Past

Why does a spinning coin come to such a sudden stop? Why does a
spinning hard-boiled egg stand up on its end? And why does the
rattleback rotate happily in one direction but not in the other?
The key mathematical aspects of these familiar dynamical phenomena,
which admit simple table-top demonstration, will be revealed.

Rapidly oscillating problems, whether differential equations or

integrals, ubiquitous in applications, are allegedly difficult to

compute. In this talk we will endeavour to persuade the audience that

this is false: high oscillation, properly understood, is good for

computation! We describe methods for differential equations, based on

Magnus and Neumann expansions of modified systems, whose efficacy

improves in the presence of high oscillation. Likewise, we analyse

generalised Filon quadrature methods, showing that also their error

sharply decreases as the oscillation becomes more rapid.

In many applications of interest, such as the conformational

dynamics of molecules, large deterministic systems can exhibit

stochastic behaviour in a relative small number of coarse-grained

variables. This kind of dimension reduction, from a large deterministic

system to a smaller stochastic one, can be very useful in understanding

the problem. Whilst the subject of statistical mechanics provides

a wealth of explicit examples where stochastic models for coarse

variables can be found analytically, it is frequently the case

that applications of interest are not amenable to analytic

dimension reduction. It is hence of interest to pursue computational

algorithms for such dimension reduction. This talk will be devoted

to describing recent work on parameter estimation aimed at

problems arising in this context.

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Joint work with Raz Kupferman (Jerusalem) and Petter Wiberg (Warwick)

A new filter method will be presented that attempts to find a feasible

point for sets of nonlinear sets of equalities and inequalities. The

method is intended to work for problems where the number of variables

or the number of (in)equalities is large, or both. No assumption is

made about convexity. The technique used is that of maintaining a list

of multidimensional "filter entries", a recent development of ideas

introduced by Fletcher and Leyffer. The method will be described, as

well as large scale numerical experiments with the corresponding

Fortran 90 module, FILTRANE.

In recent times, research into scattering of electromagnetic waves by complex objects

has assumed great importance due to its relevance to radar applications, where the

main objective is to identify targeted objects. In designing stealth weapon systems

such as military aircraft, control of their radar cross section is of paramount

importance. Aircraft in combat situations are threatened by enemy missiles. One

countermeasure which is used to reduce this threat is to minimise the radar cross

section. On the other hand, there is a demand for the enhancement of the radar cross

section of civilian spacecraft. Operators of communication satellites often request

a complicated differential radar cross section in order to assist with the tracking

of the satellite. To control the radar cross section, an essential requirement is a

capability for accurate prediction of electromagnetic scattering from complex objects.

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One difficulty which is encountered in the development of suitable numerical solution

schemes is the existence of constraints which are in excess of those needed for a unique

solution. Rather than attempt to include the constraint in the equation set, the novel

approach which is presented here involves the use of the finite element method and the

construction of a specialised element in which the relevant solution variables are

appropriately constrained by the nature of their interpolation functions. For many

years, such an idea was claimed to be impossible. While the idea is not without its

difficulties, its advantages far outweigh its disadvantages. The presenter has

successfully developed such an element for primitive variable solutions to viscous

incompressible flows and wishes to extend the concept to electromagnetic scattering

problems.

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Dr Mack has first degrees in mathematics and aeronautical engineering, plus a Masters

and a Doctorate, both in computational fluid dynamics. He has some thirty years

experience in this latter field. He pioneered the development of the innovative

solenoidal approach for the finite element solution of viscous incompressible flows.

At the time, such a radical idea was claimed in the literature to be impossible.

Much of this early research was undertaken during a six month sabbatical with the

Numerical Analysis Group at the Oxford University Computing Laboratory. Dr Mack has

since received funding from British Aerospace and the United States Department of

Defense to continue this research.

In addition to the announced topic of Pascal Matrices (abstract below) we will speak briefly about more recent work by Per-Olof Persson on generating simplicial meshes on regions defined by a function that gives the distance from the boundary. Our first goal was a short MATLAB code and we just submitted "A Simple Mesh Generator in MATLAB" to SIAM.

This is joint work with Alan Edelman at MIT and a little bit with Pascal. They had all the ideas.

Put the famous Pascal triangle into a matrix. It could go into a lower triangular L or its transpose L' or a symmetric matrix S:

These binomial numbers come from a recursion, or from the formula for i choose j, or functionally from taking powers of (1 + x).

The amazing thing is that L times L' equals S. (OK for 4 by 4) It follows that S has determinant 1. The matrices have other unexpected properties too, that give beautiful examples in teaching linear algebra. The proof of L L' = S comes 3 ways, I don't know which you will prefer:

1. By induction using the recursion formula for the matrix entries.
2. By an identity for the coefficients i+j choose j in S.
3. By applying both sides to the column vector [ 1 x x² x³ ... ]'.

The third way also gives a proof that S³ = -I but we doubt that result.

The rows of the "hypercube matrix" L² count corners and edges and faces and ... in n dimensional cubes.

Seminar moved to Week 8, 19 June 2003.

From the point of view of a numerical analyst, I will describe some algorithms for:

• clustering data points based on pairwise similarity,
• reordering a sparse matrix to reduce envelope, two-sum or bandwidth,
• reordering nodes in a range-dependent random graph to reflect the range-dependency,

and point out some connections between seemingly disparate solution techniques. These datamining problems arise across a range of disciplines. I will mention a particularly new and important application from bioinformatics concerning the analysis of gene or protein interaction data.

Immersed interface methods have been developed for a variety of

differential equations on domains containing interfaces or irregular

boundaries. The goal is to use a uniform Cartesian grid (or other fixed

grid on simple domain) and to allow other boundaries or interfaces to

cut through this grid. Special finite difference formulas are developed

at grid points near an interface that incorporate the appropriate jump

conditions across the interface so that uniform second-order accuracy

(or higher) can be obtained. For fluid flow problems with an immersed

deformable elastic membrane, the jump conditions result from a balance

between the singular force imposed by the membrane, inertial forces if

the membrane has mass, and the jump in pressure across the membrane.

A second-order accurate method of this type for Stokes flow was developed

with Zhilin Li and more recently extended to the full incompressible

Navier-Stokes equations in work with Long Lee.