
PLENARY
SPEAKER ABSTRACTS 
Luca Dieci, School of Mathematics, Georgia
Institute of Technology, Atlanta,
Filippov
sliding vector field on a surface of codimension two.
In this talk we discuss selection of a Filippov sliding vector field
in case motion is constrained on a (discontinuity) surface of codimension
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 codimension 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).
================
Jason Frank, Centrum Wiskunde & Informatica
(CWI), Amsterdam and University of Amsterdam, The Netherlands (slides
of the talk)
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 Kortewegde Vries
equation.
================
Daisuke Furihata, Cybermedia Center, Osaka University, Japan
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 CahnHilliard 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.
================
Desmond J Higham, Department of Mathematics and Statistics, University
of Strathclyde
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 EulerMaruyama method,
Giles showed that this approach improves on the computational cost of
fixedstepsize EulerMaruyama/Monte Carlo by a factor of $\epsilon(\log(\epsilon)^2$,
where $\epsilon$ is the required rootmeansquare 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 pathdependent option pricing in
mathematical finance and mean hitting time computation.
================
Linda Petzold, Department of Mechanical Engineering and Department
of Computer Science, University of California, Santa Barbara
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. Spatiotemporal
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.
================
Huazhong Tang , Department of Scientific Engineering and Computing,
Peking University
Local Oscillations in Finite Dierence Solutions of Hyperbolic
Conservation Laws (slides
of talk)
================
Mechthild Thalhammer, Department of Mathematics, Universität
of Innsbruck
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 timedependent Gross–Pitaevskii equations
arising in quantum physics for the description of multicomponent Bose–Einstein
condensates. For the considered class of problems, a variety of contributions
confirms the favourable behaviour of pseudospectral and exponential
operator splitting methods regarding efficiency and accuracy. Especially,
numerical comparisons show that higherorder 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 highaccuracy
discretisations. In particular, this includes a stability and convergence
analysis of highorder 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).
