Past Computational Mathematics and Applications Seminar

E.g., 2019-09-23
E.g., 2019-09-23
E.g., 2019-09-23
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
14 October 1999
15:00
Prof Will Light
Abstract
It has been known for some while now that every radial basis function in common usage for multi-dimensional interpolation has associated with it a uniquely defined Hilbert space, in which the radial basis function is the `minimal norm interpolant'. This space is usually constructed by utilising the positive definite nature of the radial function, but such constructions turn out to be difficult to handle. We will present a direct way of constructing the spaces, and show how to prove extension theorems in such spaces. These extension theorems are at the heart of improved error estimates in the $L_p$-norm.
  • Computational Mathematics and Applications Seminar

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