialli. INI 1 15:00 to 15:40 Afternoon Tea 15:40 to 16:30 J Nolen ([Duke])Reaction-diffusion waves in a random environment I will describe solutions of a scalar reaction diffusion equation with a spatially inhomogeneous reaction rate. When the environment is random and statistically stationary, the position of the wave is a stochastic process which has a well-defined asymptotic speed. Under suitable mixing conditions on the environment, the process may also satisfy a functional central limit theorem, depending on the form of the nonlinear term and on the initial condition. INI 1 16:30 to 17:20 B Rozovsky ([Brown])On Generalized Malliavin Calculus The Malliavin derivative, divergence operator, and the Uhlenbeck operator are extended from the traditional Gaussian setting to generalized processes. Usually, the driving random source in Malliavin calculus is assumed to be an isonormal Gaussian process on a separable Hilbert space. This process is in effect a linear combination of a countable collection of independent standard Gaussian random variables. In this talk we will discuss an extension of Malliavin calculus to nonlinear functionals of the isonormal Gaussian process as the driving random source. We will also extend the main operators of Malliavin calculus to the space of generalized random elements that arise in stochastic PDEs of various types. INI 1 17:30 to 18:30 Welcome Wine Reception 18:45 to 19:30 Dinner at Wolfson Court
 09:20 to 10:10 Asymptotic results for a class of stochastic RDEs with fast transport term and noise acting on the boundary We consider a class of stochastic reaction-diffusion equations having also a stochastic perturbation on the boundary and we show that when the diffusion rate is much larger than the rate of reaction it is possible to replace the SPDE’s by a suitable one- dimensional stochastic differential equation. We study the fluctuations around the averaged motion. INI 1 10:10 to 11:00 Perturbation theory for systems with many invariant measures: Long-time effects I will consider long time effects caused by deterministic and stochastic perturbations of dynamical systems or stochastic processes with many invariant measures. Deterministic patterns caused by random perturbations as well as stochasticity induced by deterministic perturbations of pure deterministic systems will be described. Related PDE problems will be discussed. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:20 R Dalang ([EPFL])Stochastic integrals for spde's: a comparison We present the Walsh theory of stochastic integrals with respect to martingale measures, and various extensions if this theory, alongside of the Da Prato and Zabczyk theory of stochastic integrals with respect to Hilbert-space-valued Wiener processes, and we explore the links between these theories. Somewhat surprisingly, the end results of both theories turn out to be essentially equivalent. We then show how each theory can be used to study stochastic partial differential equations, with an emphasis on the stochastic heat and wave equations driven by spatially homogeneous Gaussian noise that is white in time. We compare the solutions produced by the different theories. Authors: Robert Dalang (Ecole Polytechnique Fédérale de Lausanne), Lluis Quer-Sardanyons (Universitat Autònoma de Barcelona) INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:10 to 15:00 Stochastic heat equation with spatially-colored random forcing The aim of this talk is to show the connection between Levy processes and long term behavior of a class of stochastic heat equation. The rst part of the talk will be devoted to the case when the equation is driven by white noise. The second part of the talk will concern spatially-colored noise. INI 1 15:00 to 15:40 Afternoon Tea 15:40 to 16:30 D Khoshnevisan ([Utah])On the existence and position of the farthest peaks of a family of stochastic heat and wave equations We study the stochastic heat equation ∂tu = £u+σ(u)w in (1+1) dimensions, where w is space-time white noise, σ:R→R is Lipschitz continuous, and £ is the generator of a Lévy process. We assume that the underlying Lévy process has finite exponential moments in a neighborhood of the origin and u_0 has exponential decay at ±∞. Then we prove that under natural conditions on σ: (i) The νth absolute moment of the solution to our stochastic heat equation grows exponentially with time; and (ii) The distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. Very little else seems to be known about the location of the high peaks of the solution to the stochastic heat equation. Finally, we show that these results extend to the stochastic wave equation driven by Laplacian. This is joint work with Daniel Conus. INI 1 16:30 to 17:20 The status of the theory of stochastic viscosity sols and fully nonlinear 1st and 2nd order PDE INI 1 18:45 to 19:30 Dinner at Wolfson Court
