Past

The main goal of this talk is to demonstrate connections between the following three big areas of research: the theory of cubature formulas (numerical integration), the discrepancy theory, and nonlinear approximation. First, I will discuss a relation between results on cubature formulas and on discrepancy. In particular, I'll show how standard in the theory of cubature formulas settings can be translated into the discrepancy problem and into a natural generalization of the discrepancy problem. This leads to a concept of the r-discrepancy. Second, I'll present results on a relation between construction of an optimal cubature formula with m knots for a given function class and best nonlinear m-term approximation of a special function determined by the function class. The nonlinear m-term approximation is taken with regard to a redundant dictionary also determined by the function class. Third, I'll give some known results on the lower and the upper estimates of errors of optimal cubature formulas for the class of functions with bounded mixed derivative. One of the important messages (well known in approximation theory) of this talk is that the theory of discrepancy is closely connected with the theory of cubature formulas for the classes of functions with bounded mixed derivative.

The Riemann zeta function involves, for Re s>1, the summation of the inverse s-th powers of the integers. A class of zeta-like functions is obtained if the s-th powers of integers which contain specified digits are omitted from the summation. The numerical summation of such series, their convergence properties and analytic continuation are considered in this lecture.

Parallel sparse direct solvers are an interesting alternative to iterative methods for some classes of large sparse systems of linear equations. In the context of a parallel sparse multifrontal solver (MUMPS), we describe a new dynamic scheduling strategy aiming at balancing both the workload and the memory usage. More precisely, this hybrid approach balances the workload under memory constraints. We show that the peak of memory can be significantly reduced, while we have also improved the performance of the solver.

Then, we present preliminary work concerning a parallel out-of-core extension of the solver MUMPS, enabling to solve increasingly large simulation problems.

This is joint work with P.Amestoy, A.Guermouche, S.Pralet and E.Agullo.

The concept of the slow invariant manifold is the central idea underpinning a transition from micro to macro and model reduction in kinetic theories. We present the constructive methods of invariant manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The physical problem of reduced description is studied in the most general form as a problem of constructing the slow invariant manifold. The invariance conditions are formulated as the differential equation for a manifold immersed in the phase space. The equation of motion for immersed manifolds is obtained.

Invariant manifolds are fixed points for this equation, and slow invariant manifolds are Lyapunov stable fixed points, thus slowness is presented as stability.

A collection of methods to derive analytically and to compute numerically the slow invariant manifolds is presented. The systematic use of thermodynamic structures and of the quasi-chemical representation allows us to construct approximations which are in concordance with physical restrictions.

The following examples of applications are presented: Nonperturbative derivation of physically consistent hydrodynamics from the Boltzmann equation and from the reversible dynamics, for nudsen numbers Kn~1; construction of the moment equations for nonequilibrium media and their dynamical correction (instead of extension of the list of variables) in order to gain more accuracy in description of highly nonequilibrium flows; model reduction in chemical kinetics.

Using the dyadic parametrization of curves, and elementary theorems and

probability theory, examples are constructed of domains having bad properties on

boundary sets of large Hausdorff dimension (joint work with F.D. Lesley).

Lambda-coalescents were introduced by Pitman in (1999) and Sagitov (1999). These processes describe the evolution of particles that

undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case that the measure Lambda has the Beta$(2-\alpha,\alpha)$ they are also known to describe the genealogies of large populations where a single individual can produce a large number of offsprings. Here we use a recent result of Birkner et al. (2005) to prove that Beta-coalescents can be embedded in continuous stable random trees, for which much is known due to recent progress of Duquesne and Le Gall. This produces a number of results concerning the small-time behaviour of Beta-coalescents. Most notably, we recover an almost sure limit theorem for the number of blocks at small times, and give the multifractal spectrum corresponding to the emergence of blocks with atypical size. Also, we are able to find exact asymptotics for sampling formulae corresponding to the infinite site frequency spectrum associated with mutations in the context of population genetics.

It is suggested that topological membranes play a fundamental role

in the recently proposed topological M-theory. We formulate a topological theory

of membranes wrapping associative three-cycles in a seven-dimensional target

space with G_2 holonomy. The topological BRST rules and BRST invariant action

are constructed via the Mathai-Quillen formalism. We construct a set of local

and non-local observables for the topological membrane theory. As the BRST

cohomology of local operators turns out to be isomorphic to the de Rham

cohomology of the G_2 manifold, our observables agree with the spectrum of

d=4, N=1 G_2 compactifications of M-theory.

