Computational Mathematics and Applications
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Thu, 11/10/2007 14:00 |
Dr Omar Lakkis (University of Sussex) |
Computational Mathematics and Applications  |
Comlab |
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I will address the usage of the elliptic reconstruction technique (ERT) in a posteriori error analysis for fully discrete schemes for parabolic partial differential equations. A posteriori error estimates are effective tools in error control and adaptivity and a mathematical rigorous derivation justifies and improves their use in practical implementations. The flexibility of the ERT allows a virtually indiscriminate use of various parabolic PDE techniques such as energy methods, duality methods and heat-kernel estimates, as opposed to direct approaches which leave less maneuver room. Thanks to ERT parabolic stability techniques can be combined with different elliptic a posteriori error analysis techniques, such as residual or recovery estimators, to derive a posteriori error bounds. The method has the merit of unifying previously known approaches, as well as providing new ones and providing us with novel error bounds (e.g., pointwise norm error bounds for the heat equation). [These results are based on joint work with Ch. Makridakis and A. Demlow.] Another feature, which I would like to highlight, of the ERT is its simplifying power. It allows us to derive estimates where the analysis would be very complicated otherwise. As an example, I will illustrate its use in the context of non-conforming methods, with a special eye on discontinuous Galerkin methods. [These are recent results obtained jointly with E. Georgoulis.] |
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Thu, 18/10/2007 14:00 |
Prof Peter Benner (University of Chemnitz) |
Computational Mathematics and Applications |
Comlab |
| Model reduction (also called system reduction, order reduction) is an ubiquitous tool in the analysis and simulation of dynamical systems, control design, circuit simulation, structural dynamics, CFD, etc. In the past decades many approaches have been developed for reducing the complexity of a given model. In this introductory talk, we will survey some of the most prominent methods used for linear systems, compare their properties and highlight similarities. In particular, we will emphasize the role of recent developments in numerical linear algebra in the different approaches. Efficiently using these techniques, the range of applicability of some of the methods has considerably widened. The performance of several approaches will be demonstrated using real-world examples from a variety of engineering disciplines. | |||
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Thu, 25/10/2007 14:00 |
Dr Daniel Robinson (University of Oxford) |
Computational Mathematics and Applications |
Comlab |
| A new primal-dual augmented Lagrangian merit function is proposed that may be minimized with respect to both the primal and dual variables. A benefit of this approach is that each subproblem may be regularized by imposing explicit bounds on the dual variables. Two primal-dual variants of classical primal methods are given: a primal-dual bound constrained Lagrangian (pdBCL) method and a primal-dual l1 linearly constrained Lagrangian (pdl1-LCL) method. | |||
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Thu, 01/11/2007 14:00 |
Dr Laura Grigori (INRIA) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
| We present algorithms for dense LU and QR factorizations that minimize the cost of communication. One of today's challenging technology trends is the increased communication cost. This trend predicts that arithmetic will continue to improve exponentially faster than bandwidth, and bandwidth exponentially faster than latency. The new algorithms for dense QR and LU factorizations greatly reduce the amount of time spent communicating, relative to conventional algorithms. This is joint work with James Demmel, Mark Hoemmen, Julien Langou, and Hua Xiang. | |||
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Thu, 08/11/2007 14:00 |
Dr Daan Huybrechs (KU Leuven) |
Computational Mathematics and Applications |
Comlab |
| The evaluation of oscillatory integrals is often considered to be a computationally challenging problem. However, in many cases, the exact opposite is true. Oscillatory integrals are cheaper to evaluate than non-oscillatory ones, even more so in higher dimensions. The simplest strategy that illustrates the general approach is to truncate an asymptotic expansion, where available. We show that an optimal strategy leads to the construction of certain unconventional Gaussian quadrature rules, that converge at twice the rate of asymptotic expansions. We examine a range of one-dimensional and higher dimensional, singular and highly oscillatory integrals. | |||
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Thu, 15/11/2007 14:00 |
Prof Jan Magnus (Tilburg University) |
Computational Mathematics and Applications |
Rutherford Appleton Laboratory, nr Didcot |
The Snaer program calculates the posterior mean and variance of variables on some of which we have data (with precisions), on some we have prior information (with precisions), and on some prior indicator ratios (with precisions) are available. The variables must satisfy a number of exact restrictions. The system is both large and sparse. Two aspects of the statistical and computational development are a practical procedure for solving a linear integer system, and a stable linearization routine for ratios. We test our numerical method for solving large sparse linear least-squares estimation problems, and find that it performs well, even when the  design matrix is large (  ). |
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Thu, 22/11/2007 14:00 |
Prof Stefan Ulbrich (TU Darmstadt) |
Computational Mathematics and Applications |
Comlab |
| Adaptive discretizations and iterative multilevel solvers are nowadays well established techniques for the numerical solution of PDEs. The development of efficient multilevel techniques in the context of PDE-constrained optimization methods is an active research area that offers the potential of reducing the computational costs of the optimization process to an equivalent of only a few PDE solves. We present a general class of inexact adaptive multilevel SQP-methods for PDE-constrained optimization problems. The algorithm starts with a coarse discretization of the underlying optimization problem and provides 1. implementable criteria for an adaptive refinement strategy of the current discretization based on local error estimators and 2. implementable accuracy requirements for iterative solvers of the PDE and adjoint PDE on the current grid such that global convergence to the solution of the infinite-dimensional problem is ensured. We illustrate how the adaptive refinement strategy of the multilevel SQP-method can be implemented by using existing reliable a posteriori error estimators for the state and the adjoint equation. Moreover, we discuss the efficient handling of control constraints and describe how efficent multilevel preconditioners can be constructed for the solution of the arising linear systems. Numerical results are presented that illustrate the potential of the approach. This is joint work with Jan Carsten Ziems. | |||
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Thu, 29/11/2007 14:00 |
Dr Rodrigo Platte (University of Oxford) |
Computational Mathematics and Applications |
Comlab |
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Radial basis function (RBF) methods have been successfully used to approximate functions in multidimensional complex domains and are increasingly being used in the numerical solution of partial differential equations. These methods are often called meshfree numerical schemes since, in some cases, they are implemented without an underlying grid or mesh. The focus of this talk is on the class of RBFs that allow exponential convergence for smooth problems. We will explore the dependence of accuracy and stability on node locations of RBF interpolants. Because Gaussian RBFs with equally spaced centers are related to polynomials through a change of variable, a number of precise conclusions about convergence rates based on the smoothness of the target function will be presented. Collocation methods for PDEs will also be considered. |
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