Comlab

The aim of this talk is to describe several methods for numerically approximating

the integral of a multivariate highly oscillatory function. We begin with a review

of the asymptotic and Filon-type methods developed by Iserles and Nørsett. Using a

method developed by Levin as a point of departure we will construct a new method that

uses the same information as the Filon-type method, and obtains the same asymptotic

order, while not requiring moments. This allows us to integrate over nonsimplicial

domains, and with complicated oscillators.

High-performance computing is an important tool for computational science.

Oxford University has recently decided to invest £3M in a new supercomputing

facility which is under development now. In this seminar I will give an overview

of some activities in Oxford and provide a vision for supercomputing here.

I will discuss some of the numerical analysis software and tools,

such as Distributed Matlab and indicate some of the challenges at

the intersection of numerical analysis and high-performance computing.

The aim of this talk is to give some understanding of the theory of matrix $p$'th roots (solutions to the nonlinear matrix equation $X^{p} = A$), to explain how and how not to compute roots, and to describe some applications. In particular, an application in finance will be described concerning roots of transition matrices from Markov models.

One of the oldest approach in meshless methods for PDEs is the Interpolating Moving Least Squares (IMLS) technique developed in the 1980s. Although widely accepted by users working in fields as diverse as geoinformatics and crack dynamics I shall take a fresh look at this method and ask for the equivalent difference operators which are generated implicitly. As it turns out, these operators are optimal only in trivial cases and are "strange" in general. I shall try to exploit two different approaches for the computation of these operators.

On the other hand (and very different from IMLS), Total Variation Flow (TVF) PDEs are the most recent developments in image processing and have received much attention lately. Again I shall show that they are able to generate "strange" discrete operators and that they easily can behave badly although they may be properly implemented.

We will explore connections between the structure of high-dimensional convex polytopes and information acquisition for compressible signals. A classical result in the field of convex polytopes is that if N points are distributed Gaussian iid at random in dimension n<<N, then only order (log N)^n of the points are vertices of their convex hull. Recent results show that provided n grows slowly with N, then with high probability all of the points are vertices of its convex hull. More surprisingly, a rich "neighborliness" structure emerges in the faces of the convex hull. One implication of this phenomenon is that an N-vector with k non-zeros can be recovered computationally efficiently from only n random projections with n=2e k log(N/n). Alternatively, the best k-term approximation of a signal in any basis can be recovered from 2e k log(N/n) non-adaptive measurements, which is within a log factor of the optimal rate achievable for adaptive sampling. Additional implications for randomized error correcting codes will be presented.

This work was joint with David L. Donoho.

In this talk we present different strategies for regularization of the pure Newton method

(minimization problems)and of the Gauss-Newton method (systems of nonlinear equations).

For these schemes, we prove general convergence results. We establish also the global and

local worst-case complexity bounds. It is shown that the corresponding search directions can

be computed by a standard linear algebra technique.

We present a novel enhanced finite element method for the Darcy problem starting from the non stable

continuous $P_1 / P_0$ finite element spaces enriched with multiscale functions. The method is a departure

from the standard mixed method framework used in these applications. The methods are derived in a Petrov-Galerkin

framework where both velocity and pressure trial spaces are enriched with functions based on residuals of strong

equations in each element and edge partition. The strategy leads to enhanced velocity space with an element of

the lowest order Raviart-Thomas space and to a stable weak formulation preserving local mass conservation.

Numerical tests validate the method.

Jointly with Gabriel R Barrenechea, Universidad de Concepcion &amp;

Frederic G C Valentin, LNCC

Strong horizontal gradients of density are responsible for the occurence of a large number of (often catastrophic) flows, such as katabatic winds, dust storms, pyroclastic flows and powder-snow avalanches. For a large number of applications, the overall density contrast in the flow remains small and simulations are carried in the Boussinesq limit, where density variations only appear in the body-force term. However, pyroclastic flows and powder-snow avalanches involve much larger density contrasts, which implies that the inhomogeneous Navier-Stokes equations need to be solved, along with a closure equation describing the mass diffusion. We propose a Lagrange-Galerkin numerical scheme to solve this system, and prove optimal error bounds subject to constraints on the order of the discretization and the time-stepping. Simulations of physical relevance are then shown.

An integral part of the brain is a fluid flow system that is separate from brain tissue and the cerebral blood flow system: cerebrospinal fluid (CSF) is produced near the centre of the brain, flows out and around the brain, including around the spinal cord and is absorbed primarily in a region between the brain tissue and the skull. Hydrocephalus covers a broad range of anomalous flow and pressure situations: the normal flow path can become blocked, other problems can occur which result in abnormal tissue deformation or pressure changes. This talk will describe work that treats brain tissue as a poroelastic matrix through which the CSF can move when normal flow paths are blocked, producing tissue deformation and pressure changes. We have a number of models, the simplest treating the brain and CSF flow as having spherial symmetry ranging to more complex, fully three-dimensional computations. As well as considering acute hydrocephalus, we touch on normal pressure hydrocephalus, idiopathic intracranial hypertension and simulation of an infusion test. The numerical methods used are a combination of finite difference and finite element techniques applied to an interesting set of hydro-elastic equations.

In this talk, we report on recent activities in the development of automatic differentiation tools for Matlab and CapeML, a common intermediate language for process control, and highlight some recent AD applications. Lastly, we show the potential for parallelisation created by AD and comment on the impact on scientific computing due to emerging multicore chips which are providing substantial thread-based parallelism in a "pizza box" form factor.