Past Seminars
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Thu, 30/01/2003 14:00 |
Prof Mike Giles (University of Oxford) |
Computational Mathematics and Applications  |
Comlab |
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Fri, 24/01/2003 14:00 |
Prof Tony Chan (UCLA) |
Computational Mathematics and Applications |
Comlab |
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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.) |
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Thu, 23/01/2003 14:00 |
Prof Toby Driscoll (University of Delaware) |
Computational Mathematics and Applications |
Comlab |
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Thu, 05/12/2002 14:00 |
Various speakers |
Computational Mathematics and Applications |
Comlab |
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Thu, 28/11/2002 14:00 |
Dr Coralia Cartis (University of Cambridge) |
Computational Mathematics and Applications |
Comlab |
| 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. | |||
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Thu, 21/11/2002 14:00 |
Prof Niloufer Mackey (U.W. Michigan & University of Manchester) |
Computational Mathematics and Applications |
Comlab |
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  structured
eigenproblems. These algorithms have many desirable properties, including
parallelizability, ease of implementation, and strong stability. |
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Thu, 14/11/2002 14:00 |
Dr Andrew Cliffe (SERCO) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
| 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. | |||
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Thu, 07/11/2002 14:00 |
Dr John Mackenzie (University of Strathclyde) |
Computational Mathematics and Applications |
Comlab |
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Thu, 31/10/2002 14:00 |
Dr Arno Kuijlaars (Catholic University of Leuven) |
Computational Mathematics and Applications |
Comlab |
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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. |
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Thu, 24/10/2002 14:00 |
Prof Endre Süli (University of Oxford) |
Computational Mathematics and Applications |
Comlab |
| 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. | |||
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Thu, 17/10/2002 14:00 |
Prof Nick Higham (University of Manchester) |
Computational Mathematics and Applications |
Comlab |
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The study of the finite precision behaviour of numerical algorithms dates back at least as far as Turing and Wilkinson in the 1940s. At the start of the 21st century, this area of research is still very active. We focus on some topics of current interest, describing recent developments and trends and pointing out future research directions. The talk will be accessible to those who are not specialists in numerical analysis. Specific topics intended to be addressed include
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Thu, 10/10/2002 14:00 |
Prof Beresford Parlett (UC Berkeley) |
Computational Mathematics and Applications |
Comlab |
| We describe "avoidance of crossing" and its explanation by von Neumann and Wigner. We show Lax's criterion for degeneracy and then discover matrices whose determinants give the discriminant of the given matrix. This yields a simple proof of the bound given by Ilyushechkin on the number of terms in the expansion of the discriminant as a sum of squares. We discuss the 3 x 3 case in detail. | |||
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Thu, 13/06/2002 14:00 |
Prof Arne S. Drud (ARKI Consulting and Development) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
| The talk will discuss unsymmetric sparse LU factorization based on the Markowitz pivot selection criterium. The key question for the author is the following: Is it possible to implement a sparse factorization where the overhead is limited to a constant times the actual numerical work? In other words, can the work be bounded by o(sum(k, M(k)), where M(k) is the Markowitz count in pivot k. The answer is probably NO, but how close can we get? We will give several bad examples for traditional methods and suggest alternative methods / data structure both for pivot selection and for the sparse update operations. | |||
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Thu, 06/06/2002 14:00 |
Prof Gilbert Strang and Per-Olof Persson (MIT) |
Computational Mathematics and Applications |
Comlab |
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We discuss two filters that are frequently used to smooth data.
One is the (nonlinear) median filter, that chooses the median
of the sample values in the sliding window. This deals effectively
with "outliers" that are beyond the correct sample range, and will
never be chosen as the median. A straightforward implementation of
the filter is expensive for large windows, particularly in two dimensions
(for images).
The second filter is linear, and known as "Savitzky-Golay". It is frequently used in spectroscopy, to locate positions and peaks and widths of spectral lines. This filter is based on a least-squares fit of the samples in the sliding window to a polynomial of relatively low degree. The filter coefficients are unlike the equiripple filter that is optimal in the maximum norm, and the "maxflat" filters that are central in wavelet constructions. Should they be better known....? We will discuss the analysis and the implementation of both filters. |
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Thu, 30/05/2002 14:00 |
Dr David Gavaghan (University of Oxford) |
Computational Mathematics and Applications |
Comlab |
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Thu, 23/05/2002 14:00 |
Dr David Mayers (University of Oxford) |
Computational Mathematics and Applications |
Comlab |
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The asymptotic rate of convergence of the trapezium rule is
defined, for smooth functions, by the Euler-Maclaurin expansion.
