Past Computational Mathematics and Applications Seminar

Prof Mark Stadtherr
Abstract
Continuing advances in computing technology provide the power not only to solve increasingly large and complex process modeling and optimization problems, but also to address issues concerning the reliability with which such problems can be solved. For example, in solving process optimization problems, a persistent issue concerning reliability is whether or not a global, as opposed to local, optimum has been achieved. In modeling problems, especially with the use of complex nonlinear models, the issue of whether a solution is unique is of concern, and if no solution is found numerically, of whether there actually exists a solution to the posed problem. This presentation focuses on an approach, based on interval mathematics, that is capable of dealing with these issues, and which can provide mathematical and computational guarantees of reliability. That is, the technique is guaranteed to find all solutions to nonlinear equation solving problems and to find the global optimum in nonlinear optimization problems. The methodology is demonstrated using several examples, drawn primarily from the modeling of phase behavior, the estimation of parameters in models, and the modeling, using lattice density-functional theory, of phase transitions in nanoporous materials.
  • Computational Mathematics and Applications Seminar
22 February 2001
14:00
Dr Oliver Ernst
Abstract
This talk reviews some recent joint work with Michael Eiermann and Olaf Schneider which introduced a framework for analyzing some popular techniques for accelerating restarted Krylov subspace methods for solving linear systems of equations. Such techniques attempt to compensate for the loss of information due to restarting methods like GMRES, the memory demands of which are usually too high for it to be applied to large problems in unmodified form. We summarize the basic strategies which have been proposed and present both theoretical and numerical comparisons.
  • Computational Mathematics and Applications Seminar
Dr Colin Campbell
Abstract
Support Vector Machines are a new and very promising approach to machine learning. They can be applied to a wide range of tasks such as classification, regression, novelty detection, density estimation, etc. The approach is motivated by statistical learning theory and the algorithms have performed well in practice on important applications such as handwritten character recognition (where they currently give state-of-the-art performance), bioinformatics and machine vision. The learning task typically involves optimisation theory (linear, quadratic and general nonlinear programming, depending on the algorithm used). In fact, the approach has stimulated new questions in optimisation theory, principally concerned with the issue of how to handle problems with a large numbers of variables. In the first part of the talk I will overview this subject, in the second part I will describe some of the speaker's contributions to this subject (principally, novelty detection, query learning and new algorithms) and in the third part I will outline future directions and new questions stimulated by this research.
  • Computational Mathematics and Applications Seminar
18 January 2001
14:00
Prof Francisco Marques
Abstract
The flow in a cylinder with a rotating endwall has continued to attract much attention since Vogel (1968) first observed the vortex breakdown of the central core vortex that forms. Recent experiments have observed a multiplicity of unsteady states that coexist over a range of the governing parameters. In spite of numerous numerical and experimental studies, there continues to be considerable controversy with fundamental aspects of this flow, particularly with regards to symmetry breaking. Also, it is not well understood where these oscillatory states originate from, how they are interrelated, nor how they are related to the steady, axisymmetric basic state. \\ \\ In the aspect ratio (height/radius) range 1.6 < $\Lambda$ < 2.8, the primary bifurcation is to an axisymmetric time-periodic flow (a limit cycle). We have developed a suite of numerical techniques, exploiting the biharmonic formulation of the problem in the axisymmetric case, that allows us to compute the nonlinear time evolution, the basic state, and its linear stability in a consistent and efficient manner. We show that the basic state undergoes a succession of Hopf bifurcations and the corresponding eigenvalues and eigenvectors of these excited modes describe most of the characteristics of the observed time-dependent states. \\ \\ The primary bifurcation is non-axisymmetric, to pure rotating wave, in the ranges $\Lambda$ <1.6 and $\Lambda$ > 2.8. An efficient and accurate numerical scheme is presented for the three-dimensional Navier-Stokes equations in primitive variables in a cylinder. Using these code, primary and secondary bifurcations breaking the SO(2) symmetry are analyzed. \\ \\ We have located a double Hopf bifurcation, where an axisymmetric limit cycle and a rotating wave bifurcate simultaneously. This codimension-2 bifurcation is very rich, allowing for several different scenarios. By a comprehensive two-parameter exploration about this point we have identified precisely to which scenario this case corresponds. The mode interaction generates an unstable two-torus modulate rotating wave solution and gives a wedge-shaped region in parameter space where the two periodic solutions are both stable. \\ \\ For aspect ratios around three, experimental observations suggest that the first mode of instability is a precession of the central vortex core, whereas recent linear stability analysis suggest a Hopf bifurcation to a rotating wave at lower rotation rates. This apparent discrepancy is resolved with the aid of the 3D Navier-Stokes solver. The primary bifurcation to an m=4 traveling wave, detected by the linear stability analysis, is located away from the axis, and a secondary bifurcation to a modulated rotating wave with dominant modes m=1 and 4, is seen mainly on the axis as a precessing vortex breakdown bubble. Experiments and the linear stability analysis detected different aspects of the same flow, that take place in different spatial locations.
  • Computational Mathematics and Applications Seminar
Abstract

We combine linear algebra techniques with finite element techniques to obtain a reliable stopping criterion for the Conjugate Gradient algorithm. The finite element method approximates the weak form of an elliptic partial differential equation defined within a Hilbert space by a linear system of equations A x = b, where A is a real N by N symmetric and positive definite matrix. The conjugate gradient method is a very effective iterative algorithm for solving such systems. Nevertheless, our experiments provide very good evidence that the usual stopping criterion based on the Euclidean norm of the residual b - Ax can be totally unsatisfactory and frequently misleading. Owing to the close relationship between the conjugate gradient behaviour and the variational properties of finite element methods, we shall first summarize the principal properties of the latter. Then, we will use the recent results of [1,2,3,4]. In particular, using the conjugate gradient, we will compute the information which is necessary to evaluate the energy norm of the difference between the solution of the continuous problem, and the approximate solution obtained when we stop the iterations by our criterion.

Finally, we will present the numerical experiments we performed on a selected ill-conditioned problem.

References

  • [1] M. Arioli, E. Noulard, and A. Russo, Vector Stopping Criteria for Iterative Methods: Applications to PDE's, IAN Tech. Rep. N.967, 1995.
  • [2] G.H. Golub and G. Meurant, Matrices, moments and quadrature II; how to compute the norm of the error in iterative methods, BIT., 37 (1997), pp.687-705.
  • [3] G.H. Golub and Z. Strakos, Estimates in quadratic formulas, Numerical Algorithms, 8, (1994), pp.~241--268.
  • [4] G. Meurant, The computation of bounds for the norm of the error in the conjugate gradient algorithm, Numerical Algorithms, 16, (1997), pp.~77--87.
  • Computational Mathematics and Applications Seminar

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