Computational Mathematics and Applications
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Thu, 14/10/2004 14:00 |
Prof Kevin Burrage (University of Queensland / Oxford) |
Computational Mathematics and Applications  |
Comlab |
| A cell is a wonderously complex object. In this talk I will give an overview of some of the mathematical frameworks that are needed in order to make progress to understanding the complex dynamics of a cell. The talk will consist of a directed random walk through discrete Markov processes, stochastic differential equations, anomalous diffusion and fractional differential equations. | |||
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Thu, 21/10/2004 14:00 |
Prof Peter Lax (New York University) |
Computational Mathematics and Applications |
Comlab |
| The computation of flows of compressible fluids will be discussed, exploiting the symmetric form of the equations describing compressible flow. | |||
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Thu, 28/10/2004 14:00 |
Prof Christoph Reisinger (University of Heidelberg / OCIAM) |
Computational Mathematics and Applications |
Comlab |
| Sparse grids yield numerical solutions to PDEs with a significantly reduced number of degrees of freedom. The relative benefit increases with the dimensionality of the problem, which makes multi-factor models for financial derivatives computationally tractable. An outline of a convergence proof for the so called combination technique will be given for a finite difference discretisation of the heat equation, for which sharp error bounds can be shown. Numerical examples demonstrate that by an adaptive (heuristic) choice of the subspaces European and American options with up to thirty (and most likely many more) independent variables can be priced with high accuracy. | |||
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Thu, 04/11/2004 14:00 |
Prof Dwight Barkley (University of Warwick) |
Computational Mathematics and Applications |
Comlab |
| Plane Couette flow - the flow between two infinite parallel plates moving in opposite directions - undergoes a discontinuous transition from laminar flow to turbulence as the Reynolds number is increased. Due to its simplicity, this flow has long served as one of the canonical examples for understanding shear turbulence and the subcritical transition process typical of channel and pipe flows. Only recently was it discovered in very large aspect ratio experiments that this flow also exhibits remarkable pattern formation near transition. Steady, spatially periodic patterns of distinct regions of turbulent and laminar flow emerges spontaneously from uniform turbulence as the Reynolds number is decreased. The length scale of these patterns is more than an order of magnitude larger than the plate separation. It now appears that turbulent-laminar patterns are inevitable intermediate states on the route from turbulent to laminar flow in many shear flows. I will explain how we have overcome the difficulty of simulating these large scale patterns and show results from studies of three types of patterns: periodic, localized, and intermittent. | |||
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Thu, 11/11/2004 14:00 |
Prof Andre Weideman (University of Stellenbosch / Oxford) |
Computational Mathematics and Applications |
Comlab |
| The trapezoidal rule for numerical integration is remarkably accurate when the integrand under consideration is smooth and periodic. In this situation it is superior to more sophisticated methods like Simpson's rule and even the Gauss-Legendre rule. In the first part of the talk we discuss this phenomenon and give a few elementary examples. In the second part of the talk we discuss the application of this idea to the numerical evaluation of contour integrals in the complex plane. Demonstrations involving numerical differentiation, the computation of special functions, and the inversion of the Laplace transform will be presented. | |||
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Thu, 18/11/2004 14:00 |
Prof Angel-Victor de Miguel (London Business School) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
| An interior-point method for solving mathematical programs with equilibrium constraints (MPECs) is proposed. At each iteration of the algorithm, a single primal-dual step is computed from each subproblem of a sequence. Each subproblem is defined as a relaxation of the MPEC with a nonempty strictly feasible region. In contrast to previous approaches, the proposed relaxation scheme preserves the nonempty strict feasibility of each subproblem even in the limit. Local and superlinear convergence of the algorithm is proved even with a less restrictive strict complementarity condition than the standard one. Moreover, mechanisms for inducing global convergence in practice are proposed. Numerical results on the MacMPEC test problem set demonstrate the fast-local convergence properties of the algorithm. | |||
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Thu, 25/11/2004 14:00 |
Prof Julia Yeomans (University of Oxford) |
Computational Mathematics and Applications |
Comlab |
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Thu, 02/12/2004 14:00 |
Prof Michael Hagemann (University of Basel) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
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The use of weighted matchings is becoming increasingly standard in the
solution of sparse linear systems. While non-symmetric permutations based on these
matchings have been the state-of-the-art for
several years (especially for direct solvers), approaches for symmetric
matrices have only recently gained attention.
In this talk we discuss results of our work on using weighted matchings in the preconditioning of symmetric indefinite linear systems, following ideas introduced by Duff and Gilbert. In order to maintain symmetry, the weighted matching is symmetrized and the cycle structure of the resulting matching is used to build reorderings that form small diagonal blocks from the matched entries. For the preconditioning we investigated two approaches. One is an incomplete  preconditioning, that chooses 1x1 or 2x2 diagonal pivots
based on a simple tridiagonal pivoting criterion. The second approach
targets distributed computing, and is based on factorized sparse approximate
inverses, whose existence, in turn, is based on the existence of an
factorization. Results for a number of comprehensive test sets are given,
including comparisons with sparse direct solvers and other preconditioning
approaches. |
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