# Past 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
24 January 2003
14:00
Prof Tony Chan
Abstract

Image processing is an area with many important applications, as well as challenging problems for mathematicians. In particular, Fourier/wavelets analysis and stochastic/statistical methods have had major impact in this area. Recently, there has been increased interest in a new and complementary approach, using partial differential equations (PDEs) and differential-geometric models. It offers a more systematic treatment of geometric features of mages, such as shapes, contours and curvatures, etc., as well as allowing the wealth of techniques developed for PDEs and Computational Fluid Dynamics (CFD) to be brought to bear on image processing tasks.

I'll use two examples from my recent work to illustrate this synergy:

1. A unified image restoration model using Total Variation (TV) which can be used to model denoising, deblurring, as well as image inpainting (e.g. restoring old scratched photos). The TV idea can be traced to shock capturing methods in CFD and was first used in image processing by Rudin, Osher and Fatemi.

2. An "active contour" model which uses a variational level set method for object detection in scalar and vector-valued images. It can detect objects not necessarily defined by sharp edges, as well as objects undetectable in each channel of a vector-valued image or in the combined intensity. The contour can go through topological changes, and the model is robust to noise. The level set method was originally developed by Osher and Sethian for tracking interfaces in CFD.

(The above are joint works with Jackie Shen at the Univ. of Minnesota and Luminita Vese in the Math Dept at UCLA.)

• Computational Mathematics and Applications Seminar
5 December 2002
14:00
to
17:30
Various speakers
Abstract
2.00 pm Professor Iain Duff (RAL) Opening remarks
2.15 pm Professor M J D Powell (University of Cambridge)
Some developments of work with Alan on cubic splines
3.00 pm Professor Kevin Burrage (University of Queensland)
Stochastic models and simulations for chemically reacting systems
3.30 pm Tea/Coffee
4.00 pm Professor John Reid (RAL)
Sparse matrix research at Harwell and the Rutherford Appleton Laboratory
4.30 pm Dr Ian Jones (AEA PLC)
Computational fluid dynamics and the role of stiff solvers
5.00 pm Dr Lawrence Daniels (Hyprotech UK Ltd)
Current work with Alan on ODE solvers for HSL
• Computational Mathematics and Applications Seminar
28 November 2002
14:00
Dr Coralia Cartis
Abstract
Long-step primal-dual path-following algorithms constitute the framework of practical interior point methods for solving linear programming problems. We consider such an algorithm and a second order variant of it. We address the problem of the convergence of the sequences of iterates generated by the two algorithms to the analytic centre of the optimal primal-dual set.
• Computational Mathematics and Applications Seminar
21 November 2002
14:00
Abstract
Several real Lie and Jordan algebras, along with their associated automorphism groups, can be elegantly expressed in the quaternion tensor algebra. The resulting insight into structured matrices leads to a class of simple Jacobi algorithms for the corresponding $n \times n$ structured eigenproblems. These algorithms have many desirable properties, including parallelizability, ease of implementation, and strong stability.
• Computational Mathematics and Applications Seminar
Dr Andrew Cliffe
Abstract
A method for computing periodic orbits for the Navier-Stokes equations will be presented. The method uses a finite-element Galerkin discretisation for the spatial part of the problem and a spectral Galerkin method for the temporal part of the problem. The method will be illustrated by calculations of the periodic flow behind a circular cylinder in a channel. The problem has a simple reflectional symmetry and it will be explained how this can be exploited to reduce the cost of the computations.
• Computational Mathematics and Applications Seminar