Past

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.

An important class of nonlinear parabolic equations is the class of quasi-linear equations, i.e., equations with a leading second-order (in space) linear part (e.g., the Laplacian) and a nonlinear part which depends on the first-order spatial derivatives of the unknown function. This class contains the Navier-Stokes system of fluid dynamics, as well as "viscous" versions (or "regularized") of the Hamilton-Jacobi equation, nonlinear hyperbolic conservation laws and more. The talk will present various recent results concerning existence/uniqueness (and nonexistence/nonuniqueness) of global solutions. In addition, a new class of "Bernstein-type" estimates of derivatives will be presented. These estimates are independent of the viscosity parameter and thus lead to results concerning the "zero-viscosity" limit.

The estimation of heat kernels has been of much interest in various settings. Often, the spaces considered have some kind of uniformity in the volume growth. Recent results have shown that this is not the case for certain random fractal sets. I will present heat kernel bounds for spaces admitting a suitable resistance form, when the volume growth is not uniform, which are motivated by these examples.

We study diploid branching particle models and its behaviour when rapid

stirring, i.e. rapid exchange of particles between neighbouring spatial

sites, is added to the interaction. The particle models differ from the

``usual'' models in that they all involve two types of particles, male

and female, and branching can only occur when both types of particles

are present. We establish the existence of nontrivial stationary

distributions for various models when birth rates are sufficiently large.

Most people acquire their

Abstract 1 Another Orthogonal Matrix

A householder reflection and a suitable product of Givens rotations are two well known examples of an orthogonal matrix with given first column. We present another way to produce such a matrix and apply it to produce a "fast Givens" method to compute the R factor of A, A = QR. This approach avoids the danger of under/overflow.
(joint work with Eric Barszcz)

Abstract 2 An application of Pfaff's Theorem (on skew-symmetric matrices)

There are no constraints on the eigenvalues of a product of two real symmetric matrices but what about the product of two real skew-symmetric matrices?
(joint work with A Dubrulle)