University of Oxford

Convolution is widely-used and fundamental mathematical operation
in signal processing, statistics, and PDE theory.

Unfortunately the CONV() method in Chebfun for convolving two chebfun 
objects has long been one of the most disappointingly slow features of 
the project. In this talk we will present a new algorithm, which shows 
performance gains on the order of a factor 100.

The key components of the new algorithm are:
* a convolution theorem for Legendre polynomials 
* recurrence relations satisfied by spherical Bessel functions
* recent developments in fast Chebyshev-Legendre transforms [1]

Time-permitting, we shall end with an application from statistics,
using the fact that the probability distribution of the sum of two 
independent random variables is the convolution of their individual 
distributions.

[1] N. Hale and A. Townsend, "A fast, simple, and stable Chebyshev-
Legendre transform using an asymptotic formula”, SISC (to appear).

The splitting-up method is a powerful tool to solve (SP)DEs by dividing the equation into a set of simpler equations that are easier to handle. I will speak about how such splitting schemes can be derived and extended by insights from the theory of rough paths.

Finally, I will discuss numerics for real-world applications that appear in the management of risk and engineering applications like nonlinear filtering.

Sampling a $d$-dimensional continuous signal (say a semimartingale) $X:[0,T] \rightarrow \mathbb{R}^d$ at times $D=(t_i)$, we follow the recent papers [Gyurko-Lyons-Kontkowski-Field-2013] and [Lyons-Ni-Levin-2013] in constructing a lead-lag path; to be precise, a piecewise-linear, axis-directed process $X^D: [0,1] \rightarrow
\mathbb{R}^{2d}$ comprised of a past and future component. Lifting $X^D$ to its natural rough path enhancement, we can consider the question of convergence as
the latency of our sampling becomes finer.

TBA

We will present three different recent applications of cell motion in biology: (i) Movement of epithelial sheets and rosette formation, (ii) neural crest cell migrations, (iii) acid-mediated cancer cell invasion. While the talk will focus primarily on the biological application, it will be shown that all of these processes can be represented by reaction-diffusion equations with nonlinear diffusion term.

A new benchmark, High Performance Conjugate Gradient (HPCG), finally was introduced recently for the Top500 list and the Green500 list. This will draw more attention to performance of sparse iterative solvers on distributed supercomputers and energy efficiency of hardware and software. At the same time, this will more widely promote the concept that communications are the bottleneck of performance of iterative solvers on distributed supercomputers, here we will go a little deeper, discussing components of communications and discuss which part takes a dominate share. Also discussed are mathematics tricks to detect some metrics of an underlying supercomputer.