I am going to show that all self-scaled barriers for the
cone of symmetric positive semidefinite matrices are of the form
$X\mapsto -c_1\ln\det X +c_0$ for some constants $c_1$ > $0,c_0 \in$ \RN.
Equivalently one could state say that all such functions may be
obtained via a homothetic transformation of the universal barrier
functional for this cone. The result shows that there is a certain
degree of redundancy in the axiomatic theory of self-scaled barriers,
and hence that certain aspects of this theory can be simplified. All
relevant concepts will be defined. In particular I am going to give
a short introduction to the notion of self-concordance and the
intuitive ideas that motivate its definition.
Interval arithmetic is a way to produce guaranteed enclosures of the
results of numerical calculations. Suppose $f(x)$ is a real
expression in real variables $x= (x_1, \ldots, x_n)$, built up from
the 4 basic arithmetic operations and other 'standard functions'. Let
$X_1, \ldots, X_n$ be (compact) real intervals. The process of {\em
interval evaluation} of $f(X_1, ..., X_n)$ replaces each real
operation by the corresponding interval operation wherever it occurs
in $f$, e.g. $A \times B$ is the smallest interval containing $\{a
\times b \mid a \in A, b \in B\}$.
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As is well known, it yields a guaranteed enclosure for the true range
$\{f(x_1, \ldots, x_n) \mid x_1 \in X_1, \ldots, x_n \in X_n\}$,
provided no exceptions such as division by (an interval containing)
zero occur during evaluation.
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Interval arithmetic takes set inputs and produces set outputs. Noting
this, we show there is a consistent way to extend arithmetic to $R^* =
R \cup \{-\infty, +\infty\}$, such that interval evaluation continues
to give enclosures, and there are {\em no exceptions}. The basic
ideas are: the usual set-theory meaning of evaluating a relation at a
set; and taking topological closure of the graph of a function in a
suitable $(R^{*})^n$. It gives rigorous meaning to intuitively
sensible statements like $1/0 = \{-\infty, +\infty\}$, $0/0 = R^*$
(but $(x/x)_{|x=0} = 1$), $\sin(+\infty) = [-1,1]$, etc.
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A practical consequence is that an exception-free floating-point
interval arithmetic system is possible. Such a system is implemented
at hardware level in the new Sun Fortran compiler, currently on
beta-release.
Boundary layers are often studied with no pressure gradient
or with an imposed pressure gradient. Either of these assumptions
can lead to difficulty in obtaining solutions. A major advance in fluid
dynamics last century (1969) was the development of a triple deck
formulation for boundary layers where the pressure is not
specified but emerges through an interaction between
boundary layer and the inviscid outer flow. This has given rise to
new computational problems and computations have in turn
fed ideas back into theoretical developments. In this survey talk
based on my new book, I will look at three problems:
flow past a plate, flow separation and flow in channels
and discuss the interaction between theory and computation
in advancing boundary layer theory.
This talk reviews some theoretical and practical aspects
of incompressible flow modelling using finite element approximations
of the (Navier-) Stokes equations.
The infamous Q1-P0 velocity/pressure mixed finite element approximation
method is discussed. Two practical ramifications of the inherent
instability are focused on, namely; the convergence of the approximation
with and without regularisation, and the behaviour of fast iterative
solvers (of multigrid type) applied to the pressure Poisson system
that arises when solving time-dependent Navier-Stokes equations
using classical projection methods.
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This is joint work with David Griffiths from the University of Dundee.
We examine the convergence characteristics of iterative methods based
on a new preconditioning operator for solving the linear systems
arising from discretization and linearization of the Navier-Stokes
equations. With a combination of analytic and empirical results, we
study the effects of fundamental parameters on convergence. We
demonstrate that the preconditioned problem has an eigenvalue
distribution consisting of a tightly clustered set together with a
small number of outliers. The structure of these distributions is
independent of the discretization mesh size, but the cardinality of
the set of outliers increases slowly as the viscosity becomes smaller.
These characteristics are directly correlated with the convergence
properties of iterative solvers.