Filippov sliding vector field on a surface of co-dimension two.
In this talk we discuss selection of a Filippov sliding vector field in case motion is constrained on a (discontinuity) surface of co-dimension 2. We show that a systematic selection process is possible, and justified under natural attractivity assumptions. The relation of this selection process to a certain regularization approach is outlined. We also discuss cases in which the flow of the system leaves the surface tangentially, and contrast this situation to the case in which the surface ceases to be attractive, but motion on it continues to be well defined. Finaly, we briefly discuss difficulties and open problems for the case of a surface of co-dimension greater than 2. This talk is based on collaborations with Luciano Lopez and Cinzia Elia, (Univ. of Bari), and Nicola Guglielmi (Univ. of L’Aquila).

Statistical measure corrections for discretized PDEs

Time series data from long numerical simulations of partial differential equations are frequently subjected to statistical analysis. However the choice of numerical method typically implies some bias on the statistical distributions. In fluid dynamics, for example, only a few specialized numerical discretizations are able to approximate the equilibrium statistical mechanics distributions predicted by recent theories. In molecular dynamics, thermostats have been developed to perturb the trajectories of a simulation such that on a long time scale, they sample a given equilibrium measure, for example, the Gibbs distribution. We discuss the application of thermostats to a point vortex model of ideal fluids, and to the Korteweg-de Vries equation.

Discrete Variational Derivative Method-- one of structure preserving methods for PDEs --(slides of the talk)

We have proposed a kind of structure preserving methods, discrete variational derivative method, to construct numerical schemes for PDEs. The key idea of this method is to discretize the variational structure of PDEs using rigorous definitions of discrete variational derivative. Based on this method we can construct numerical schemes that preserve energy dissipation/conservation properties for a lot of problems, e.g., the Cahn--Hilliard equation, the KdV equation and the nonlinear Schroedinger equation.

We will talk about the basic framework of this method, show some applications and indicate recent developments of it. We think that applicability of this method to arbitrary shaped domains is interesting in particular.

Multilevel Monte Carlo in Stochastic Simulation (slides of talk)

Monte Carlo is an intuitively natural approach that lies at the heart of most computations involving randomness. The multilevel Monte Carlo technique was proposed by Mike Giles in 2008, with related ideas introduced by Stefan Heinrich in 1998. The multilevel technique applies to the case where samples from the target distribution are computed via a numerical discretization. In this context, asking for a more accurate sample requires us to refine the mesh, which increases the computational cost of the sample. The multilevel idea involves carefully combining a range of samples, computed at different discretization levels. At one extreme, many cheap samples are used to get the broad picture, and at the other extreme very few expensive samples are used to fill in the high resolution details. In the context of computing expected values of solutions to stochastic differential equations (SDEs) via the Euler-Maruyama method, Giles showed that this approach improves on the computational cost of fixed-stepsize Euler-Maruyama/Monte Carlo by a factor of $\epsilon(\log(\epsilon)^2$, where $\epsilon$ is the required root-mean-square accuracy. The technique is perfectly general; unlike typical variance reduction tricks it does not rely on specific properties of the SDE. In this talk I will discuss some recent extensions of the multilevel technique to other problems in stochastic computation, including path-dependent option pricing in mathematical finance and mean hitting time computation.

Spatial Stochastic Simulation of Polarization in Yeast Mating
In microscopic systems formed by living cells, the small numbers of some reactant molecules can result in dynamical behavior that is discrete and stochastic rather than continuous and deterministic. Spatio-temporal gradients and patterns play an important role in many of these systems. In this lecture we report on recent progress in the development of computational methods and software for spatial stochastic simulation. Then we describe a spatial stochastic model of polarisome formation in mating yeast. The new model is built on simple mechanistic components, but is able to achieve a highly polarized phenotype with a relatively shallow input gradient, and to track movement in the gradient. The spatial stochastic simulations are able to reproduce experimental observations to an extent that is not possible with deterministic simulation.

Favourable space and time discretisations for nonlinear Schrödinger equations
In this talk, I shall address the issue of efficient numerical methods for the space and time discretisation of nonlinear Schrödinger equations such as systems of coupled time-dependent Gross–Pitaevskii equations arising in quantum physics for the description of multi-component Bose–Einstein condensates. For the considered class of problems, a variety of contributions confirms the favourable behaviour of pseudo-spectral and exponential operator splitting methods regarding efficiency and accuracy. Especially, numerical comparisons show that higher-order splitting
methods outperform standard time integration methods when smaller tolerances are required or long term computations are carried out. My main objective is to study the quantitative and qualitative behaviour of these high-accuracy discretisations. In particular, this includes a stability and convergence analysis of high-order exponential operator splitting methods for nonlinear evolutionary Schrödinger equations. In this regard, I will discuss an approach that remains suitable in the presence of unbounded nonlinear operators and for small values of an additional critical parameter and illustrate the theoretical results by numerical examples. Furthermore, as in the absence of an adaptive local error control in space and time, the reliability of the numerical solution and the performance of the space and time discretisation strongly depends on the experienced scientist selecting the space and time grid in advance, I will exemplify different approaches for the reliable time integration of Gross–Pitaevskii systems on the basis of a local error control for splitting methods.
The presented results were obtained in collaboration with Winfried Auzinger (Technische UniversitätWien, Austria), Stéphane Descombes (Université de Nice, France), and Othmar Koch (UniversitätWien, Austria).