Computational Mathematics and Applications
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Thu, 12/10/2006 14:00 |
Prof Thomas Sonar (TU Braunschweig) |
Computational Mathematics and Applications  |
Comlab |
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One of the oldest approach in meshless methods for PDEs is the Interpolating Moving Least Squares (IMLS) technique developed in the 1980s. Although widely accepted by users working in fields as diverse as geoinformatics and crack dynamics I shall take a fresh look at this method and ask for the equivalent difference operators which are generated implicitly. As it turns out, these operators are optimal only in trivial cases and are "strange" in general. I shall try to exploit two different approaches for the computation of these operators. On the other hand (and very different from IMLS), Total Variation Flow (TVF) PDEs are the most recent developments in image processing and have received much attention lately. Again I shall show that they are able to generate "strange" discrete operators and that they easily can behave badly although they may be properly implemented. |
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Thu, 19/10/2006 14:00 |
Prof Nick Higham (University of Manchester) |
Computational Mathematics and Applications |
Comlab |
The aim of this talk is to give some understanding of the theory of matrix  'th roots (solutions to the nonlinear matrix equation  ), to explain how and how not to compute roots, and to describe some applications. In particular, an application in finance will be described concerning roots of transition matrices from Markov models. |
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Thu, 26/10/2006 14:00 |
Dr Anne Trefethen (OeRC) |
Computational Mathematics and Applications |
Comlab |
| High-performance computing is an important tool for computational science. Oxford University has recently decided to invest £3M in a new supercomputing facility which is under development now. In this seminar I will give an overview of some activities in Oxford and provide a vision for supercomputing here. I will discuss some of the numerical analysis software and tools, such as Distributed Matlab and indicate some of the challenges at the intersection of numerical analysis and high-performance computing. | |||
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Thu, 02/11/2006 14:00 |
Mr Sheehan Olver (University of Cambridge) |
Computational Mathematics and Applications |
Comlab |
| The aim of this talk is to describe several methods for numerically approximating the integral of a multivariate highly oscillatory function. We begin with a review of the asymptotic and Filon-type methods developed by Iserles and Nørsett. Using a method developed by Levin as a point of departure we will construct a new method that uses the same information as the Filon-type method, and obtains the same asymptotic order, while not requiring moments. This allows us to integrate over nonsimplicial domains, and with complicated oscillators. | |||
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Thu, 09/11/2006 14:00 |
Dr Hou-Dou Qi (University of Southampton) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
| The talk starts with a general introduction of the convex quadratic semidefinite programming problem (QSDP), followed by a number of interesting examples arising from finance, statistics and computer sciences. We then discuss the concept of primal nondegeneracy for QSDP and show that some QSDPs are nondegenerate and others are not even under the linear independence assumption. The talk then moves on to the algorithmic side by introducing the dual approach and how it naturally leads to Newton's method, which is quadratically convergent for degenerate problems. On the implementation side of the Newton method, we stress that direct methods for the linear equations in Newton's method are impossible simply because the equations are quite large in size and dense. Our numerical experiments use the conjugate gradient method, which works quite well for the nearest correlation matrix problem. We also discuss difficulties for us to find appropriate preconditioners for the linear system encountered. The talk concludes in discussing some other available methods and some future topics. | |||
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Thu, 16/11/2006 14:00 |
Prof Tim Phillips (University of Cambridge) |
Computational Mathematics and Applications |
Comlab |
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Thu, 23/11/2006 14:00 |
Prof Philippe Toint (University of Namur) |
Computational Mathematics and Applications |
Comlab |
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Many large-scale optimization problems arise in the context of the discretization of infinite dimensional applications. In such cases, the description of the finite-dimensional problem is not unique, but depends on the discretization used, resulting in a natural multi-level description. How can such a problem structure be exploited, in discretized problems or more generally? The talk will focus on discussing this issue in the context of unconstrained optimization and in relation with the classical multigrid approach to elliptic systems of partial differential equations. Both theoretical convergence properties of special purpose algorithms and their numerical performances will be discussed. Perspectives will also be given. Collaboration with S. Gratton, A. Sartenaer and M. Weber. |
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Thu, 30/11/2006 14:00 |
Dr Martin Van Gijzen (Delft University of Technology) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
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Joint work with Yogi Erlangga and Kees Vuik. Shifted Laplace preconditioners have attracted considerable attention as a technique to speed up convergence of iterative solution methods for the Helmholtz equation. In this paper we present a comprehensive spectral analysis of the Helmholtz operator preconditioned with a shifted Laplacian. Our analysis is valid under general conditions. The propagating medium can be heterogeneous, and the analysis also holds for different types of damping, including a radiation condition for the boundary of the computational domain. By combining the results of the spectral analysis of the preconditioned Helmholtz operator with an upper bound on the GMRES-residual norm we are able to provide an optimal value for the shift, and to explain the mesh-depency of the convergence of GMRES preconditioned with a shifted Laplacian. We illustrate our results with a seismic test problem. |
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