The extension to other methods, such as Gauss rules, is straightforward;
this talk begins with some special cases, such as Periodic functions, and
functions with various singularities.
Convergence rates for ODEs (Initial and Boundary value problems) and for PDEs are available, but not so well known. Extension to singular problems seems to require methods specific to each situation. Some of the results are unexpected - to me, anyway. |
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Thu, 16/05/2002 14:00 |
Dr Victor Pereyra (Weidlinger Associates) |
Computational Mathematics and Applications |
Comlab |
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In the past few years we have developed some expertise in solving optimization
problems that involve large scale simulations in various areas of Computational
Geophysics and Engineering. We will discuss some of those applications here,
namely: inversion of seismic data, characterization of piezoelectrical crystals
material properties, optimal design of piezoelectrical transducers and
opto-electronic devices, and the optimal design of steel structures.
A common theme among these different applications is that the goal functional is very expensive to evaluate, often, no derivatives are readily available, and some times the dimensionality can be large. Thus parallelism is a need, and when no derivatives are present, search type methods have to be used for the optimization part. Additional difficulties can be ill-conditioning and non-convexity, that leads to issues of global optimization. Another area that has not been extensively explored in numerical optimization and that is important in real applications is that of multiobjective optimization. As a result of these varied experiences we are currently designing a toolbox to facilitate the rapid deployment of these techniques to other areas of application with a minimum of retooling. |
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Thu, 09/05/2002 14:00 |
Prof Valeria Simoncini (University of Bologna) |
Computational Mathematics and Applications |
Comlab |
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Thu, 02/05/2002 14:00 |
Prof Tim Barth (NASA Ames) |
Computational Mathematics and Applications |
Comlab |
A-Posteriori Error estimates for high order Godunov finite
volume methods are presented which exploit the two solution
representations inherent in the method, viz. as piecewise
constants  and cell-wise  -th order reconstructed
functions  . The analysis provided here applies
directly to schemes such as MUSCL, TVD, UNO, ENO, WENO or any
other scheme that is a faithful extension of Godunov's method
to high order accuracy in a sense that will be made precise.
Using standard duality arguments, we construct exact error
representation formulas for derived functionals that are
tailored to the class of high order Godunov finite volume
methods with data reconstruction, . We then consider
computable error estimates that exploit the structure of higher
order Godunov finite volume methods. The analysis technique used
in this work exploits a certain relationship between higher
order Godunov methods and the discontinuous Galerkin method.
Issues such as the treatment of nonlinearity and the optional
post-processing of numerical dual data are also discussed.
Numerical results for linear and nonlinear scalar conservation
laws are presented to verify the analysis. Complete details can
be found in a paper appearing in the proceedings of FVCA3,
Porquerolles, France, June 24-28, 2002. |
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Thu, 25/04/2002 14:00 |
Dr Stefano Salvini (NAG Ltd.) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
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SMP (Symmetric Multi-Processors) hardware technologies are very popular
with vendors and end-users alike for a number of reasons. However, true
shared memory parallelism has experienced somewhat slower to take up
amongst the scientific-programming community. NAG has been at the
forefront of SMP technology for a number of years, and the NAG SMP
Library has shown the potential of SMP systems.
At the very high end, SMP hardware technologies are used as building blocks of modern supercomputers, which truly consist of clusters of SMP systems, for which no dedicated model of parallelism yet exists. The aim of this talk is to introduce SMP systems and their potential. Results from our work at NAG will also be introduced to show how SMP parallelism, based on a shared memory paradigm, can be used to very good effect and can produce high performance, scalable software. The talk also aims to discuss some aspects of the apparent slow take up of shared memory parallelism and the potential competition from PC (i.e. Intel)-based cluster technology. The talk then aims to explore the potential of SMP technology within "hybrid parallelism", i.e. mixed distributed and shared memory modes, illustrating the point with some preliminary work carried out by the author and others. Finally, a number of potential future challenges to numerical analysts will be discussed. The talk is aimed at all who are interested in SMP technologies for numerical computing, irrespective of any previous experience in the field. The talk aims to stimulate discussion, by presenting some ideas, backing these with data, not to stifle it in an ocean of detail! |
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