Past

We show that the solutions of stochastic differential equations converge in

the rough path metric as the coefficients of these equations converge in a

suitable lipschitz norm. We then use this fact to obtain results about

differential equations driven by the Brownian rough path.

It is now known that the overall behaviour of a simple random walk (SRW) on

supercritical (p>p_c) percolation cluster in Z^d is similiar to that of the SRW

in Z^d. The critical case (p=p_c) is much harder, and one needs to define the

'incipient infinite cluster' (IIC). Alexander and Orbach conjectured in 1982

that the return probability for the SRW on the IIC after n steps decays like

n^{2/3} in any dimension. The easiest case is that of trees; this was studied by

Kesten in 1986, but we can now revisit this problem with new techniques.

We use the partial differential inclusion method to establish existence of

infinitely many weak solutions to the one-dimensional version of the

Perona-Malek anisotropic diffusion model in the theory of image processing. We

consider the homogeneous Neumann problem as the model requires.

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The Yang-Mills energy is a non-negative functional on the space of connections on a principal bundle over a Riemannian manifold. At a heuristical level, this energy determines a Gibbs measure which is called the Yang-Mills measure. When the manifold is a surface, a stochastic process can be constructed - at least in two different ways - which is a sensible candidate for the random holonomy of a connection distributed according to the Yang-Mills measure. This process is constructed by using some specifications given by physicists of its distribution, namely some of its finite-dimensional marginals, which of course physicists have derived from the Yang-Mills energy, but by non-rigorous arguments. Without assuming any familiarity with this stochastic process, I will present a large deviations result which is the first rigorous link between the Yang-Mills energy and the Yang-Mills measure.

Current methods for globalizing Newton's Method for solving systems of nonlinear equations fall back on steps biased towards the steepest descent direction (e.g. Levenberg/Marquardt, Trust regions, Cauchy point dog-legs etc.), when there is difficulty in making progress. This can occasionally lead to very slow convergence when short steps are repeatedly taken.

This talk looks at alternative strategies based on searching curved arcs related to Davidenko trajectories. Near to manifolds on which the Jacobian matrix is singular, certain conjugate steps are also interleaved, based on identifying a Pareto optimal solution.

Preliminary limited numerical experiments indicate that this approach is very effective, with rapid and ultimately second order convergence in almost all cases. It is hoped to present more detailed numerical evidence when the talk is given. The new ideas can also be incorporated with more recent ideas such as multifilters or nonmonotonic line searches without difficulty, although it may be that there is no longer much to gain by doing this.