Comlab

No seminar today

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.

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.

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.

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.

The sphere is an important setting for applied mathematics, yet the underlying approximation theory and numerical analysis needed for serious applications (such as, for example, global weather models) is much less developed than, for example, for the cube.

\\

\\

This lecture will apply recent developments in approximation theory on the sphere to two different problems in scientific computing.

\\

\\

First, in geodesy there is often the need to evaluate integrals using data selected from the vast amount collected by orbiting satellites. Sometimes the need is for quadrature rules that integrate exactly all spherical polynomials up to a specified degree $n$ (or equivalently, that integrate exactly all spherical harmonies $Y_{\ell ,k}(\theta ,\phi)$ with $\ell \le n).$ We shall demonstrate (using results of M. Reimer, I. Sloan and R. Womersley in collaboration with

W. Freeden) that excellent quadrature rules of this kind can be obtained from recent results on polynomial interpolation on the sphere, if the interpolation points (and thus the quadrature points) are chosen to be points of a so-called extremal fundamental system.

\\

\\

The second application is to the scattering of sound by smooth three-dimensional objects, and to the inverse problem of finding the shape of a scattering object by observing the pattern of the scattered sound waves. For these problems a methodology has been developed, in joint work with I.G. Graham, M. Ganesh and R. Womersley, by applying recent results on constructive polynomial approximation on the sphere. (The scattering object is treated as a deformed sphere.)

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.