Annunziato, Mario (Salerno): On a finite
difference method for piecewise deterministic processes with
memory and its parallel algorithm implementation
We treat the problem of finding the distribution function for a
class of piecewise deterministic processes with memory, by numerically
solving the associated Liouville-Master Equation. That equation is
composed of a linear system of hyperbolic PDEs and of boundary
conditions that depend on the integral over the interior of the
solution domain.
We illustrate a numerical scheme of the first order, resulting by a
combination of the upwind scheme and of a direct quadrature. We show a
Courant-Friedrichs-Lewy condition that both ensures convergence and
the momotonic property (positivity) of the numerical solution. We
describe the parallel algorithm, implemented by using the MPI library,
and the results of the performance tests for a known problem.
Please directly contact the speaker for the slides of this talk.
Buckwar, Evelyn (HU Berlin): Weak
convergence of the Euler-Maruyama scheme for stochastic delay
differential equations
We consider weak approximations of solutions of stochastic delay
differential equations with discrete delays. We discuss the
Euler-Maruyama scheme for this type of equations and show that it has
order of weak convergence 1, as in the case of stochastic ordinary
differential equations, i.e., equations without delay. Although the
set-up is non-anticipating, our approach uses the Malliavin calculus
and the anticipating stochastic analysis techniques of Nualart and
Pardoux. Numerical experiments illustrate the theoretical
findings. The results presented are joint work with Rachel Kuske,
Salah Mohammed and Tony Shardlow.
Dirr, Nicolas (MPI Leipzig): Interfaces in
heterogeneous environment
We describe results concerning qualitative properties
of equations that model interfaces (e.g. phase boundaries)
in periodic or random media. The evolution is governed by
a gradient flow for the interfacial energy which is perturbed
by stochastic or periodic terms (heterogeneities)on a small scale.
Aspects of the effective behavior on the larger scale are derived.
Please directly contact the speaker for the slides of this talk.
Gentz, Barbara (Bielefeld):
Desynchronisation of coupled bistable oscillators
perturbed by additive white noise
We consider a chain of N overdamped bistable oscillators with
nearest-neighbour coupling. Each site is perturbed by an
independent white noise, modeling the influence of a heat
reservoir. This system is described by a set of stochastic
differential equations on R^{Z/N Z}.
For sufficiently large coupling, synchronisation is observed. We
show that, as the coupling decreases, the system desynchronises in
a sequence of symmetry-breaking bifurcations, corresponding to a
gradual increase in the number of local minima of the potential
landscape. Large-deviation techniques provide estimates on
activation energy as well as on optimal transition paths and times
from one synchronised state to other.
Joint work with Nils Berglund (CPT, Marseille) and Bastien
Fernandez (CPT, Marseille).
Grecksch, Wilfried (Halle):
An infinite-dimensional fractional linear-quadratic
regulator problem
We consider a fractional analogue of the infinite-dimensional linear quadratic
regulator problem. The state process is governed by a linear stochastic evolution
equation with additive fractional Brownian noise. The state process is
completely observed. The admissible controls are closed-loop
controls. The optimal control is constructed explecitely which
minimizes a quadratic goal functional. Adjoint equations and the so
called Gaussian fundamental martingale are applied.
Grün, Günther (Erlangen):
Thin-film flow influenced by thermal fluctuations
We will be concerned with the effects thermal fluctuations have on
thin-film (de)wetting. Starting from incompressible Navier-Stokes
equations with noise,
we use long-wave approximation and Fokker-Planck-type arguments to
derive a fourth-order degenerate parabolic stochastic partial
differential equation – the stochastic thin-film equation. We propose
a discretization scheme and give both formal and numerical evidence
for our conjecture that thermal fluctuations may resolve discrepancies
with respect to time-scales of dewetting which have been observed
recently in comparing physical experiments and deterministic numerical
simulations. This is joint work with K. Mecke and M. Rauscher.
Please directly contact the speaker for the slides of this talk.
Higham, Des
(Strathclyde): Multi-Level Monte Carlo
Many problems require the expected value of a functional
of the solution to a stochastic differential equation (SDE).
A notable application area is option valuation
in mathematical finance. A simple and widely used computational approach
is to simulate the SDE numerically, for example, with the Euler-Maruyama
method, and to regard each approximate solution path as a sample in a Monte
Carlo algorithm. In this case there are two sources of error:
(1) Discretization: Euler-Maruyama does not follow paths exactly, and
(2) Statistical: the sample mean does not match the true expected
value.
By balancing the contribution of these two types of error we obtain an
algorithm that requires O(ε^{-3}) effort to achieve an
accuracy of O(ε).
Recently, in the report "Multi-level Monte Carlo path simulation",
Report NA-06/03, Oxford University Computing Laboratory, 2006,
Mike Giles showed the remarkable result that the computational
effort can be reduced to O(ε^{-2} (log ε)^{2}).
This new `multi-level' Monte Carlo algorithm uses an approach that
bears some resemblance to multigrid methods and more generally to
Richardson extrapolation. Surprisingly, both the
analysis and the actual performance of the algorithm depend heavily on the
property of strong convergence.
