Past Seminars
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Mon, 22/09/2003 12:00 |
Dr Tim Dokchister (Durham University) |
Number Theory Seminar  |
L3 |
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Tue, 12/08/2003 15:00 |
Anna de Mier (UPC Barcelona) |
Combinatorial Theory Seminar |
L3 |
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Thu, 07/08/2003 16:00 |
Professor Brian Raines (Baylor University) |
Analytic Topology in Mathematics and Computer Science |
Higman Room |
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Thu, 07/08/2003 15:45 |
Gregory F. Lawler (Cornell University, USA) |
Stochastic Analysis Seminar |
L3 |
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Mon, 28/07/2003 14:15 |
Steve Evans (University of California, Berkeley) |
Stochastic Analysis Seminar |
L3 |
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Thu, 19/06/2003 14:00 |
Dr Austin Mack (University of Technology) |
Computational Mathematics and Applications |
Comlab |
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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. |
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Thu, 19/06/2003 14:00 |
Prof Philippe Toint (University of Namur) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
| 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. | |||
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Thu, 12/06/2003 14:00 |
Prof Gilbert Strang (MIT) |
Computational Mathematics and Applications |
Comlab | ||||||||||||||||||||||
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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. 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. |
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Thu, 05/06/2003 14:00 |
Computational Mathematics and Applications |
Comlab | |
| Seminar moved to Week 8, 19 June 2003. | |||
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Thu, 29/05/2003 14:00 |
Prof Des Higham (University of Strathclyde) |
Computational Mathematics and Applications |
Comlab |
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From the point of view of a numerical analyst, I will describe some algorithms for:
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. |
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Thu, 22/05/2003 14:00 |
Prof Randy LeVeque (University of Washington) |
Computational Mathematics and Applications |
Comlab |
| 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. | |||
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Thu, 15/05/2003 14:00 |
Prof Nancy Nichols (University of Reading) |
Computational Mathematics and Applications |
Comlab |
| 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. | |||
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Thu, 08/05/2003 14:00 |
Dr Shaun Forth (Shrivenham) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
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Thu, 01/05/2003 14:00 |
Dr Danny Ralph (University of Cambridge) |
Computational Mathematics and Applications |
Comlab |
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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. |
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Thu, 13/03/2003 14:00 |
Dr Stefan Scholtes (University of Cambridge) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
| 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. | |||
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Thu, 06/03/2003 14:00 |
Dr Keith Briggs (BTexact Technologies) |
Computational Mathematics and Applications |
Comlab |
| 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. | |||
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Thu, 27/02/2003 14:00 |
Prof Jeremy Astley (University of Southampton) |
Computational Mathematics and Applications |
Comlab |
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Thu, 20/02/2003 14:00 |
Prof Jean-Paul Berrut (University of Fribourg) |
Computational Mathematics and Applications |
Comlab |
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The pseudospectral method for solving boundary value problems on the interval
consists in replacing the solution by an interpolating polynomial in Lagrangian
form between well-chosen points and collocating at those same points.
Due to its globality, the method cannot handle steep gradients well (Markov's inequality). We will present and discuss two means of improving upon this: the attachment of poles to the ansatz polynomial, on one hand, and conformal point shifts on the other hand, both optimally adapted to the problem to be solved. |
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Thu, 13/02/2003 14:00 |
Dr Tony Garratt (Cambridge) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
| Dynamic optimisation is a tool that enables the process industries to compute optimal control strategies for important chemical processes. Aspen DynamicsTM is a well-established commercial engineering software package containing a dynamic optimisation tool. Its intuitive graphical user interface and library of robust dynamic models enables engineers to quickly and easily define a dynamic optimisation problem including objectives, control vector parameterisations and constraints. However, this is only one part of the story. The combination of dynamics and non-linear optimisation can create a problem that can be very difficult to solve due to a number of reasons, including non-linearities, poor initial guesses, discontinuities and accuracy and speed of dynamic integration. In this talk I will begin with an introduction to process modelling and outline the algorithms and techniques used in dynamic optimisation. I will move on to discuss the numerical issues that can give us so much trouble in practice and outline some solutions we have created to overcome some of them. | |||
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Thu, 06/02/2003 14:00 |
Prof Nick Trefethen (University of Oxford) |
Computational Mathematics and Applications |
Comlab |
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Many questions of interest to both mathematicians and physicists relate
to the behavior of eigenvalues and eigenmodes of the Laplace operator
on a polygon. Algorithmic improvements have revived the old "method
of fundamental solutions" associated with Fox, Henrici and Moler; is it
going to end up competitive with the state-of-the-art method of Descloux,
Tolley and Driscoll? This talk will outline the numerical issues but
give equal attention to applications including "can you hear the shape
of a drum?", localization of eigenmodes, eigenvalue avoidance, and the
design of drums that play chords.
This is very much work in progress – with graduate student Timo Betcke. |
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