University of Oxford

Accurate methods for the first-order advection equation, used for example in tracking contaminants in fluids, usually exploit the theory of characteristics. Such methods are described and contrasted with methods that do not make use of characteristics.

Then the second-order wave equation, in the form of a first-order system, is considered. A review of the one-dimensional theory using solutions of various Riemann problems will be provided. In the special case that the medium has the ‘Goupillaud’ property, that waves take the same time to travel through each layer, one can derive exact solutions even when the medium is spatially heterogeneous. The extension of this method to two-dimensional problems will then be discussed. In two-dimensions it is not apparent that exact solutions can be found, however by exploiting a generalised Goupillaud property, it is possible to calculate approximate solutions of high accuracy, perhaps sufficient to be of benchmark quality. Some two-dimensional simulations, using exact one-dimensional solutions and operator splitting, will be described and a numerical evaluation of accuracy will be given.

In this talk I will review mathematical models used to describe the dynamics of ice sheets, and highlight some current areas of active research.  Melting of glaciers and ice sheets causes an increase in global sea level, and provides many other feedbacks on isostatic adjustment, the dynamics of the ocean, and broader climate patterns.  The rate of melting has increased in recent years, but there is still considerable uncertainty over  why this is, and whether the increase will continue.  Central to these questions is understanding the physics of how the ice intereacts with the atmosphere, the ground on which it rests, and with the ocean at its margins.  I will given an overview of the fluid mechanical problems involved and the current state of mathematical/computational modelling.  I will focus particularly on the issue of changing lubrication due to water flowing underneath the ice, and discuss how we can use models to rationalise observations of ice speed-up and slow-down.

Adaptive cubic regularization methods have recently emerged as a credible alternative to line search and trust-region for smooth nonconvex optimization, with optimal complexity amongst second-order methods. Here we consider a general class of adaptive regularization methods, that use first- or higher-order local Taylor models of the objective regularized by a(ny) power of the step size. We investigate the worst-case complexity/global rate of convergence of these algorithms, in the presence of varying (unknown) smoothness of the objective. We find that some methods automatically adapt their complexity to the degree of smoothness of the objective; while others take advantage of the power of the regularization step to satisfy increasingly better bounds with the order of the models. This work is joint with Nick Gould (RAL) and Philippe Toint (Namur).

Computing distinct solutions of differential equations -- Patrick Farrell

Abstract: TBA

Triangles and equations -- Yufei Zhao

Abstract: I will explain how tools in graph theory can be useful for understanding certain problems in additive combinatorics, in particular the existence of arithmetic progressions in sets of integers. 

TBA