 09:20 to 10:10 Rough viscosity solutions and applications to SPDEs INI 1 10:10 to 11:00 Stochastic Cahn-Hilliard equation with singularities and reflections We study the stochastic Cahn-Hilliard equation with an additive space-time white noise. We consider the physical potential with a double logarithmic singularity in -1 and +1 in a one-dimensionnal domain. Since the singularities are not strong enough to prevent the solution from going out the physical domain [-1,1], we add two reflection measures in the boundary. We show that the system has a unique invariant measure in order to obtain existence and uniqueness of stationary solution. We also prove some results about ergodicity, exponential mixing and integration by parts formula. This is a joint work with Arnaud Debussche (ENS Cachan Bretagne). INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:20 The Hybrid Monte Carlo Algorithm on Hilbert Space Hybrid Monte Carlo methods are a class of algorithms for sampling probability measures defined via a density with respect to Lebesgue measure. However, in many applications the probability measure of interest is on an infinite dimensional Hilbert space and is defined via a density with respect to a Gaussian measure. I will show how the Hybrid Monte Carlo methodology can be extended to this Hilbert space setting. A key building block is the study of measure preservation properties for certain semilinear partial differential equations of Hamiltonian type, and approximation of these equations by volume-preserving integrators. Joint work with A. Beskos (UCL), F. Pinski (Cincinnati) and J.-M. Sanz-Serna (Valladolid). INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:10 to 15:00 E Faou ([IRISA/INRIA])Weak backward error analysis for stochastic differential equations Backward error analysis is a powerful tool to understand the long time behavior of discrete approximations of deterministic differential equations. Roughly speaking, it can be shown that a discrete numerical solution associated with an ODE can be interpreted as the exact solution of a modified ODE over extremely long time with respect to the time discretization parameter. In this work, we consider numerical simulations of SDEs and we show a weak backward error analysis result in the sense that the generator associated with the numerical solution coincides with the solution of a modified Kolmogorov equation up to high order terms with respect to the stepsize. In the case where the SDE possesses a unique invariant measure with exponentially mixing properties, this implies that the numerical solution remains exponentially mixing for a modified quasi invariant measure over very long time. This is a joint work with Arnaud Debussche (ENS Cachan Bretagne). INI 1 15:00 to 15:40 Afternoon Tea 15:40 to 16:30 The Numerical Approximation for SPDEs driven by Levy Processes The topic of the talk is the numerical approximation of spdes driven by Levy noise. Here the main emphasis will be on the topic how to simulate a Levy walk. INI 1 16:30 to 18:30 Poster Session 19:30 to 22:00 Conference Dinner at Gonville & Caius College
 09:20 to 10:10 Solution of SPDEs with applications in porous media We consider the numerical approximation of a general second order semi-linear parabolic stochastic partial differential equation (SPDE) driven by space-time noise. We introduce time-stepping schemes that use a linear functional of the noise and analyse a finite element discretization in space. We present convergence results and illustrate the work with examples motivated from realistic porous media flow. INI 1 10:10 to 11:00 Kink stochastics Localised coherent structures are a striking feature of noisy, nonlinear, spatially-extended systems. In one space dimension with local bistability, coherent structures are kinks. At late times, a steady-state density is dynamically maintained: kinks are nucleated in pairs, diffuse and annihilate on collision. Long-term averages can be calculated using the transfer-integral method, developed in the 1970s, giving exact results that can be compared with large-scale numerical solutions of SPDE. More recently, the equivalence between the stationary density (in space) of an SPDE and that of a suitably-chosen diffusion process (in time) has been used, by a different community of researchers, to perform sampling of bridge diffusions. In this talk, diffusion-limited reaction is the name given to a reduced model of the SPDE dynamics, in which kinks are treated as point particles. Some quantities, such as the mean number of particles per unit length, can be calculated exactly. