Schedule of Workshop 5: High-Dimensional Aspects of Stochastic PDEs

Stephan Dahlke: Adaptive Wavelet Methods for SPDEs: Theoretical Analysis and Practical Realization
In the first part of the talk, we introduce new (spatial) noise models which are based on wavelet expansions. The approach provides an explicit control on the Besov smoothness of the realizations. We study different linear and nonlinear approximation schemes and discuss adaptive wavelet algorithms for stochastic elliptic equations based on these new random functions. The second part of the talk is concerned with the theoretical foundation of adaptive numerical schemes. It is well-known that the order of approximation that can be achieved by adaptive and other nonlinear methods is determined by the regularity of the exact solution in a specific scale of Besov spaces. In contrast, the approximation order of nonadaptive (uniform) methods is determined by the Sobolev smoothness. Therefore, to justify the use of adaptive schemes, sufficiently high Besov smoothness compared to the Sobolev regularity has to be established. We show that for linear stochastic evolution equations in Lipschitz domains the spatial Besov regularity of the solution is commonly much higher than its Sobolev smoothness, so that the use of adaptive schemes is completely justified.

Philipp Dörsek: Splitting and cubature for S(P)DEs: a semigroup perspective
We consider the approximation of the marginal distributions of solutions of stochastic partial differential equations by splitting and cubature methods. The main difficulty is the correct choice of test functions. Hence, we introduce Banach spaces B of functions defined on the state space of the stochastic partial differential equation to which the classical Feller condition for strong continuity of Markov semigroups generalizes. The infinitesimal generator of the Markov semigroup induced by the solution of a stochastic partial differential equation is characterized as a differential operator in infinite dimensions.
These results are applied to the numerical analysis of the Ninomiya-Victoir scheme. The possibility of extrapolation and the extension of the approach to cubature is discussed. As applications, the Heath-Jarrow-Morton equation of interest rate theory and the stochastic Navier-Stokes equation in the sense of M. Hairer and J. Mattingly are simulated. Numerical experiments confirm our theoretical findings, and underline the applicability of the suggested approach.
Parts of this work are joint with J. Teichmann and D. Veluscek.

Helmut Harbrecht: A fast deterministic method for stochastic interface problems
In this work, we propose a fast deterministic numerical method to solve stochastic elliptic interface problems with random interfaces. Shape calculus is employed to derive a shape-type Taylor expansion in the framework of the asymptotic perturbation approach. Given the mean field and the two-point correlation function of the random interface variation, we can quantify the mean field and variance of the random solution in terms of certain orders of the perturbation magnitude by solving a deterministic elliptic interface problem and its tensorized counterpart with respect to the reference interface. Error estimates are derived for interface-resolved finite element approximation in both physical and stochastic dimensions. We discuss sparse grid and low-rank approximations to compute the two-point correlation function of the random solution. In particular, a fast finite difference scheme is proposed which uses a low-rank approximation based on the pivoted Cholesky decomposition. Numerical experiments are presented to validate and quantity the advantages of the proposed method.

Viet Ha Hoang: Analytic regularity and polynomial approximation of stochastic, parametric elliptic multiscale PDEs
A class of second order, elliptic PDEs in divergence form with stochastic and anisotropic conductivity coefficients and n known, separated microscopic length scales in a d dimensional bounded domain is considered. Neither stationarity nor ergodicity of these coefficients is assumed. Sufficient conditions are given for the random solution to converge as the microscopic scales converge to zero to a stochastic, elliptic one-scale limit problem in a tensorized domain of dimension (n+1)d. It is shown that this stochastic limit problem admits best N-term "polynomial chaos" type approximations which converge at a rate that is determined by the summability of the random inputs' Kahunen-Loeve expansion. The convergence of the polynomial chaos expansion is shown to hold almost surely and uniformly with respect to the microscopic scale parameters. Regularity results for the stochastic, one-scale limiting problem are established. This is a joint work with Ch. Schwab.

Kyeong-Hun Kim: An L_p-theory of stochastic PDEs with random fractional Laplacian operator
In this talk, we introduce an L_p-theory of a class of parabolic stochastic equations with random fractional Laplacian operator. The driving noises of the equations are general Levy processes. Uniqueness and existence results in Sobolev spaces will be introduced.

Annika Lang: Multi-level Monte Carlo Finite Element method for parabolic stochastic partial differential equation
We analyze the convergence and complexity of Multi-Level Monte Carlo (MLMC) discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales.
We show, under regularity assumptions on the solution that are minimal under certain criteria, that the judicious combination of piecewise linear, continuous multi-level Finite Element discretizations in space and Euler--Maruyama discretizations in time yields mean square convergence of order one in space and of order 1/2 in time to the expected value of the mild solution. The complexity of the multi-level estimator is shown to scale log-linearly with respect to the corresponding work to generate a single solution path on the finest mesh, resp. of the corresponding deterministic parabolic problem on the finest mesh. Examples are provided for Levy driven SPDEs as well as equations for randomly forced surface diffusions.

