Past Computational Mathematics and Applications Seminar

3 February 2000
14:00
Prof Henk van der Vorst
Abstract
Krylov subspace methods offer good possibilities for the solution of large sparse linear systems of equations.For general systems, some of the popular methods often show an irregular type of convergence behavior and one may wonder whether that could be improved or not. Many suggestions have been made for improvement and the question arises whether these corrections are cosmetic or not. There is also the question whether the irregularity shows inherent numerical instability. In such cases one should take extra care in the application of smoothing techniques. We will discuss strategies that work well and strategies that might have been expected to work well.
  • Computational Mathematics and Applications Seminar
27 January 2000
14:00
Prof Neil Sandham
Abstract
This work forms part of a larger research project to develop efficient low-dissipative high-order numerical techniques for high-speed turbulent flow simulation, including shock wave interactions with turbulence. The requirements on a numerical method are stringent.For the turbulence the method must be capable of resolving accurately a wide range of length scales, whilst for shock waves the method must be stable and not generate excessive local oscillations. Conventional methods are either too dissipative, or incapable of shock capturing. Higher-order ENO, WENO or hybrid schemes are too expensive for practical computations. Previous work of Yee, Sandham & Djomehri (1999) developed high-order shock-capturing schemes which minimize the use of numerical dissipation away from shock waves. The objective of the present study is to further minimize the use of numerical dissipation for shock-free compressible turbulence simulations.
  • Computational Mathematics and Applications Seminar
20 January 2000
14:00
Prof Bruce Christianson
Abstract
In 1983 Pantoja described a stagewise construction of the exact Newton direction for a discrete time optimal control problem. His algorithm requires the solution of linear equations with coefficients given by recurrences involving second derivatives, for which accurate values are therefore required. \\ \\ Automatic differentiation is a set of techniques for obtaining derivatives of functions which are calculated by a program, including loops and subroutine calls, by transforming the text of the program. \\ \\ In this talk we show how automatic differentiation can be used to evaluate exactly the quantities required by Pantoja's algorithm, thus avoiding the labour of forming and differentiating adjoint equations by hand. \\ \\ The cost of calculating the newton direction amounts to the cost of solving one set of linear equations, of the order of the number of control variables, for each time step. The working storage cost can be made smaller than that required to hold the solution.
  • Computational Mathematics and Applications Seminar
11 November 1999
15:00
Dr Andy Wathen
Abstract
The general importance of preconditioning in combination with an appropriate iterative technique for solving large scale linear(ised) systems is widely appreciated. For definite problems (where the eigenvalues lie in a half-plane) there are a number of preconditioning techniques with a range of applicability, though there remain many difficult problems. For indefinite systems (where there are eigenvalues in both half-planes), techniques are generally not so well developed. Constraints arise in many physical and mathematical problems and invariably give rise to indefinite linear(ised) systems: the incompressible Navier-Stokes equations describe conservation of momentum in the presence of viscous dissipation subject to the constraint of conservation of mass, for transmission problems the solution on an interior domain is often solved subject to a boundary integral which imposes the exterior field, in optimisation the appearance of constraints is ubiquitous... \\ \\ We will describe two approaches to preconditioning such constrained systems and will present analysis and numerical results for each. In particular, we will describe the applicability of these techniques to approximations of incompressible Navier-Stokes problems using mixed finite element approximation.
  • Computational Mathematics and Applications Seminar
4 November 1999
15:00
Abstract
As an alternative to floating-point, several workers have proposed the use of a logarithmic number system, in which a real number is represented as a fixed-point logarithm. Multiplication and division therefore proceed in minimal time with no rounding error. However, the system can only offer an overall advantage if addition and subtraction can be performed with speed and accuracy at least equal to that of floating-point, but this has hitherto been difficult to achieve. We will present a number of original techniques by which this has now been accomplished. We will then demonstrate by means of simulations that the logarithmic system offers around twofold improvements in speed and accuracy, and finally will describe a new European collaborative project which aims to develop a logarithmic microprocessor during the next three years.
  • Computational Mathematics and Applications Seminar
28 October 1999
15:00
Abstract
We show that Sorensen's (1992) implicitly restarted Arnoldi method (IRAM) (including its block extension) is non-stationary simultaneous iteration in disguise. By using the geometric convergence theory for non-stationary simultaneous iteration due to Watkins and Elsner (1991) we prove that an implicitly restarted Arnoldi method can achieve a super-linear rate of convergence to the dominant invariant subspace of a matrix. We conclude with some numerical results the demonstrate the efficiency of IRAM.
  • Computational Mathematics and Applications Seminar

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