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:20 Finite element approximation of the Cahn-Hilliard-Cook equation We study the Cahn-Hilliard equation perturbed by additive colored noise also known as the Cahn-Hilliard-Cook equation. We show almost sure existence and regularity of solutions. We introduce spatial approximation by a standard finite element method and prove error estimates of optimal order on sets of probability arbitrarily close to $1$. We also prove strong convergence without known rate. This is joint work with Mihaly Kovacs, University of Otago, New Zealand, and Ali Mesforush, Chalmers University of Technology, Sweden. INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:10 to 15:00 Stochastic perturbation of scalar conservation laws In this joint work with Arnaud Debussche, we show that the Cauchy Problem for a randomly forced, periodic multi-dimensional scalar first-order conservation law with additive or multiplicative noise is well-posed: it admits a unique solution, characterized by a kinetic formulation of the problem, which is the limit of the solution of the stochastic parabolic approximation. INI 1 15:00 to 15:40 Afternoon Tea 15:40 to 16:30 A Szepessy (KTH NADA)Stochastic molecular dynamics Starting from the Schrödinger equation for nuclei-electron systems I will show two stochastic molecular dynamics effects derived from a Gibbs distribution: - when the ground state has a large spectral gap a precise Langevin equation for molecular dynamics approximates observables from the Schrödinger equation - if the gap is smaller in some sense, the temperature also gives a precise correction to the ab initio ground state potential energy. The two approximation results holds with a rate depending on the spectral gap and the ratio of nuclei and electron mass. I will also give an example of coarse-graining this stochastic Langevin molecular dynamics equation to obtain a continuum stochastic partial differential equation for phase transitions. INI 1 16:30 to 17:20 On the stochastic nonlinear Schrodinger equation We consider a non linear Schrodinger equation on a compact manifold of dimension d subject to some multiplicative random perturbation. Using some stochastic Strichartz inequality, we prove the existence and uniqueness of a maximal solution in H^1 under some general conditions on the diffusion coefficient. Under stronger conditions on the noise, the nonlinearity and the diffusion coefficient, we deduce the existence of a global solution when d=2. This is a joint work with Z. Brzezniak. INI 1 18:45 to 19:30 Dinner at Wolfson Court
 09:20 to 10:10 Stability analysis for numerical methods applied to systems of SODEs An important issue arising in the analysis of numerical methods for approximating the solution of a differential equation is concerned with the ability of the methods to preserve the asymptotic properties of equilibria. For stochastic ordinary differential equations and numerical methods applied to them investigations in this direction have mainly focussed on studying scalar test equations so far. In this talk I will give an overview over recently obtained stability results for systems of SODEs and several types of methods. INI 1 10:10 to 11:00 M Sanz-Solé ([Barcelona])Hitting probabilities for systems of stochastic waves We will give some criteria which yield upper and lower bounds for the hitting probabilities of random fields in terms of Hausdorff measure an Bessel-Riesz capacity, respectively. Firstly, the results will be applied to systems of stochastic wave equations in arbitrary spatial dimension, driven by a multidimensional additive Gaussian noise, white in time and colored in space. In a second part, we shall consider spatial dimensions $k\le 3$. We will report on work in progress concerning some extensions to systems driven by multiplicative noise. This is joint work with R. Dalang (EPFL, Switzerland). INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:20 M Katsoulakis ([Massachusetts])Multi-level Coarse Grained Monte Carlo methods Microscopic systems with complex interactions arise in numerous applications such as micromagnetics, epitaxial growth and polymers. In particular, many-particle, microscopic systems with combined short- and long-range interactions are ubiquitous in science and engineering applications exhibiting rich mesoscopic morphologies. In this work we propose an efficient Markov Chain Monte Carlo method for sampling equilibrium distributions for stochastic lattice models. We design a Metropolis-type algorithm with proposal probability transition matrix based on the coarse graining approximating measures. The method is capable of handling correctly long and short range interactions while accelerating computational simulations. It is proved theoretically and numerically that the proposed algorithm samples correctly the desired microscopic measure, has comparable mixing properties with the classical microscopic Metropolis algorithm and reduces the computational cost due to coarse-grained representations of the microscopic interactions. We also discuss extensions to Kinetic Monte Carlo algorithms. This is a joint work with E. Kalligianaki (Oak Ridge National Lab, USA) and P. Plechac (University of Tennessee & Oak Ridge National Lab, USA). INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:10 to 15:00 J Voss ([Leeds])Discretising Burgers' SPDE with Small Noise/Viscosity INI 1 15:00 to 15:50 Stochastic order methods for stochastic traveling waves INI 1 15:50 to 16:20 Afternoon Tea 18:45 to 19:30 Dinner at Wolfson Court