# Past 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
20 February 2003
14:00
Prof Jean-Paul Berrut
Abstract
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.
• Computational Mathematics and Applications Seminar
Dr Tony Garratt
Abstract
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.
• Computational Mathematics and Applications Seminar
6 February 2003
14:00
Prof Nick Trefethen
Abstract
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.
• Computational Mathematics and Applications Seminar