In this talk I will give an overview of the algorithms used by Chebfun to numerically compute polynomial and trigonometric minimax approximations of continuous functions. I'll also present Chebfun's capabilities to compute best approximations on compact subsets of an interval and how these methods can be used to design digital filters.

In this talk we will explore the convergence of Krylov methods when used to solve $Lu = f$ where $L$ is an unbounded linear operator.  We will show that for certain problems, methods like Conjugate Gradients and GMRES still converge even though the spectrum of $L$ is unbounded. A theoretical justification for this behavior is given in terms of polynomial approximation on unbounded domains.    

Sparse matrices occur in numerical simulations throughout science and engineering. In particular, it is often desirable to solve systems of the form Ax=b, where A is a sparse matrix with 100,000+ rows and columns. The order that the rows and columns occur in can have a dramatic effect on the viability of a direct solver e.g., the time taken to find x, the amount of memory needed, the quality of x,... We shall consider symmetric matrices and, with the help of playdough, explore how best to order the rows/columns using a nested dissection strategy. Starting with a straightforward strategy, we will discover the pitfalls and develop an adaptive strategy with the aim of coping with a large variety of sparse matrix structures.

Some of the talk will involve the audience playing with playdough, so bring your inner child along with you!

We review a class of systems of non-linear PDEs, derived from the Cahn--Hilliard and Ohta--Kawasaki functionals, that describe the energy evolution of diblock copolymers. These are long chain molecules that can self assemble into repeating patterns as they cool. We are particularly interested in finite element numerical methods that approximate these PDEs in the two-phase (in which we model the polymer only) and three-phase (in which we imagine the polymer surrounded by, and interacting with, a void) cases.

We present a brief derivation of the underlying models, review a class of numerical methods to approximate them, and showcase some early results from our codes.

The talk will present ongoing work on medical image reconstruction from x-ray scanners. A suitable method for reconstruction of these undersampled systems is compressed sensing. The presentation will show respective reconstruction methods and their analysis. Furthermore, work in progress about extensions of the standard approach will be shown.

Our everyday usage of the Internet generates huge amounts of data on how humans collect and exchange information worldwide. In this talk, I will outline recent work in which we investigate whether data from sources such as Google, Wikipedia and Flickr can be used to gain new insight into real world human behaviour. I will provide case studies from a range of domains, including disease detection, crowd size estimation, and evaluating whether the beauty of the environment we live in might affect our health.