Finite Element Methods for Stochastic PDEs

Stig Larsson (Chalmers, Gothenburg)

We study evolution partial differential equations driven by noise, for example, the stochastic heat equation, the stochastic wave equation, or the stochastic Cahn-Hilliard equation, where the source term (right-hand side) is perturbed by a noise term. The noise can be uncorrelated in space and time (white noise) or spatially correlated.
We begin by briefly presenting an abstract framework in which such equations can be given a rigorous meaning. The framework is based on the theory of semigroups of bounded linear operators in Hilbert space. The noise is described in terms of a Hilbert space valued Wiener process and the stochastic integral with respect to such a process plays an important role. We present an introduction to the existence and regularity theory for such equations.
The equations are discretized by a standard finite element method in the spatial variable and by the Euler-Maruyama method in the temporal variable. The discrete equations are set in the same abstract framework.
We prove convergence of so-called strong and weak type. The linear case is emphasized but nonlinear equations are briefly discussed at the end.

1. The Wiener process. Stochastic integral.
2. The stochastic heat, wave and Cahn-Hilliard equations. Existence and regularity.
3. Finite element and Euler-Maruyama approximation. Strong convergence for linear equations.
4. Weak convergence for linear equations.
5. Nonlinear equations.

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Optimisation and control of PDEs

Michael Hintermuller (Humboldt, Berlin)

The lectures provide an introduction to optimization problems involving either elliptic or
parabolic differential equations and pointwise constraints on the control and/or the
state or its derivative. The derivation of first order optimality conditions based on
concepts from non-smooth analysis will be discussed and solution algorithms in function space
are introduced and analysed. The latter will have a focus on non-smooth Newton and
path-following techniques. Finally, discrete concepts are introduced and issues related
to mesh independence and adaptive discretization are discussed. The structure of the lectures is as
follows:

1. First order optimality conditions, nonlinear complementarity systems and equivalent forms.
2. Newton-differential and its calculus, semismoothness and generalized Newton methods.
3. Discrete concepts, mesh independence and adaptive discretization.

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Multiscale Methods for SDEs and PDEs

Greg Pavliotis (Imperial College, London)

In these lectures we will present techniques for analyzing deterministic and stochastic systems with multiple scales. Our focus will be on two types of problems, namely partial differential equations (PDEs) with rapidly oscillating periodic coefficients and singularly perturbed stochastic differential equations (SDEs), i.e. systems of SDEs with two characteristic widely separated time scales. The analysis of these problems is based on the theory of homogenization and on singular perturbation theory for PDEs. The main goal of these lectures is the presentation of these techniques and their application to PDEs with rapidly oscillating coefficients and to singularly perturbed SDEs.The course will consist of four 90min long lectures:

1. Homogenization theory for second order elliptic PDEs with periodic coefficients.
2. Homogenization theory of parabolic and transport PDEs. Connection with SDEs.
3. Averaging and homogenization for fast/slow systems of SDEs.
4. Multiscale analysis for stochastic PDEs with quadratic nonlinearities.