Computational Mathematics and Applications
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Thu, 29/04/2010 14:00 |
Prof Dominique Orban (Ecole Polytechnique de Montréal and GERAD) |
Computational Mathematics and Applications  |
Rutherford Appleton Laboratory, nr Didcot |
| Interior-point methods for linear and convex quadratic programming require the solution of a sequence of symmetric indefinite linear systems which are used to derive search directions. Safeguards are typically required in order to handle free variables or rank-deficient Jacobians. We propose a consistent framework and accompanying theoretical justification for regularizing these linear systems. Our approach is akin to the proximal method of multipliers and can be interpreted as a simultaneous proximal-point regularization of the primal and dual problems. The regularization is termed "exact" to emphasize that, although the problems are regularized, the algorithm recovers a solution of the original problem. Numerical results will be presented. If time permits we will illustrate current research on a matrix-free implementation. This is joint work with Michael Friedlander, University of British Columbia, Canada | |||
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Thu, 06/05/2010 14:00 |
Prof Roland Herzog (Chemnitz University of Technology) |
Computational Mathematics and Applications |
3WS SR |
| We consider saddle point problems arising as (linearized) optimality conditions in elliptic optimal control problems. The efficient solution of such systems is a core ingredient in second-order optimization algorithms. In the spirit of Bramble and Pasciak, the preconditioned systems are symmetric and positive definite with respect to a suitable scalar product. We extend previous work by Schoeberl and Zulehner and consider problems with control and state constraints. It stands out as a particular feature of this approach that an appropriate symmetric indefinite preconditioner can be constructed from standard preconditioners for those matrices which represent the inner products, such as multigrid cycles. Numerical examples in 2D and 3D are given which illustrate the performance of the method, and limitations and open questions are addressed. | |||
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Thu, 13/05/2010 14:00 |
Dr Francisco Bernal (OCCAM, University of Oxford) |
Computational Mathematics and Applications |
3WS SR |
| Meshless (or meshfree) methods are a relatively new numerical approach for the solution of ordinary- and partial differential equations. They offer the geometrical flexibility of finite elements but without requiring connectivity from the discretization support (ie a mesh). Meshless methods based on the collocation of radial basis functions (RBF methods) are particularly easy to code, and have a number of theoretical advantages as well as practical drawbacks. In this talk, an adaptive RBF scheme is presented for a novel application, namely the solution of (a rather broad class of) delayed- and neutral differential equations. | |||
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Thu, 20/05/2010 14:00 |
Dr Jan Van lent (UWE Bristol) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
| In the eighteenth century Gaspard Monge considered the problem of finding the best way of moving a pile of material from one site to another. This optimal transport problem has many applications such as mesh generation, moving mesh methods, image registration, image morphing, optical design, cartograms, probability theory, etc. The solution to an optimal transport problem can be found by solving the Monge-Ampère equation, a highly nonlinear second order elliptic partial differential equation. Leonid Kantorovich, however, showed that it is possible to analyse optimal transport problems in a framework that naturally leads to a linear programming formulation. In recent years several efficient methods have been proposed for solving the Monge-Ampère equation. For the linear programming problem, standard methods do not exploit the special properties of the solution and require a number of operations that is quadratic or even cubic in the number of points in the discretisation. In this talk I will discuss techniques that can be used to obtain more efficient methods. Joint work with Chris Budd (University of Bath). | |||
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Thu, 27/05/2010 14:00 |
Prof Mahadevan Ganesh (Colorado School of Mines) |
Computational Mathematics and Applications |
3WS SR |
| We discuss a class of high-order spectral-Galerkin surface integral algorithms with specific focus on simulating the scattering of electromagnetic waves by a collection of three dimensional deterministic and stochastic particles. | |||
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Thu, 03/06/2010 14:00 |
Dr Garth Wells (University of Cambridge) |
Computational Mathematics and Applications |
3WS SR |
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Thu, 10/06/2010 14:00 |
Prof Gil Strang (MIT) |
Computational Mathematics and Applications |
3WS SR |
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Thu, 17/06/2010 14:00 |
Prof Joseph Ward (Texas A&M University) |
Computational Mathematics and Applications |
3WS SR |
| This talk will focus on highly localized basis functions which exist for certain kernels and spaces associated with these kernels. Such kernels include certain radial basis functions (RBFs), their restrictions to spheres (SBFs), and their restrictions to more general manifolds embeddable in Rd. The first part of the talk will be of an introductory nature. It will discuss radial basis functions and their restriction to manifolds which give rise to various kernels on these manifolds. The talk will then focus on the development (for certain kernels) of highly localized Lagrange functions which serve as effective bases: i.e., bases which are stable and local. Scaled versions of these bases will then be used to establish the stability of the L2 minimization operator in Lp, 1 ≤ p ≤ ∞, thus obtaining a multivariate analogue of a result of de Boor. Since these bases are scalable with the data, they have potential uses beyond approximation including meshless methods and, more generally, computations of a multiresolution nature. The talk is primarily based on joint work with T. Hangelbroek, F. J. Narcowich and X. Sun. | |||
