Past Forthcoming Seminars

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
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
Dr Stefan Scholtes
Abstract
Traditional optimisation theory and -methods on the basis of the Lagrangian function do not apply to objective or constraint functions which are defined by means of a combinatorial selection structure. Such selection structures can be explicit, for example in the case of "min", "max" or "if" statements in function evaluations, or implicit as in the case of inverse optimisation problems where the combinatorial structure is induced by the possible selections of active constraints. The resulting optimisation problems are typically neither convex nor smooth and do not fit into the standard framework of nonlinear optimisation. Users typically treat these problems either through a mixed-integer reformulation, which drastically reduces the size of tractable problems, or by employing nonsmooth optimisation methods, such as bundle methods, which are typically based on convex models and therefore only allow for weak convergence results. In this talk we argue that the classical Lagrangian theory and SQP methodology can be extended to a fairly general class of nonlinear programs with combinatorial constraints. The paper is available at http://www.eng.cam.ac.uk/~ss248/publications.
  • Computational Mathematics and Applications Seminar
6 March 2003
14:00
Dr Keith Briggs
Abstract
Is it possible to construct a computational model of the real numbers in which the sign of every computed result is corrected determined? The answer is yes, both in theory and in practice. The resulting viewpoint contrasts strongly with the traditional floating point model. I will review the theoretical background and software design issues, discuss previous attempts at implementation and finally demonstrate my own python and C++ codes.
  • Computational Mathematics and Applications Seminar

Pages