University of Oxford

In this talk I will describe a multiple frequency approach to the boundary control of Helmholtz and Maxwell equations. We give boundary conditions and a finite number of frequencies such that the corresponding solutions satisfy certain non-zero constraints inside the domain. The suitable boundary conditions and frequencies are explicitly constructed and do not depend on the coefficients, in contrast to the illuminations given as traces of complex geometric optics solutions. This theory finds applications in several hybrid imaging modalities. Some examples will be discussed.

State-space models are a very popular class of time series models which have found thousands of applications in engineering, robotics, tracking, vision,  econometrics etc. Except for linear and Gaussian models where the Kalman filter can be used, inference in non-linear non-Gaussian models is analytically intractable.  Particle methods are a class of flexible and easily parallelizable simulation-based algorithms which provide consistent approximations to these inference problems. The aim of this talk is to introduce particle methods and to present the most recent developments in this area.

This talk introduces a random linear model to investigate the memory bandwidth barrier effect on current shared memory computers. Based on the fact that floating-point operations can be hidden by implicit compiling techniques, the runtime for memory intensive applications can be modelled by memory reference time plus a random term. The random term due to cache conflicts, data reuse and other environmental factors is proportional to memory reference volume. Statistical techniques are used to quantify the random term and the runtime performance parameters. Numerical results based on thousands representative matrices from various applications are presented, compared, analysed and validated to confirm the proposed model. The model shows that a realistic and fair metric for performance of iterative methods and other memory intensive applications should consider the memory bandwidth capability and memory efficiency.

The Euler-Maclaurin formula is a quadrature rule based on corrections to the trapezoid rule using odd derivatives at the end-points of the function being integrated. It appears that no one has ever thought about a related function approximation that will give us the Euler-Maclaurin quadrature rule, i.e., just like we can derive Newton-Cotes quadrature by integrating polynomial approximations of the function, we investigate, what function approximation will integrate exactly to give the corresponding Euler-Maclaurin quadrature. It turns out, that the right function approximation is a combination of a trigonometric interpolant and a polynomial.

To make the method more practical, we also look at the closely related Newton-Gregory quadrature, which is very similar to the Euler-Maclaurin formula but instead of derivatives, uses finite differences. Following almost the same procedure, we find another mixed function approximation, derivative free, whose exact integration yields the Newton-Gregory quadrature rule.

Compressed sensing and matrix completion are techniques by which simplicity in data can be exploited for more efficient data acquisition. For instance, if a matrix is known to be (approximately) low rank then it can be recovered from few of its entries. The design and analysis of computationally efficient algorithms for these problems has been extensively studies over the last 8 years. In this talk we present a new algorithm that balances low per iteration complexity with fast asymptotic convergence. This algorithm has been shown to have faster recovery time than any other known algorithm in the area, both for small scale problems and massively parallel GPU implementations. The new algorithm adapts the classical nonlinear conjugate gradient algorithm and shows the efficacy of a linear algebra perspective to compressed sensing and matrix completion.

We give a new proof of the Frankl-Rödl theorem on set systems with a forbidden intersection. Our method extends to codes with forbidden distances, where over large alphabets our bound is significantly better than that obtained by Frankl and Rödl. One consequence of our result is a Frankl-Rödl type theorem for permutations with a forbidden distance. Joint work with Peter Keevash.

Networks arise pervasively in biology, physics, technology, social science, and myriad other areas. An ordinary network consists of a collection of entities (called nodes) that interact via edges. "Multilayer networks" are a more general representation that can be used when nodes are connected to each other via multiple types of edges or a network changes in time.  In this talk, I will discuss how to find dense sets of nodes called "communities" in multilayer networks and some applications of community structure to problems in neuroscience and finance.

Sobolev spaces are the standard framework in which to analyse weak (variational) formulations of PDEs or integral equations and their numerical solution (e.g. using the Finite Element Method or the Boundary Element Method). There are many different ways to define Sobolev spaces on a given domain, for example via integrability of weak derivatives, completions of spaces of smooth functions with respect to certain norms, or restriction from spaces defined on a larger domain. For smooth (e.g. Lipschitz) domains things many of these definitions coincide. But on rough (e.g. fractal) domains the picture is much more complicated. In this talk I'll try to give a flavour of the sort of interesting behaviour that can arise, and what implications this behaviour has for a "practical" example, namely acoustic wave scattering by fractal screens.