After describing Giles' algorithm, we will give some new strong convergence
results that justify the use of the algorithm for various types of
financial option valuation. This is joint work with Mike Giles and
Xuerong Mao.
Imkeller, Peter (HU Berlin):
Meta-stability in stochastic differential equations
induced by Levy noise
A spectral analysis of the time series representing average
temperatures during the last ice age featuring the
Dansgaard-Oeschger events reveals an α-stable noise
component with an α ∼ 1.78. Based on this observation,
papers in the physics literature attempted a qualitative
interpretation by studying diffusion equations that describe
simple dynamical systems perturbed by small Lévy noise. We study
exit and transition problems for solutions of stochastic
differential equations of this type. Due to the heavy-tail nature
of the α-stable component of the noise, the results differ
strongly from the well known case of purely Gaussian
perturbations.
Kloeden, Peter (Frankfurt): The pathwise
numerical approximation of stationary solutions of
semilinear stochastic evolution
Under a one-sided dissipative Lipschitz
condition on its drift, a stochastic evolution equation with additive noise of
the reaction-diffusion type is shown to have a unique stochastic stationary
solution which pathwise attracts all other solutions. A similar situation
holds for each Galerkin approximation and each implicit Euler scheme applied
to these Galerkin approximations. Moreover, the stationary solution of the
Euler schemes converges pathwise to that of the Galerkin system as the
stepsize tends to zero and the stationary solutions of the Galerkin systems
converge pathwise to that of the evolution equation as the dimension
increases. The analysis is carried out on random partial and ordinary
differential equations obtained from their stochastic counterparts by
substraction of appropriate Ornstein-Uhlenbeck stationary solutions.
Kuksin, Sergei (Edinburgh): Statistical
hydrodynamics in thin 3D layers
I will consider the Navier-Stokes equation in an ε-thin 3D
layer. The equation is perturbed by a non-degenerate random force. I
will show that, firstly, when ε<<1 the equation has a unique
stationary measure which describes asymptotic in time statistical
properties of solutions and, secondly, after the averaging in the thin
direction this measure converges (as ε goes to zero) to a
unique stationary measure for the Navier-Stokes equation on the
corresponding 2D surface. Thus, this 2D equation describes asymptotic
in time and limiting in ε statistical properties of the
Navier-Stokes flow in the narrow 3D domain.
Larsson, Stig (Göteborg): Finite
element approximation of parabolic stochastic
PDEs
We consider a linear parabolic stochastic differential equation in
several spatial variables forced by additive colored space-time noise.
The equations are discretized in the spatial variable by a finite
element method. In order to capture the multiscale behavior of the
colored noise we use a hierarchical wavelet basis for the finite element
space. We prove strong convergence estimates.
Lord,
Gabriel (Edinburgh): An exponential type scheme for stochastic PDE
We discretize a stochastic parabolic PDE using an exponential type
integrator in time. In space we use a Galerkin approximation and investigate
how many modes are required for an efficient scheme.
Loy, Matthias (Tübingen): Exponential
integrators for the linear stochastic Schroedinger equation
Certain Magnus integrators and Trotter splittings are applied to the
(spatial) pseudo-spectral discretized linear stochastic Schrödinger
equation. Under the assumption of commutator- and energy-bounds strong
error behaviour for the presented methods are discussed. Further, it
is shown that the Trotter splitting methods are in general preferable.
Maier-Paape, Stanislaus (Aachen):
Phase Separation in Stochastic Cahn-Hilliard
Models
The Cahn-Hilliard equation is one of the fundamental models for
phase separation dynamics in metal alloys. On a qualitative level,
it can successfully describe phenomena such as spinodal
decomposition and nucleation. Yet, as deterministic partial
differential equation it does not account for thermal fluctuations
or similar random effects. In this survey talk we describe some
dynamical aspects of a stochastic version of the model due to
Cook. These include recent results on spinodal decomposition, as
well as a brief discussion of nucleation and its relation to the
deterministic attractor structure. In addition, differences
between the deterministic and the stochastic dynamics are discussed.
Malham, Simon
(Edinburgh) Stochastic Lie group integrators
We present Lie group integrators for nonlinear
stochastic differential equations with non-commutative vector fields
whose solution evolves on a smooth manifold.
Given a Lie group action that generates transport along the manifold,
we pull back the stochastic flow on the manifold to the Lie group
via the action, and subsequently pull back the flow
to the corresponding Lie algebra via the exponential map.
We construct an approximation to the stochastic flow in the Lie algebra
via closed operations and then push back to the Lie group and
then to the manifold, thus ensuring our approximation lies in
the manifold. We also present an order one numerical
integration scheme based on the Castell–Gaines
exponential Lie series approach that is uniformly
more accurate than the Milstein scheme.
The lecture is based on some joint works with Etienne Pardoux (Marseille,
Fance) and Lucian Maticiuc (Iasi, Romania).
