Past Numerical Analysis Group Internal Seminar

25 February 2014
14:00
Abstract

The classical theory of Gaussian quadrature assumes a positive weight function. This implies many desirable properties of the rule: Guaranteed existence and uniqueness of the orthogonal polynomials whose zeros are the nodes of the rule, nodes that are contained in the interval of integration, as well as positive quadrature weights, which implies that the rule is stable. There has been little research on polynomials that are orthogonal with respect to non-positive weight functions, although these could be interesting for, for example, oscillatory quadrature problems. In this talk I will present some of the few results we have on this, as well as some weird and wonderful conjectures.
 

  • Numerical Analysis Group Internal Seminar
18 February 2014
14:30
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.

  • Numerical Analysis Group Internal Seminar
18 February 2014
14:00
Abstract

When applied to an inequality constrained optimization problem, interior point methods generate iterates that belong to the interior of the set determined by the constraints, thus avoiding/ignoring the combinatorial aspect of the solution. This comes at the cost of difficulty in predicting the optimal active constraints that would enable termination.  We propose the use of controlled perturbations to address this challenge. Namely, in the context of linear programming with only nonnegativity constraints as the inequality constraints, we consider perturbing the nonnegativity constraints so as to enlarge the feasible set. Theoretically, we show that if the perturbations are chosen appropriately, the solution of the original problem lies on or close to the central path of the perturbed problem and that a primal-dual path-following algorithm applied to the perturbed problem is able to predict the optimal active-set of the original problem when the duality gap (for the perturbed problem) is not too small. Encouraging preliminary numerical experience is obtained when comparing the perturbed and unperturbed interior point algorithms' active-set predictions for the purpose of cross-over to simplex.

  • Numerical Analysis Group Internal Seminar
11 February 2014
14:30
Mason Alexander Porter
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.

  • Numerical Analysis Group Internal Seminar
11 February 2014
14:00
David Hewett
Abstract

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. 

  • Numerical Analysis Group Internal Seminar
4 February 2014
14:30
Patrick Farrell
Abstract

Quantifying the uncertainty in computational simulations is one of the central challenges confronting the field of computational science and engineering today. The uncertainty quantification of inverse problems is neatly addressed in the Bayesian framework, where instead of seeking one unique minimiser of a regularised misfit functional, the entire posterior probability distribution is to be characterised. In this talk I review the deep connection between deterministic PDE-constrained optimisation techniques and Bayesian inference for inverse problems, discuss some recent advances made in the Bayesian viewpoint by adapting deterministic techniques, and mention directions for future research.

  • Numerical Analysis Group Internal Seminar
4 February 2014
14:00
Jeffrey D. Blanchard
Abstract

Composite dilation wavelets are affine systems which extend the notion of wavelets by incorporating a second set of dilations.  The addition of a second set of dilations allows the composite system to capture directional information in addition to time and frequency information.  We classify admissible dilation groups at two extremes: frequency localization through minimally supported frequency composite dilation wavelets and time localization through crystallographic Haar-type composite dilation wavelets. 

  • Numerical Analysis Group Internal Seminar
28 January 2014
14:30
Abstract

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).

  • Numerical Analysis Group Internal Seminar
28 January 2014
14:00
Rachael Tappenden
Abstract

The accurate and efficient solution of linear systems $Ax=b$ is very important in many engineering and technological applications, and systems of this form also arise as subproblems within other algorithms. In particular, this is true for interior point methods (IPM), where the Newton system must be solved to find the search direction at each iteration. Solving this system is a computational bottleneck of an IPM, and in this talk I will explain how preconditioning and deflation techniques can be used, to lessen this computational burden.  This work is joint with Jacek Gondzio.

  • Numerical Analysis Group Internal Seminar
21 January 2014
14:00
Iain Smears
Abstract

Hamilton—Jacobi—Bellman (HJB) equations are a class of fully nonlinear second-order partial differential equations (PDE) of elliptic or parabolic type that originate from Stochastic Optimal Control Theory. These PDE are fully nonlinear in the sense that the nonlinear terms include the second partial derivatives of the unknown solution; this strong nonlinearity severely restricts the range of numerical methods that are known to be convergent. These problems have traditionally been solved with low order monotone schemes, often of finite difference type, which feature certain limitations in terms of efficiency and practicability.
In this summary talk of my DPhil studies, we will be interested in the development of hp-version discontinuous Galerkin finite element methods (DGFEM) for the class of HJB equations that satisfy a Cordès condition. First, we will show the novel techniques of analysis used to find a stable and convergent scheme in the elliptic setting, and then we will present recent work on their extension to parabolic problems. The resulting method is very nonstandard, provably of high order, and it even allows for exponential convergence under hp-refinement. We present numerical experiments showing the accuracy, computational efficiency and flexibility of the scheme

  • Numerical Analysis Group Internal Seminar

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