Past Computational Mathematics and Applications Seminar

12 October 2000
14:00
Prof Howard Elman
Abstract
We examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the Navier-Stokes equations. With a combination of analytic and empirical results, we study the effects of fundamental parameters on convergence. We demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. The structure of these distributions is independent of the discretization mesh size, but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller. These characteristics are directly correlated with the convergence properties of iterative solvers.
  • Computational Mathematics and Applications Seminar
Dr Steven Benbow
Abstract
The talk will focus on solution methods for augmented linear systems of the form \\ \\ $[ A B ][x] = [b] [ B' 0 ][y] [0]$. \\ \\ Augmented linear systems of this type arise in several areas of numerical applied mathematics including mixed finite element / finite difference discretisations of flow equations (Darcy flow and Stokes flow), electrical network simulation and optimisation. The general properties of such systems are that they are large, sparse and symmetric, and efficient solution techniques should make use of the block structure inherent in the system as well as of these properties. \\ \\ Iterative linear solution methods will be described that attempt to take advantage of the structure of the system, and observations on augmented systems, in particular the distribution of their eigenvalues, will be presented which lead to further iterative methods and also to preconditioners for existing solution methods. For the particular case of Darcy flow, comments on properties of domain decomposition methods of additive Schwarz type and similarities to incomplete factorisation preconditioners will be made.
  • Computational Mathematics and Applications Seminar
25 May 2000
14:00
Dr Raphael Hauser
Abstract
I am going to show that all self-scaled barriers for the cone of symmetric positive semidefinite matrices are of the form $X\mapsto -c_1\ln\det X +c_0$ for some constants $c_1$ > $0,c_0 \in$ \RN. Equivalently one could state say that all such functions may be obtained via a homothetic transformation of the universal barrier functional for this cone. The result shows that there is a certain degree of redundancy in the axiomatic theory of self-scaled barriers, and hence that certain aspects of this theory can be simplified. All relevant concepts will be defined. In particular I am going to give a short introduction to the notion of self-concordance and the intuitive ideas that motivate its definition.
  • Computational Mathematics and Applications Seminar

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