# Past Computational Mathematics and Applications Seminar

16 October 2003
14:00
Prof Andrew Stuart
Abstract
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. \\ \\ Joint work with Raz Kupferman (Jerusalem) and Petter Wiberg (Warwick)
• Computational Mathematics and Applications Seminar
19 June 2003
14:00
Dr Austin Mack
Abstract
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. \\ \\ 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. \\ \\ 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.
• Computational Mathematics and Applications Seminar
Prof Philippe Toint
Abstract
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.
• Computational Mathematics and Applications Seminar
12 June 2003
14:00
Prof Gilbert Strang
Abstract

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:

 [ 1 0 0 0 ] [ 1 1 1 1 ] [ 1 1 1 1] L = [ 1 1 0 0 ] L' = [ 0 1 2 3 ] S = [ 1 2 3 4] [ 1 2 1 0 ] [ 0 0 1 3 ] [ 1 3 6 10] [ 1 3 3 1 ] [ 0 0 0 1 ] [ 1 4 10 20]

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.

• Computational Mathematics and Applications Seminar
5 June 2003
14:00
Abstract
Seminar moved to Week 8, 19 June 2003.
• Computational Mathematics and Applications Seminar
29 May 2003
14:00
Prof Des Higham
Abstract

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.

• Computational Mathematics and Applications Seminar
22 May 2003
14:00
Prof Randy LeVeque
Abstract
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.
• Computational Mathematics and Applications Seminar
15 May 2003
14:00
Prof Nancy Nichols
Abstract
Feedback design for a second order control system leads to an eigenstructure assignment problem for a quadratic matrix polynomial. It is desirable that the feedback controller not only assigns specified eigenvalues to the second order closed loop system, but also that the system is robust, or insensitive to perturbations. We derive here new sensitivity measures, or condition numbers, for the eigenvalues of the quadratic matrix polynomial and define a measure of robustness of the corresponding system. We then show that the robustness of the quadratic inverse eigenvalue problem can be achieved by solving a generalized linear eigenvalue assignment problem subject to structured perturbations. Numerically reliable methods for solving the structured generalized linear problem are developed that take advantage of the special properties of the system in order to minimize the computational work required.
• Computational Mathematics and Applications Seminar
1 May 2003
14:00
Dr Danny Ralph
Abstract
Electricity markets facilitate pricing and delivery of wholesale power. Generators submit bids to an Independent System Operator (ISO) to indicate how much power they can produce depending on price. The ISO takes these bids with demand forecasts and minimizes the total cost of power production subject to feasibility of distribution in the electrical network. \\ \\ Each generator can optimise its bid using a bilevel program or mathematical program with equilibrium (or complementarity) constraints, by taking the ISOs problem, which contains all generators bid information, at the lower level. This leads immediately to a game between generators, where a Nash equilibrium - at which each generator's bid maximises its profit provided that none of the other generators changes its bid - is sought. \\ \\ In particular, we examine the idealised model of Berry et al (Utility Policy 8, 1999), which gives a bilevel game that can be modelled as an "equilibrium problem with complementarity constraints" or EPCC. Unfortunately, like bilevel games, EPCCs on networks may not have Nash equilibria in the (common) case when one or more of links of the network is saturated (at maximum capacity). Nevertheless we explore some theory and algorithms for this problem, and discuss the economic implications of numerical examples where equilibria are found for small electricity networks.
• Computational Mathematics and Applications Seminar