Past Seminars

Thu, 07/06/2001
14:00 		Prof Mike J D Powell (University of Cambridge) Computational Mathematics and Applications Add to calendar Rutherford Appleton Laboratory, nr Didcot
Some properties of thin plate spline interpolation
Let the thin plate spline radial basis function method be applied to interpolate values of a smooth function $ f(x) $, $ x \!\in\! {\cal R}^d $. It is known that, if the data are the values $ f(jh) $, $ j \in {\cal Z}^d $, where $ h $ is the spacing between data points and $ {\cal Z}^d $ is the set of points in $ d $ dimensions with integer coordinates, then the accuracy of the interpolant is of magnitude $ h^{d+2} $. This beautiful result, due to Buhmann, will be explained briefly. We will also survey some recent findings of Bejancu on Lagrange functions in two dimensions when interpolating at the integer points of the half-plane $ {\cal Z}^2 \cap \{ x : x_2 \!\geq\! 0 \} $. Most of our attention, however, will be given to the current research of the author on interpolation in one dimension at the points $ h {\cal Z} \cap [0,1] $, the purpose of the work being to establish theoretically the apparent deterioration in accuracy at the ends of the range from $ {\cal O} ( h^3 ) $ to $ {\cal O} ( h^{3/2} ) $ that has been observed in practice. The analysis includes a study of the Lagrange functions of the semi-infinite grid $ {\cal Z} \cap \{ x : x \!\geq\! 0 \} $ in one dimension.
Thu, 24/05/2001
14:00 		Dr Helen Byrne (University of Nottingham) Computational Mathematics and Applications Add to calendar Comlab
Solid tumour growth: modelling and simulations
Thu, 17/05/2001
14:00 		Dr Lawrence Daniels and Dr Iain Strachan (Hyprotech) Computational Mathematics and Applications Add to calendar Rutherford Appleton Laboratory, nr Didcot
On the robust solution of process simulation problems
In this talk we review experiences of using the Harwell Subroutine Library and other numerical software codes in implementing large scale solvers for commercial industrial process simulation packages. Such packages are required to solve problems in an efficient and robust manner. A core requirement is the solution of sparse systems of linear equations; various HSL routines have been used and are compared. Additionally, the requirement for fast small dense matrix solvers is examined.
Thu, 10/05/2001
14:00 		Prof Maciej Zworksi (UC Berkeley) Computational Mathematics and Applications Add to calendar Comlab
Pseudospectra and solvability of PDEs
Thu, 03/05/2001
14:00 		Dr Remy Abgrall (University of Bordeaux) Computational Mathematics and Applications Add to calendar Comlab
Upwind residual distributive schemes for compressible flows
Thu, 15/03/2001
14:00 		Prof Ian Sloan (University of New South Wales) Computational Mathematics and Applications Add to calendar Comlab
Scientific computing for problems on the sphere - applying good approximations on the sphere to geodesy and the scattering of sound
The sphere is an important setting for applied mathematics, yet the underlying approximation theory and numerical analysis needed for serious applications (such as, for example, global weather models) is much less developed than, for example, for the cube.

This lecture will apply recent developments in approximation theory on the sphere to two different problems in scientific computing.

First, in geodesy there is often the need to evaluate integrals using data selected from the vast amount collected by orbiting satellites. Sometimes the need is for quadrature rules that integrate exactly all spherical polynomials up to a specified degree $ n $ (or equivalently, that integrate exactly all spherical harmonies $ Y_{\ell ,k}(\theta ,\phi) $ with $ \ell \le n). $ We shall demonstrate (using results of M. Reimer, I. Sloan and R. Womersley in collaboration with W. Freeden) that excellent quadrature rules of this kind can be obtained from recent results on polynomial interpolation on the sphere, if the interpolation points (and thus the quadrature points) are chosen to be points of a so-called extremal fundamental system.