Stig Larsson: A fully discrete scheme for the stochastic wave equation
We consider the stochastic wave equation on a bounded domain with smooth or polygonal boundary in several spatial dimensions driven by additive Gaussian noise. We use a standard continuous finite element method for spatial approximation and a rational approximation scheme, such as the Crank-Nicolson or diagonal Pade scheme, for time discretization. We prove weak and strong error estimates and show how to choose the time stepping method to match the spatial approximation order when the spatial regularity is low. This is a joint work with Mihaly Kovacs (Otago) and Fredrik Lindgren (Chalmers).

Kijung Lee: On L_p-theory of Stochastic Linear Parabolic Equations/Systems
In this talk we discuss the key ideas on L_p-theory on stochastic parabolic equations and systems in an informal manner; we use simple equations and systems to avoid technical complexity. We talk over the followings:

• A heat equation with stochastic force and its solution.
• Estimation of the solution using BDG inequality and a generalized Littlewood-Paley inequality.
• Half space domain and weights.
• A warning on systems.
• Stochastic parabolic systems.

Mikhail Lifshits: Approximation of additive random fields depending on large number of parameters
This is a joint work with Marguerite Zani (Paris-12).
Let X(t,ω), (t,ω) ∈ [0,1]^d x Ω be an additive random field, that is a sum of independent random fields, each term depending on its own group of coordinates. We investigate the complexity of finite rank approximation X(t,ω) ≈ ∑_k=1,...,n ξ_k(ω) φ_k(t).
The results obtained in asymptotic setting d → ∞, as suggested H. Wo'zniakowski, provide quantitative version of dimension curse phenomenon: we show that the number of terms in the series needed to obtain a given relative approximation error depends exponentially on d and find the explosion coefficients.
Some related open problems will be also considered.

Felix Lindner: Spatial Sobolev regularity for the stochastic heat equation on polygonal domain
Based on a classical result by P. Grisvard we show that the solution of the stochastic heat equation with zero Dirichlet boundary condition on a polygonal domain can be decomposed into a regular part with maximal spatial Sobolev regularity and a singular part whose spatial Sobolev regularity is limited due to the shape of the domain. In the context of adaptive approximation of SPDEs this complements a recent result about the spatial Besov regularity of parabolic SPDEs on Lipschitz domains (cf. the talk by S. Dahlke).

Olivier Le Maitre: Proper Generalized Decomposition for Linear and Non-Linear Stochastic problems
Proper Generalized Decomposition (PGD, Nouy 2007) for stochastic problem aims at finding solutions of pdes with stochastic coefficients by means of suitable reduced bases approximations. We shall review the essential theoretical results supporting PGD, in the case of symmetric definite operators, and discuss different algorithms for the construction of the PGD. efficiency of the method will be illustrated on elliptic equations and its computational complexity will be contrasted with the classical Galerkin Polynomial Chaos method. Extension of PGD to non-linear problems, in particular the incompressible Navier-Stokes equation, will be discussed and recent simulation results will be shown.

Siddhartha Mishra: Multi-Level Monte Carlo methods for quantifying uncertainty in hyperbolic PDEs
We quantify statistical uncertainty in solutions of hyperbolic PDEs by using improved sampling techniques, namely the Multi-level Monte Carlo (MLMC) method. We present the numerical method along with convergence and complexity estimates. A novel static load balancing algorithm that allows for the method to scale on a large number of processors is also described. Examples from shallow-water, Euler and MHD equations are presented. The talk is based on joint work with C. Schwab and J. Sukys (ETH Zurich).

Fabio Nobile: Stochastic Galerkin and Collocation approximations of PDEs with random coefficients
We consider the problem of numerically approximating statistical moments of the solution of a partial differential equation (PDE), whose input data (coefficients, forcing terms, boundary conditions, geometry, etc.) are uncertain and described by a finite or countably infinite number of random variables. This situation includes the case of infinite dimensional random fields suitably expanded in e.g Karhunen-Loeve or Fourier expansions.
We focus on polynomial chaos approximations of the solution with respect to the underlying random variables and review common techniques to practically compute such polynomial approximation by Galerkin projection, Collocation on sparse grids or regression methods from random evaluations.
We discuss in particular the proper choice of the polynomial space both for linear elliptic PDEs with random diffusion coefficient and second order hyperbolic equations with random piecewise constant wave speed. Numerical results showing the effectiveness and limitations of the approaches will be presented as well.

Andreas Prohl: Numerical Analysis of the stochastic incompressible Navier-Stokes equations
I will discuss implementable space-time discretizations of the stochastic incompressible Navier-Stokes equations. In 3D, martingale solutions are constructed by a discretization that is based on the implicit Euler method, and LBB-stable finite elements. The convergence proof rests on a discrete energy law, and uniform control of higher moments of increments of approximates.
In 2D, strong solutions with improved regularity properties are approximated with certain rates by means of new time-splitting schemes which properly address the interplay of general noise and pressure; general LBB-stable finite elements lead to suboptimal convergence behavior for the same reason, which favors exactly divergence free finite elements instead.
This is joint work with Z. Brzezniak (U York) and E. Carelli (U Tuebingen).