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Term 1
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05.10.2010
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V. JERDJEVIC (U. TORONTO)
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Optimal Control Problems on Lie Groups and Integrable Systems
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12.10.2010
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S. REICH (U. POTSDAM)
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Taming
unpredictability: combining model dynamics with observations
Abstract:
The talk is primarily dedicated to the problem of data assimilation.
Data
assimilation is a technique for combining mathematical models of
physical systems with measurements in order
to estimate either the state of
the system or the parameters of the
model. In this talk we will view data
assimilation as a continuous deformation process
of the underlying
probability density functions (PDFs) and their approximation by empirical measures.
After
a discussion of the general mathematical methodology we will discuss our
approach
in more detail for variance minimizing filters such
as ensemble
Kalman filters.
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19.10.2010
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G. GOTTWALD (U. Sydney)
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Title:
"A variance constraining ensemble Kalman
filter: How to improve forecast using climatic data of unobserved
variables."
Abstract:
Data
assimilation aims to solve one of the fundamental problems of numerical
weather prediction - estimating the optimal state of the atmosphere given a
numerical model of the dynamics, and sparse, noisy observations of the
system. A standard tool in attacking this filtering problem is the Kalman filter.
We
consider the problem when only partial observations are available. In
particular we consider the situation where the observational space consists
of variables which are directly observable with
known observational error, and of variables of which only their climatic
variance and mean are given. We derive the corresponding Kalman filter in a variational
setting.
We
analyze the variance constraining Kalman filter
(VCKF) filter for a simple linear toy model and determine its range of
optimal performance. We explore the variance constraining Kalman filter in an ensemble transform setting for the
Lorenz-96 system, and show that incorporating the information on the
variance on some un-observable variables can improve the skill and also
increase the stability of the data assimilation procedure.
Using
methods from dynamical systems theory we then systems where the un-observed
variables evolve deterministically but chaotically on a fast time scale.
This
is joint work with Lewis Mitchell and Sebastian Reich
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26.10.2010
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D.
HENRY (DUBLIN CITY UNIVERSITY)
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Title:
"Infinite propagation speed for a two component Camassa-Holm
equation."
Abstract: In this talk we will examine
the rate of propagation of a class of solutions of an integrable
two component generalisation
of the celebrated Camassa-Holm equation. We
examine the propagation behaviour of compactly
supported solutions, namely whether solutions which are initially compactly
supported will retain this property throughout their time of evolution. In
the negative case, where we show that solutions have an infinite speed of
propagation, we present a description of how the solutions retain weaker
properties throughout their existence time, namely they decay at an
exponentially fast rate for the duration of their existence.
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02.11.2010
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B. GODDARD (IC)
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Non-adiabatic
transitions through avoided crossings.
Abstract
The
photo-dissociation of diatomic molecules is one of the paradigmatic
chemical reactions of quantum chemistry. The associated mathematical
problem is the study of non-adiabatic transitions at avoided crossings in a
two-level system, with one effective spatial degree of freedom. Such
problems are highly multi-scale; the transmitted wavefunction
is typically very small. This leads to great difficulty in performing
accurate numerical calculations, and an alternative method is
required. Using superadiabatic representations
(which will be defined), and an approximation of the dynamics near the
crossing region, we obtain an explicit formula for the transmitted wavefunction. Our results agree extremely well with
high precision ab-initio calculations.
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09.11.2010
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A.M. STUART (U. WARWICK)
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Connections
Bewtween (S)PDEs
and MCMC in a Hilbert Space
I
will show that a range of inverse problems, when formulated using the
Bayesian
approach, lead to the problem of sampling a
probability measure on a Banach space.
When
the prior is a Gaussian random field, then the posterior measure is
defined via its density wrt
the Gaussian. I will discuss properties of the posterior
measure and show how these properties can
be used to design efficient
MCMC-based
sampling algorithms which are robust under refinement of finite
dimensional approximation of the Banach space. In particular I will demonstrate
the central role of stochastic PDEs in the construction of these methods.
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16.11.2010
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K.
MATTHIES (U. BATH)
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Hard-sphere
dynamics and Boltzmann equations
abstract:
The
derivation of the continuum models from deterministic atomistic
descriptions is a longstanding and fundamental
challenge. In particular
the emergence of irreversible
macroscopic evolution from reversible
deterministic microscopic evolution is still not
fully understood. We
study a classic system: N balls that
interact with each other via a
hard-core potential and show rigorously
that in the case of kinetic
annihilation (particles annihilate each other
upon collision) the
asymptotic behavior as N tends to infinity is
correctly described by the
Boltzmann
equation without gain-term for non-concentrated initial
distributions. The mean-field description fails,
when there are
concentrations in the space or the velocity
coordinates. Extension
towards the full collision case are
outlined. Joint work with Florian
Theil.
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23.11.2010
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S. WHITTINGTON
(U. TORONTO)
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Title: Counting knotted curves and surfaces
in lattices
Simple closed curves can be knotted in 3-space and these knots
can be seen experimentally in circular DNA molecules, where the
entanglements affect cellular processes. It has been known for over twenty years that the knot
probability
goes to unity as the size increases. It is natural to ask what happens in higher
dimensions. 2-spheres can be
knotted in 4-space and one might ask if the knot probability goes to unity
as the area
of the 2-sphere increases.
This seminar will first review the situation in three dimensions, then discuss why the higher dimensional case is more
difficult and why the lower dimensional argument fails in 4-space. Finally we shall discuss some
results in this direction where the 2-sphere is confined to a tube in the
4-dimensional hypercubic lattice.
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30.11.2010
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R.
CRASTER (IC)
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Title:
High frequency homogenization
Abstract:
It is highly desirable to be able to create continuum equations that embed
a known microstructure through effective
or averaged quantities such as wavespeeds
or shear moduli. The methodology for achieving
this at low frequencies and for waves long relative to the microstructure
is well-known and such static or quasi-static
theories are well developed. However, at high frequencies the multiple
scattering by the elements of the microstructure, which is now of a similar
scale to the wavelength,
has apparently prohibited any homogenization theory.
Recently we have developed an asymptotic approach that overcomes this
limitation and continuum equations are developed even though the
microstructure and wavelength are of the same order.
The
general theory will be described and applications to continuum, discrete
and frame lattice structures will be outlined. The results and methodology are confirmed versus various
illustrative exact/ numerical calculations.
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07.12.2010
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Ch. ORTNER (OXFORD U.)
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Atomistic/Continuum
Hybrid Methods: Construction, Ghost Forces, Consistency
Low
energy equilibria of crystalline materials are
typically characterised by localized defects that
interact with their environment through long-range elastic fields. By
coupling atomistic models of the defects with continuum models for the
elastic far field one can, in principle, obtain models with near-atomistic
accuracy at significantly reduced computational cost. However, several
pitfalls need to be overcome to find a reliable coupling mechanism.
Possibly the most widely discussed among these pitfalls are the so-called
Òghost forcesÓ that typically arise in energy-based atomistic/continuum
coupling mechanisms.
In
this talk I will first describe the construction and of
energy-based atomistic/continuum coupling methods with and without ghost
forces. I will then explain the resulting model errors due to different
types of interface treatment. To this end I will first review some results
on one-dimensional model problems and then describe some recent results in
higher dimensions. I will show that, in 1D, absence of ghost forces does
imply Òhigh accuracyÓ of the coupling scheme but that in 2D/3D this is not
clear at this point.
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14.12.2010
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V.
SMYSHLYAEV (UCL and U. Bath)
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Title: TBA
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