We present an existence and uniqueness result for the backward stochastic
differential equation involving subdifferential operators in Hilbert
spaces:
If D is an open and bounded subset of R^{d} with a sufficiently
smooth boundary Γ and β ⊂ R^{2} is a maximal
monotone operator, then, as first examples for the problem (P), we
have the multivalued Neumann Dirichlet backward stochastic partial
differential equations:
Ritter, Klaus (Darmstadt): Optimal
Approximation for a Class of Stochastic Heat
Equations
We study numerical approximation of stochastic heat
equations
driven by nuclear or by space-time white noise on
the spatial domain ]0,1[^{d}.
The error of an approximation is defined in L_{2}-sense,
and the compuational cost of an algorithm is measured by
the number of evaluations of the one-dimensional
components of W.
We are interested in the following basic question: what
is the minimal error e(n) that can be achieved
by any algorithm with cost at most n.
The sequence of minimal errors
quantifies the intrinsic difficulty of our computational
problem.
We determine the asymptotic behaviour of e(n) and we
present almost optimal algorithms, i.e, algorithms
with error close to e(n) and cost bounded by n.
In the case of nuclear noise it is crucial to
use a non-uniform time discretization in order to achieve
optimality.
Schmalfuß, Björn
(Paderborn): Dynamics for numerical schemes for
SPDE
We introduce a numerical scheme for a SPDE with additive white
noise. This numerical scheme defines a random dynamical system. To
investigate the long–time behavior of such a numerical scheme the
term pullback attractor is introduced. The existence of this type
of attractor for the numerical scheme is proved and relations to
the attractor of the original spde are discussed.
Please directly contact the speaker for the slides of this talk.
Shardlow,
Tony (Manchester): Stochastic PDEs and excitable media
I will discuss some stochastic PDEs that arise in modelling excitable
media. Two effects will be discussed: nucleation of waves occurs due
to background noise and we show how the type of wave that is nucleated
depends on system parameters. Waves are found to propagate in a larger
region of parameter space when noise is present: we investigate
numerically.
Please directly contact the speaker for the slides of this talk.
Sickenberger, Thorsten (HU Berlin):
Adaptive Methods for SDAEs with small noise
In this talk the numerical approximation of solutions of Ito
stochastic differential algebraic equations (SDAEs) with small
noise is considered. SDAEs arise from transient noise analysis in
circuit simulation.
We discuss adaptive linear multi-step methods and their
mean-square convergence. For the case of small noise we present a
strategy for controlling the step-size. It is based on estimating
the mean-square local errors by an ensemble of solution paths that
are computed simultaneously. This leads to step-size sequences
that are identical for all paths.
Test results illustrate the performance of the presented methods.
(joint work with Renate Winkler)
Stannat, Wilhelm (Darmstadt): Particle
filters and the Kushner-Stratonovitch equation
The Kushner-Stratonovitch equation is a measure-valued stochastic
partial differential equation arising in stochastic filtering theory
and describing the conditional distribution of a Markovian
signal observed through additive noise. The solution of the
Kushner-Stratonovitch equation can be
approximated by weighted particle systems (called particle filters in
this context) with a mutation/selection
mechanism. Using a variational approach, we discuss two theoretical
issues concerning the mean squared
error of the approximation and demonstrate the use of a ground state
transform to increase its efficiency.
Stuart, Andrew (Warwick): MCMC Methods for
Sampling Conditioned Diffusions
There are a wide variety of applications which can
be cast as sampling problems for conditioned SDEs (diffusion
processes). Examples include nonlinear filtering in signal
processing, data assimilation in the ocean/atmosphere sciences,
data interpolation in econometrics, and finding transition pathways
in molecular systems. In all these examples the object
to sample is a path in time, and is hence infinite dimensional. We
describe abstract MCMC methods for sampling such problems, based
on writing the target measure as a change of measure from a Gaussian
process. We generalize the independence sampler, Metropolis adjusted
Langevin algorithm and Random Walk Metropolis algorithm to this path
space setting. We give an overview of the subject area, describing the
analytical, statistical and computational challenges, and illustrating
applicability of the techniques being developed.
Collaboration with:
Alex Beskos, Martin Hairer, Jochen Voss (Warwick), David White (Warwick),
Gareth Roberts (Lancaster)
Zouraris, Georgios (Kreta): Finite element
methods for a fourth order stochastic parabolic equation
We consider, as the main problem,
an initial- and Dirichlet boundary- value
problem for a fourth-order linear stochastic parabolic equation,
in one space dimension, forced by an additive space-time
white noise.
Since the solution of the problem
is not regular, following
[Allen, Novosel and Zhang, Stochastics Stochastics Rep., vol.64, 1998],
we replace the space-time white noise by a properly
chosen discrete space-time white noise kernel
to get a modified fourth-order linear
stochastic parabolic problem.
Fully-discrete approximations to the solution
of the modified problem are constructed by using,
for discretization in space, a standard Galerkin
finite element method and, for time-stepping,
the Backward Euler method.
We derive a priori estimates
(i) for the difference between the solution of the main problem and the
solution of the modified problem,
and
(ii) for the numerical method approximation
error to the solution of the modified problem.
Please directly contact the speaker for the slides of this talk.