Past

Moving hydrodynamic boundaries (waves and bubbles, for example) produce acoustic signatures.

It is well known that the computation of a few extreme eigenvalues, and the corresponding eigenvectors, of a symmetric matrix A can be rewritten as computing extrema of the Rayleigh quotient of A. However, since the Rayleigh quotient is a homogeneous function of degree zero, its extremizers are not isolated. This difficulty can be remedied by restricting the search space to a well-chosen manifold, which brings the extreme eigenvalue problem into the realm of optimization on manifolds. In this presentation, I will show how a recently-proposed generalization of trust-region methods to Riemannian manifolds applies to this problem, and how the resulting algorithms compare with existing ones.

I will also show how the Joint Diagonalization problem (that is, approximately diagonalizing a collection of symmetric matrices via a congruence transformation) can be tackled by a differential geometric approach. This problem has an important application in Independent Component Analysis.

For atomistic (and related) material models, global minimization

gives the wrong qualitative behaviour; a theory of equilibrium

solutions needs to be defined in different terms. In this talk, a

process based on gradient flow evolutions is presented, to describe

local minimization for simple atomistic models based on the Lennard-

Jones potential. As an application, it is shown that an atomistic

gradient flow evolution converges to a gradient flow of a continuum

energy, as the spacing between the atoms tends to zero. In addition,

the convergence of the resulting equilibria is investigated, in the

case of both elastic deformation and fracture.

Levy trees are random continuous trees that are obtained as

scaling limits of Galton-Watson trees. Continuous tree means here real tree, that is a certain class of path-connected metric spaces without cycles. This class of random trees contains in particular the continuum random tree of Aldous that is the limit of the uniform random tree with N vertices and egde length one over the square root of N when N goes to infinity. In this talk I give a precise definition of the Levy trees and I explain some interesting fractal properties of these trees. This talk is based on joint works with J-F Le Gall and M. Winkel available on arxiv : math.PR/0501079 (published in

PTRF) math.PR/0509518 (preprint)

math.PR/0509690 (preprint).

I will present two-sided estimates for the heat kernel of the elliptic

Schr

Taking a view common in the finite element analysis, we interpret

the first N terms of the usual Fourier series solution as the exact

solution of an approximating problem in a subspace spanned by the

eigenfunctions of the underlying Sturm Liouville problem. This view

leads to a consistent solution technique for the heat, wave and

Poisson's equation, and allows an analysis of the error caused by

truncating the Fourier series. Applications to a variety of problems

will be discussed to demonstrate that the analytic approach remains a

valuable complement to purely numerical methods.

The talk is intended for students with an interest in actually

solving partial differential equations. It assumes a standard

background in undergraduate mathematics but not necessarily prior

exposure to the subject. The goal is to show that there is more to

separation of variables than is apparent from standard texts on

engineering mathematics.

It is by now well known that one cannot HEAR the shape of a

drum: There are many known examples of isospectral yet not isometric "drums". Recently we discovered that the sequences of integers formed by counting the nodal domains of successive eigenfunctions encode geometrical information, which can also be used to resolve spectral ambiguities. I shall discuss these sequences and indicate how the information stored in the nodal sequences can be deciphered.

We discuss a method for solving very large structured symmetric indefinite equation systems arising in optimization with interior point methods.

Many real-life economic models involve system dynamics, spatial distribution or uncertainty and lead to large-scale optimization problems. Such problems usually have a hidden structure: they are constructed by replication of some small generic block. The linear algebra subproblems which arise in optimization algorithms for such problems involve matrices which are not only sparse, but they additionally display a block-structure with many smaller blocks sparsely distributed in the large matrix.

We have developed a structure-exploiting parallel interior point solver for optimization problems. Its design uses object-orientated programming techniques. The progress OOPS (Object-Orientated Parallel Solver: http://www.maths.ed.ac.uk/~gondzio/parallel/solver.html) on a number of different computing platforms and achieves scalability on a number of different computing platforms. We illustrate its performance on a collection of problems with sizes reaching 10⁹ variables arising from asset liability management and portfolio optimization.

This is a joint work with Andreas Grothey.

Given a family of independent events in a probability space, the probability

that none of the events occurs is of course the product of the probabilities

that the individual events do not occur. If there is some dependence between the

events, however, then bounding the probability that none occurs is a much less

trivial matter. The Lov

/notices/events/abstracts/stochastic-analysis/mt05/m

In Clarendon Lab

A self-interacting random walk is a random process evolving in an environment depending on its past behaviour.

The notion of Edge-Reinforced Random Walk (ERRW) was introduced in 1986 by Coppersmith and Diaconis [2] on a discrete graph, with the probability of a move along an edge being proportional to the number of visits to this edge. In the same spirit, Pemantle introduced in 1988 [5] the Vertex-Reinforced Random Walk (VRRW), the probability of move to an adjacent vertex being then proportional to the number of visits to this vertex (and not to the edge leading to the vertex). The Self-Interacting Diffusion (SID) is a continuous counterpart to these notions.

Although introduced by similar definitions, these processes show some significantly different behaviours, leading in their understanding to various methods. While the study of ERRW essentially requires some probabilistic tools, corresponding to some local properties, the comprehension of VRRW and SID needs a joint understanding of on one hand a dynamical system governing the general evolution, and on the other hand some probabilistic phenomena, acting as perturbations, and sometimes changing the nature of this dynamical system.

The purpose of our talk is to present our recent results on the subject [1,3,4,6].

Bibliography

[1] M. Bena