The second application is to the scattering of sound by smooth three-dimensional objects, and to the inverse problem of finding the shape of a scattering object by observing the pattern of the scattered sound waves. For these problems a methodology has been developed, in joint work with I.G. Graham, M. Ganesh and R. Womersley, by applying recent results on constructive polynomial approximation on the sphere. (The scattering object is treated as a deformed sphere.)
Thu, 08/03/2001
14:00 		Prof Wilhelm Heinrichs (University of Essen) Computational Mathematics and Applications Add to calendar Comlab
Spectral multigrid methods for the Navier-Stokes equations
Thu, 01/03/2001
14:00 		Prof Mark Stadtherr (University of Notre Dame) Computational Mathematics and Applications Add to calendar Rutherford Appleton Laboratory, nr Didcot
Reliable process modelling and optimisation using interval analysis
Continuing advances in computing technology provide the power not only to solve increasingly large and complex process modeling and optimization problems, but also to address issues concerning the reliability with which such problems can be solved. For example, in solving process optimization problems, a persistent issue concerning reliability is whether or not a global, as opposed to local, optimum has been achieved. In modeling problems, especially with the use of complex nonlinear models, the issue of whether a solution is unique is of concern, and if no solution is found numerically, of whether there actually exists a solution to the posed problem. This presentation focuses on an approach, based on interval mathematics, that is capable of dealing with these issues, and which can provide mathematical and computational guarantees of reliability. That is, the technique is guaranteed to find all solutions to nonlinear equation solving problems and to find the global optimum in nonlinear optimization problems. The methodology is demonstrated using several examples, drawn primarily from the modeling of phase behavior, the estimation of parameters in models, and the modeling, using lattice density-functional theory, of phase transitions in nanoporous materials.
Thu, 22/02/2001
14:00 		Dr Oliver Ernst (Bergakademie Freiberg) Computational Mathematics and Applications Add to calendar Comlab
Acceleration strategies for restarted minimum residual methods
This talk reviews some recent joint work with Michael Eiermann and Olaf Schneider which introduced a framework for analyzing some popular techniques for accelerating restarted Krylov subspace methods for solving linear systems of equations. Such techniques attempt to compensate for the loss of information due to restarting methods like GMRES, the memory demands of which are usually too high for it to be applied to large problems in unmodified form. We summarize the basic strategies which have been proposed and present both theoretical and numerical comparisons.
Thu, 15/02/2001
14:00 		Dr David Griffiths (University of Dundee) Computational Mathematics and Applications Add to calendar Comlab
Mixed finite element methods, stability and related issues
Thu, 08/02/2001
14:00 		Dr Colin Campbell (University of Bristol) Computational Mathematics and Applications Add to calendar Rutherford Appleton Laboratory, nr Didcot
Support Vector machines and related kernel methods
Support Vector Machines are a new and very promising approach to machine learning. They can be applied to a wide range of tasks such as classification, regression, novelty detection, density estimation, etc. The approach is motivated by statistical learning theory and the algorithms have performed well in practice on important applications such as handwritten character recognition (where they currently give state-of-the-art performance), bioinformatics and machine vision. The learning task typically involves optimisation theory (linear, quadratic and general nonlinear programming, depending on the algorithm used). In fact, the approach has stimulated new questions in optimisation theory, principally concerned with the issue of how to handle problems with a large numbers of variables. In the first part of the talk I will overview this subject, in the second part I will describe some of the speaker's contributions to this subject (principally, novelty detection, query learning and new algorithms) and in the third part I will outline future directions and new questions stimulated by this research.
Thu, 01/02/2001
14:00 		Prof James Binney (University of Oxford) Computational Mathematics and Applications Add to calendar Comlab
Simulating galaxy formation
Thu, 25/01/2001
14:00 		Prof K W Morton (University of Bath and University of Oxford) Computational Mathematics and Applications Add to calendar Comlab
The system wave equation - a generic hyperbolic problem?
Thu, 18/01/2001
14:00 		Prof Francisco Marques (University Politecnica de Catalunya) Computational Mathematics and Applications Add to calendar Comlab
Instabilities, symmetry breaking and mode interactions in an enclosed swirling flow
The flow in a cylinder with a rotating endwall has continued to attract much attention since Vogel (1968) first observed the vortex breakdown of the central core vortex that forms. Recent experiments have observed a multiplicity of unsteady states that coexist over a range of the governing parameters. In spite of numerous numerical and experimental studies, there continues to be considerable controversy with fundamental aspects of this flow, particularly with regards to symmetry breaking. Also, it is not well understood where these oscillatory states originate from, how they are interrelated, nor how they are related to the steady, axisymmetric basic state.

In the aspect ratio (height/radius) range 1.6 < $ \Lambda $ < 2.8, the primary bifurcation is to an axisymmetric time-periodic flow (a limit cycle). We have developed a suite of numerical techniques, exploiting the biharmonic formulation of the problem in the axisymmetric case, that allows us to compute the nonlinear time evolution, the basic state, and its linear stability in a consistent and efficient manner. We show that the basic state undergoes a succession of Hopf bifurcations and the corresponding eigenvalues and eigenvectors of these excited modes describe most of the characteristics of the observed time-dependent states.

The primary bifurcation is non-axisymmetric, to pure rotating wave, in the ranges $ \Lambda $ <1.6 and $ \Lambda $ > 2.8. An efficient and accurate numerical scheme is presented for the three-dimensional Navier-Stokes equations in primitive variables in a cylinder. Using these code, primary and secondary bifurcations breaking the SO(2) symmetry are analyzed.

We have located a double Hopf bifurcation, where an axisymmetric limit cycle and a rotating wave bifurcate simultaneously. This codimension-2 bifurcation is very rich, allowing for several different scenarios. By a comprehensive two-parameter exploration about this point we have identified precisely to which scenario this case corresponds. The mode interaction generates an unstable two-torus modulate rotating wave solution and gives a wedge-shaped region in parameter space where the two periodic solutions are both stable.

For aspect ratios around three, experimental observations suggest that the first mode of instability is a precession of the central vortex core, whereas recent linear stability analysis suggest a Hopf bifurcation to a rotating wave at lower rotation rates. This apparent discrepancy is resolved with the aid of the 3D Navier-Stokes solver. The primary bifurcation to an m=4 traveling wave, detected by the linear stability analysis, is located away from the axis, and a secondary bifurcation to a modulated rotating wave with dominant modes m=1 and 4, is seen mainly on the axis as a precessing vortex breakdown bubble. Experiments and the linear stability analysis detected different aspects of the same flow, that take place in different spatial locations.

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

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