Past

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...

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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.