Mon, 02 Jun 2014

14:15 - 15:15
Oxford-Man Institute

We consider the short time asymptotics of the heat content $E(s)$ of a domain $D$ of $\mathbb{R}^d$, where $D$ has a random boundary.

PHILIPPE CHARMOY
(University of Oxford)
Abstract

When $\partial D$ is spatially homogeneous, we show that we can recover the lower and upper Minkowski dimensions of $\partial D$ from the sort time behaviour of $E(s)$. Furthermore, when the Minkowski dimension exists, finer geometric fluctuations can be recovered and $E(s)$ is controlled by $s^\alpha e^{f(\log(1/s))}$ for small $s$, for some $\alpha \in (0, \infty)$ and some regularly varying function $f$. The function $f$ is not constant is general and carries some geometric information.

When $\partial D$ is statistically self-similar, the Minkowski dimension and content of $\partial D$ typically exist and can be recovered from $E(s)$. Furthermore, $E(s)$ has an almost sure expansion $E(s) = c s^{\alpha} N_\infty + o(s^\alpha)$ for small $s$, for some $c$ and $\alpha \in (0, \infty)$ and some positive random variable $N_\infty$ with unit expectation arising as the limit of some martingale. In some cases, we can show that the fluctuations around this almost sure behaviour are governed by a central limit theorem, and conjecture that this is true more generally.

This is based on joint work with David Croydon and Ben Hambly.

Mon, 28 Apr 2014

15:45 - 16:45
Oxford-Man Institute

The decay rate of the expected signature of a stopped Brownian motion

NI HAO
(University of Oxford)
Abstract

In this presentation, we focus on the decay rate of the expected signature of a stopped Brownian motion; more specifically we consider two types of the stopping time: the first one is the Brownian motion up to the first exit time from a bounded domain $\Gamma$, denoted by $\tau_{\Gamma}$, and the other one is the Brownian motion up to $min(t, \tau_{\Gamma\})$. For the first case, we use the Sobolev theorem to show that its expected signature is geometrically bounded while for the second case we use the result in paper (Integrability and tail estimates for Gaussian rough differential equation by Thomas Cass, Christian Litterer and Terry Lyons) to show that each term of the expected signature has the decay rate like 1/ \sqrt((n/p)!) where p>2. The result for the second case can imply that its expected signature determines the law of the signature according to the paper (Unitary representations of geometric rough paths by Ilya Chevyrev)

Thu, 29 May 2014

16:00 - 17:00
L5

The algebraicity of sieved sets and rational points on curves

Miguel Walsh
(University of Oxford)
Abstract
We will discuss some connections between the polynomial method, sieve theory, inverse problems in arithmetic combinatorics and the estimation of rational points on curves. Our motivating questions will be to classify those sets that are irregularly distributed in residue classes and to understand how many rational points of bounded height can a curve of fixed degree have.
Fri, 02 May 2014

12:00 - 13:00
C6

Using multiple frequencies to satisfy local constraints in PDE and applications to hybrid inverse problems

Giovanni Alberti
(University of Oxford)
Abstract

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.

Tue, 11 Mar 2014

14:00 - 15:00
L5

Particle Methods for Inference in Non-linear Non-Gaussian State-Space Models

Arnaud Doucet
(University of Oxford)
Abstract

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.

Tue, 04 Mar 2014

14:00 - 15:00
L5

Towards realistic performance for iterative methods on shared memory machines

Shengxin (Jude) Zhu
(University of Oxford)
Abstract

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.

Tue, 04 Mar 2014

14:00 - 14:30
L5

Euler-Maclaurin and Newton-Gregory Interpolants

Mohsin Javed
(University of Oxford)
Abstract

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.

Tue, 18 Feb 2014

14:30 - 15:00
L5

Conjugate gradient iterative hard thresholding for compressed sensing and matrix completion

Ke Wei
(University of Oxford)
Abstract

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.

Tue, 11 Feb 2014

14:30 - 15:30
L6

Frankl-Rödl type theorems for codes and permutations

Eoin Long
(University of Oxford)
Abstract

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.

Tue, 11 Feb 2014

14:30 - 15:00
L5

Community Structure in Multilayer Networks

Mason Alexander Porter
(University of Oxford)
Abstract

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.

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