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# Anfahrtsplan

Das Institut für Stochastik befindet sich im Science Park 2, 6. Stockwerk ... mehr zu Anfahrtsplan (Titel)

# Weiterführende Links

- SDE-Net (Neues Fenster)
- Johann Radon Institute for Computational and Applied Mathematics (RICAM) (Neues Fenster)
- Österreichische Mathematische Gesellschaft (Neues Fenster)
- Deutsche Mathematiker-Vereinigung (Neues Fenster)

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# Program

*Andrea Barth (University of Stuttgart)*

Title: **Simulation of infinite-dimensional Levy-Processes
**

Abstract: In various applications, stochastic partial differential equations are not driven by Gaussian noise but rather by one whose marginal distributions have heavier tails. Unlike the case of an infinite-dimensional Gaussian process, a general infinite-dimensional L\'evy process cannot be built from independent, one-dimensional L\'evy processes and still admit these one-dimensional distributions as its marginals. In this talk I introduce an approach to construct time-dependent random fields that have marginal distributions which follow certain L\'evy measures. I show convergence of the method and wrap up with some numerical examples.

This is joint work with Andreas Stein (University of Stuttgart).

*David Cohen (University of Umea)*

Title:

**A fully discrete approximation of the one-dimensional stochastic heat equation**

Abstract: A fully discrete approximation of one-dimensional nonlinear stochastic heat equations driven by multiplicative noise is presented. A standard finite difference approximation is used in space and a stochastic exponential method is used for the temporal approximation. This explicit time integrator allows for error bounds in $L^p(\Omega)$, uniformly in time and space. Furthermore, uniform almost sure convergence of the numerical solution is proved. Numerical experiments are presented and confirm the theoretical results.

The presentation is based on an ongoing joint work with Rikard Anton (Umea University) and

Lluis Quer-Sardanyons (Universitat Autonoma de Barcelona).

*Ludovic Goudenege (CNRS)*

Title:

**Numerical computation about metastable states of phase separation models**

Abstract: For many phase separation models, there is a huge literature about computation of deterministic stable and unstable states. In a probabilistic setting and under some regime -for instance in the large deviation asymptotic- these states play an important role in the computation of quantities of interest. Maybe the most important are the expected exit times of region near metastable states. In this talk I will present estimator, schemes and ideas about numerical computation of these quantities. Precisely I will first present the generalization of adaptive multilevel splitting algorithm in a general framework. Then I will explain how to use it to create an estimator of expected exit times or expected length of reactive trajectories. Finally I will present some numerical result in the context of phase separation models.

Gabriel Lord (Heriot Watt University)

Gabriel Lord (Heriot Watt University)

Title:

**Efficient time discretisation of parabolic SPDEs**

Abstract: We introduce adaptive time stepping techniques to control growth in the numerical solution of SPDEs. This can be thought of as an alternative to proving moment bounds for the numerical method and to using a fixed step taming method.

Ideas and the convergence result will be illustrated with some numerical experiments that also show that the adaptivity leads to more accurate solutions.

If time permits we will introduce a new exponential based method for time stepping for SPDEs with multiplicative noise which have an improved rate of convergence in specific circumstances.

*David Silvester (University of Manchester)*

Title:

**An adaptive algorithm for PDE problems with random data**

Abstract: We present a new adaptive algorithm for computing stochastic

Galerkin finite element approximations for a class of elliptic PDE

problems with random data. Specifically, we assume that the

underlying differential operator has affine dependence on a large,

possibly infinite, number of random parameters. Stochastic Galerkin

approximations are then sought in the tensor product space

$X \otimes {\cal P}$, where $X$ is a finite element space associated

with a physical domain and ${\cal P}$ is a set of multivariate polynomials

over a finite-dimensional manifold in the (stochastic) parameter space.

Our adaptive strategy is based on computing two error estimators

(the spatial estimator and the stochastic one) that reflect the two distinct

sources of discretisation error and, at the same time, provide effective

estimates of the error reduction for the corresponding enhanced approximations.

In particular, our algorithm adaptively `builds' a polynomial space over a

low-dimensional manifold in the infinitely-dimensional parameter space such

that the discretisation error is reduced most efficiently (in the energy norm).

Convergence of the adaptive algorithm is demonstrated numerically.

This is joint work with Alex Bespalov (University of Birmingham) and

Catherine Powell (University of Manchester)

*David Siska (University of Edinburgh)*

Title:

**L^p-estimates and regularity for SPDEs with monotone semilinearity**

Abstract: We prove L^p-estimates for semilinear stochastic partial differential equations (on bounded domains) with monotone semilinear term. These are used together with known results for linear SPDEs to obtain regularity results for such equations. These are interior estimates in Sobolev norms and estimates up to the boundary in weighted Sobolev spaces. This is joint work with Neelima.

Aretha Teckentrup (University of Edinburgh)

Aretha Teckentrup (University of Edinburgh)

Title:

**Quasi- and Multilevel Monte Carlo Methods for Computing Posterior Expectations**

Abstract: The parameters in mathematical models for physical processes are often impossible to determine fully or accurately, and are hence subject to uncertainty. By modelling the input parameters as stochastic processes, it is possible to quantify the induced uncertainty in the model outputs. Based on available information, a prior distribution is assigned to the input parameters. If in addition some dynamic data (or observations) related to the model outputs are available, a better representation of the parameters can be obtained by conditioning the prior distribution on these data, leading to the posterior distribution in the Bayesian framework.

In most situations, the posterior distribution is intractable in the sense that the normalising constant is unknown and exact sampling is unavailable. Using Bayes’ Theorem, we show that posterior expectations of quantities of interest can be written as the ratio of two prior expectations involving the quantity of interest and the likelihood of the observed data. These prior expectations can then be computed using Quasi-Monte Carlo and multilevel Monte Carlo methods.

In this talk, we give a convergence and complexity analysis of the resulting ratio estimators, and demonstrate their effectiveness on a typical model problem in uncertainty quantification.

Michael Tretyakov (University of Nottingham)

Michael Tretyakov (University of Nottingham)

Title:

**Uncertainty Quantification for moving boundary problems**

Abstract: The main motivation for the considered UQ problem comes from modelling of one of the main manufacturing processes for producing advanced composites -- resin transfer molding (RTM). We consider one-dimensional and two-dimensional models of the stochastic resin transfer molding process,

which are formulated as random moving boundary problems. We study their properties, analytically in the one-dimensional case and numerically in the two-dimensional case. We show how variability of time to fill depends on correlation lengths and smoothness of a random permeability field.

We will also briefly discuss experimental results for fibre preforms manufactured with Automated Dry Fibre Placement (ADFP) in a laboratory as well as Bayesian inversion for RTM. The talk is based on works with Minho Park, Marco Iglesias, Mikhail Matveev, Arthur Jones, Andy Long, Frank Ball.