# Past 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
31 October 2002
14:00
Dr Arno Kuijlaars
Abstract
The convergence of Krylov subspace methods like conjugate gradients depends on the eigenvalues of the underlying matrix. In many cases the exact location of the eigenvalues is unknown, but one has some information about the distribution of eigenvalues in an asymptotic sense. This could be the case for linear systems arising from a discretization of a PDE. The asymptotic behavior then takes place when the meshsize tends to zero. \\ \\ We discuss two possible approaches to study the convergence of conjugate gradients based on such information. The first approach is based on a straightforward idea to estimate the condition number. This method is illustrated by means of a comparison of preconditioning techniques. The second approach takes into account the full asymptotic spectrum. It gives a bound on the asymptotic convergence factor which explains the superlinear convergence observed in many situations. This method is mathematically more involved since it deals with potential theory. I will explain the basic ideas.
• Computational Mathematics and Applications Seminar
24 October 2002
14:00
Prof Endre Süli
Abstract
We develop an algorithm for estimating the local Sobolev regularity index of a given function by monitoring the decay rate of its Legendre expansion coefficients. On the basis of these local regularities, we design and implement an hp--adaptive finite element method based on employing discontinuous piecewise polynomials, for the approximation of nonlinear systems of hyperbolic conservation laws. The performance of the proposed adaptive strategy is demonstrated numerically.
• Computational Mathematics and Applications Seminar