Computational Mathematics and Applications (past)
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Thu, 22/02/2001 14:00 |
Dr Oliver Ernst (Bergakademie Freiberg) |
Computational Mathematics and Applications  |
Comlab |
| 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. | |||
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Thu, 15/02/2001 14:00 |
Dr David Griffiths (University of Dundee) |
Computational Mathematics and Applications |
Comlab |
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Thu, 08/02/2001 14:00 |
Dr Colin Campbell (University of Bristol) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
| 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. | |||
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Thu, 01/02/2001 14:00 |
Prof James Binney (University of Oxford) |
Computational Mathematics and Applications |
Comlab |
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Thu, 25/01/2001 14:00 |
Prof K W Morton (University of Bath and University of Oxford) |
Computational Mathematics and Applications |
Comlab |
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Thu, 18/01/2001 14:00 |
Prof Francisco Marques (University Politecnica de Catalunya) |
Computational Mathematics and Applications |
Comlab |
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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 <  < 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 <1.6 and > 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. |
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