Through the advent of enhanced medical imaging computational modelling can now be applied to anatomically correct arterial geometries. However many flow feautures under physiological and pathological flow paraemeters, even in idealised problems, are relatively poorly understood. A commonly studied idealisation of an arterial blockage or stenosis, potentially generated by atherosclerosis, is a sinusoidally varying constricted tube. Although most physiological flow conditions are typically laminar, in this configuration turbulent flow states can arise due to the local increase in sectional Reynolds number. To examine the onset of turbulence in this geometry, under a simple non-reversing pulsatile flows, we have applied Floquet stability analysis and direct
numerical simulation.
As illustrated in the above figure, a period-doubling absolute instability mode associated with alternating tilting of the vortex rings that are ejected out of the stenosis/constriction during each pulse. This primary instability occurs for relatively large reduced velocities associated with long pulse periods (or low Womersley numbers). For lower reduced velocities the primary instability typically manifests itself as azimuthal waves (Widnall instability modes) of low wavenumber that grow on each vortex ring. We have also observed the shear layer of the steady axisymmetric flow is convectively unstable at still shorter temporal periods.
In this presentation we shall outline the challenges of modelling vascular flow problems with a particular focus on idealised stenotic flow. After briefly outlining the numerical analysis methods we shall discuss the flow investigations outlined above and their relation to more classical vortex instabilities.

Stochastic differential equations (SDEs) with nonglobally Lipschitz coefficients

possessing unique solutions make up a very important class in applications. For

instance, Langevin-type equations and gradient systems with noise belong to this

class. At the same time, most numerical methods for SDEs are derived under the

global Lipschitz condition. If this condition is violated, the behaviour of many

standard numerical methods in the whole space can lead to incorrect conclusions.

This situation is very alarming since we are forced to refuse many effective

methods and/or to resort to some comparatively complicated numerical procedures.

We propose a new concept which allows us to apply any numerical method of weak

approximation to a very broad class of SDEs with nonglobally Lipschitz

coefficients. Following this concept, we discard the approximate trajectories

which leave a sufficiently large sphere. We prove that accuracy of any method of

weak order p is estimated by $\varepsilon+O(h^{p})$, where $\varepsilon$ can be

made arbitrarily small with increasing the radius of the sphere. The results

obtained are supported by numerical experiments. The concept of rejecting

exploding trajectories is applied to computing averages with respect to the

invariant law for Langevin-type equations. This approach to computing ergodic

limits does not require from numerical methods to be ergodic and even convergent

in the nonglobal Lipschitz case. The talk is based on joint papers with G.N.

Milstein.

A classical result due to Kunita says that if the coefficients are global

Lipschitzian, then the s.d.e defines a global flow of homeomorphisms. In this

talk, we shall prove that under suitable growth on Lipschitz constants, the sde

define still a global flow.

I aim to give some review of how generalised geometries provide a natural

framework for describing supersymmetric string backgrounds. In particular I

will focus on a rewriting of type II supergravity in terms of generalised

structures. Hitchin functions appear naturally along with generalised

extensions of the Gukov-Vafa-Witten superpotential.

In this talk, a network topology is presented that allows both peer-to-peer and non-local traffic in a cellular based TDD-CDMA system known as opportunity driven multiple access (ODMA). The key to offering appropriate performance of peer-to-peer communication in such a system relies on the use of a routing algorithm which minimises interference. This talk will discuss the constraints and limitations on the capacity of such a system using a variety of routing techniques. A congestion based routing algorithm will be presented that attempts to minimize the overall power of the system as well as providing a measure of feasibility. This technique provides the lowest required transmit power in all circumstances, and the highest capacity in nearly all cases studied. All of the routing algorithms considered allocate TDD time slots on a first come first served basis according to a set of pre-defined rules. This fact is utilised to enable the development of a combined routing and resource allocation algorithm for TDD-CDMA relaying. A novel method of time slot allocation according to relaying requirements is then developed.

Two measures of assessing congestion are presented based on matrix norms. One is suitable for a current interior point solution, the other is more elegant but is not currently suitable for efficient minimisation and thus practical implementation.

Image Inpainting turns the art of image restoration, retouching, and disocclusion into a computer-automated trade. Mathematically, it may be viewed as an interpolation problem in BV, SBV, or other fancy function spaces thought suitable for digital images. It has recently drawn the attention of the numerical PDE community, which has led to some impressive results. However, stability restrictions of the suggested explicit schemes so far yield computing times that are next to prohibitive in the realm of interactive digital image processing. We address this issue by constructing an appropriatecontinuous energy functional that combines the possibility of a fast discrete minimization with high perceptible quality of the resulting inpainted images.

The talk will survey the background of the inpainting problem and prominent PDE-based methods before entering the discussion of the suggested new energy functional. Many images will be shown along the way, in parts with online demonstrations.

This is joint work with my student Thomas März.