Past Numerical Analysis Group Internal Seminar

17 May 2016
14:30
Abstract

Diffusion processes are commonly used in image processing. In particular, complex diffusion models have been successfully applied in medical imaging denoising. The interpretation of a complex diffusion equation as a cross-diffusion system motivates the introduction of more general models of this type and their study in the context of image processing. In this talk we will discuss the use of nonlinear cross-diffusion systems to perform image restoration. We will analyse the well-posedness, scale-space properties and
long time behaviour of the models along with their performance to treat image filtering problems. Examples of application will be highlighted.

  • Numerical Analysis Group Internal Seminar
10 May 2016
14:30
Nick Trefethen
Abstract

Low-rank compression of matrices and tensors is a huge and growing business.  Closely related is low-rank compression of multivariate functions, a technique used in Chebfun2 and Chebfun3.  Not all functions can be compressed, so the question becomes, which ones?  Here we focus on two kinds of functions for which compression is effective: those with some alignment with the coordinate axes, and those dominated by small regions of localized complexity.

 

  • Numerical Analysis Group Internal Seminar
10 May 2016
14:00
Abstract

We establish necessary and sufficient conditions for linear convergence of operator splitting methods for a general class of convex optimization problems where the associated fixed-point operator is averaged. We also provide a tight bound on the achievable convergence rate. Most existing results establishing linear convergence in such methods require restrictive assumptions regarding strong convexity and smoothness of the constituent functions in the optimization problem. However, there are several examples in the literature showing that linear convergence is possible even when these properties do not hold. We provide a unifying analysis method for establishing linear convergence based on linear regularity and show that many existing results are special cases of our approach.

  • Numerical Analysis Group Internal Seminar
3 May 2016
14:30
Abstract
We propose several optimal preconditioners for systems defined by some functions $g$ of Toeplitz matrices $T_n$. In this paper we are interested in solving $g(T_n)x=b$ by the preconditioned conjugate method or the preconditioned minimal residual method, namely in the cases when $g(T_n)$ are the analytic functions $e^{T_n}$, $\sin{T_n}$ and $\cos{T_n}$. Numerical results are given to show the effectiveness of the proposed preconditioners.
  • Numerical Analysis Group Internal Seminar
3 May 2016
14:00
Abstract

This talk will be a geophysicist's view on the emerging properties of a numerical model representing the Earth's climate and volcanic activity over the past million years.

The model contains a 2D ice sheet (Glen's Law solved with a semi-implicit scheme), an energy balance for the atmosphere and planet surface (explicit), and an ODE for the time evolution of CO2 (explicit).

The dependencies between these models generate behaviour similar to weakly coupled nonlinear oscillators.

  • Numerical Analysis Group Internal Seminar
26 April 2016
14:30
Leslie Hogben
Abstract

Low-rank compression of matrices and tensors is a huge and growing business.  Closely related is low-rank compression of multivariate functions, a technique used in Chebfun2 and Chebfun3.  Not all functions can be compressed, so the question becomes, which ones?  Here we focus on two kinds of functions for which compression is effective: those with some alignment with the coordinate axes, and those dominated by small regions of localized complexity.

  • Numerical Analysis Group Internal Seminar
26 April 2016
14:00
Yuji Nakatsukasa
Abstract

An important observation in compressed sensing is the exact recovery of an l0 minimiser to an underdetermined linear system via the l1 minimiser, given the knowledge that a sparse solution vector exists. Here, we develop a continuous analogue of this observation and show that the best L1 and L0 polynomial approximants of a corrupted function (continuous analogue of sparse vectors) are equivalent. We use this to construct best L1 polynomial approximants of corrupted functions via linear programming. We also present a numerical algorithm for computing best L1 polynomial approximants to general continuous functions, and observe that compared with best L-infinity and L2 polynomial approximants, the best L1 approximants tend to have error functions that are more localized.

Joint work with Alex Townsend (MIT).

  • Numerical Analysis Group Internal Seminar
8 March 2016
14:30
Dave Hewett
Abstract

The Faraday cage effect is the phenomenon whereby electrostatic and electromagnetic fields are shielded by a wire mesh "cage". Nick Trefethen, Jon Chapman and I recently carried out a mathematical analysis of the two-dimensional electrostatic problem with thin circular wires, demonstrating that the shielding effect is not as strong as one might infer from the physics literature. In this talk I will present new results generalising the previous analysis to the electromagnetic case, and to wires of arbitrary shape. The main analytical tool is the asymptotic method of multiple scales, which is used to derive continuum models for the shielding, involving homogenized boundary conditions on an effective cage boundary. In the electromagnetic case one observes interesting resonance effects, whereby at frequencies close to the natural frequencies of the equivalent solid shell, the presence of the cage actually amplifies the incident field, rather than shielding it. We discuss applications to radiation containment in microwave ovens and acoustic scattering by perforated shells. This is joint work with Ian Hewitt.

  • Numerical Analysis Group Internal Seminar

Pages