Forthcoming events in this series
Some properties of thin plate spline interpolation
Abstract
Let the thin plate spline radial basis function method be applied to
interpolate values of a smooth function $f(x)$, $x \!\in\! {\cal R}^d$.
It is known that, if the data are the values $f(jh)$, $j \in {\cal Z}^d$,
where $h$ is the spacing between data points and ${\cal Z}^d$ is the
set of points in $d$ dimensions with integer coordinates, then the
accuracy of the interpolant is of magnitude $h^{d+2}$. This beautiful
result, due to Buhmann, will be explained briefly. We will also survey
some recent findings of Bejancu on Lagrange functions in two dimensions
when interpolating at the integer points of the half-plane ${\cal Z}^2
\cap \{ x : x_2 \!\geq\! 0 \}$. Most of our attention, however, will
be given to the current research of the author on interpolation in one
dimension at the points $h {\cal Z} \cap [0,1]$, the purpose of the work
being to establish theoretically the apparent deterioration in accuracy
at the ends of the range from ${\cal O} ( h^3 )$ to ${\cal O} ( h^{3/2}
)$ that has been observed in practice. The analysis includes a study of
the Lagrange functions of the semi-infinite grid ${\cal Z} \cap \{ x :
x \!\geq\! 0 \}$ in one dimension.
Pulmonary airway closure: A large-displacement fluid-structure-interaction problem
On the robust solution of process simulation problems
Abstract
In this talk we review experiences of using the Harwell Subroutine
Library and other numerical software codes in implementing large scale
solvers for commercial industrial process simulation packages. Such
packages are required to solve problems in an efficient and robust
manner. A core requirement is the solution of sparse systems of linear
equations; various HSL routines have been used and are compared.
Additionally, the requirement for fast small dense matrix solvers is
examined.
Upwind residual distributive schemes for compressible flows
Iterative nonlinear inverse problems: theory and numerical examples
Scientific computing for problems on the sphere - applying good approximations on the sphere to geodesy and the scattering of sound
Abstract
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.)
Spectral multigrid methods for the Navier-Stokes equations
Reliable process modelling and optimisation using interval analysis
Abstract
Continuing advances in computing technology provide the power not only to solve
increasingly large and complex process modeling and optimization problems, but also
to address issues concerning the reliability with which such problems can be solved.
For example, in solving process optimization problems, a persistent issue
concerning reliability is whether or not a global, as opposed to local,
optimum has been achieved. In modeling problems, especially with the
use of complex nonlinear models, the issue of whether a solution is unique
is of concern, and if no solution is found numerically, of whether there
actually exists a solution to the posed problem. This presentation
focuses on an approach, based on interval mathematics,
that is capable of dealing with these issues, and which
can provide mathematical and computational guarantees of reliability.
That is, the technique is guaranteed to find all solutions to nonlinear
equation solving problems and to find the global optimum in nonlinear
optimization problems. The methodology is demonstrated using several
examples, drawn primarily from the modeling of phase behavior, the
estimation of parameters in models, and the modeling, using lattice
density-functional theory, of phase transitions in nanoporous materials.
Acceleration strategies for restarted minimum residual methods
Abstract
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.
Mixed finite element methods, stability and related issues
Support Vector machines and related kernel methods
Abstract
Support Vector Machines are a new and very promising approach to
machine learning. They can be applied to a wide range of tasks such as
classification, regression, novelty detection, density estimation,
etc. The approach is motivated by statistical learning theory and the
algorithms have performed well in practice on important applications
such as handwritten character recognition (where they currently give
state-of-the-art performance), bioinformatics and machine vision. The
learning task typically involves optimisation theory (linear, quadratic
and general nonlinear programming, depending on the algorithm used).
In fact, the approach has stimulated new questions in optimisation
theory, principally concerned with the issue of how to handle problems
with a large numbers of variables. In the first part of the talk I will
overview this subject, in the second part I will describe some of the
speaker's contributions to this subject (principally, novelty
detection, query learning and new algorithms) and in the third part I
will outline future directions and new questions stimulated by this
research.
The system wave equation - a generic hyperbolic problem?
Instabilities, symmetry breaking and mode interactions in an enclosed swirling flow
Abstract
The flow in a cylinder with a rotating endwall has continued to
attract much attention since Vogel (1968) first observed the vortex
breakdown of the central core vortex that forms. Recent experiments
have observed a multiplicity of unsteady states that coexist over a
range of the governing parameters. In spite of numerous numerical and
experimental studies, there continues to be considerable controversy
with fundamental aspects of this flow, particularly with regards to
symmetry breaking. Also, it is not well understood where these
oscillatory states originate from, how they are interrelated, nor how
they are related to the steady, axisymmetric basic state.
\\
\\
In the aspect ratio (height/radius) range 1.6 2.8. An efficient and
accurate numerical scheme is presented for the three-dimensional
Navier-Stokes equations in primitive variables in a cylinder. Using
these code, primary and secondary bifurcations breaking the SO(2)
symmetry are analyzed.
\\
\\
We have located a double Hopf bifurcation, where an axisymmetric limit
cycle and a rotating wave bifurcate simultaneously. This codimension-2
bifurcation is very rich, allowing for several different scenarios. By
a comprehensive two-parameter exploration about this point we have
identified precisely to which scenario this case corresponds. The mode
interaction generates an unstable two-torus modulate rotating wave
solution and gives a wedge-shaped region in parameter space where the
two periodic solutions are both stable.
\\
\\
For aspect ratios around three, experimental observations suggest that
the first mode of instability is a precession of the central vortex
core, whereas recent linear stability analysis suggest a Hopf
bifurcation to a rotating wave at lower rotation rates. This apparent
discrepancy is resolved with the aid of the 3D Navier-Stokes
solver. The primary bifurcation to an m=4 traveling wave, detected by
the linear stability analysis, is located away from the axis, and a
secondary bifurcation to a modulated rotating wave with dominant modes
m=1 and 4, is seen mainly on the axis as a precessing vortex breakdown
bubble. Experiments and the linear stability analysis detected
different aspects of the same flow, that take place in different
spatial locations.
A spectral Petrov-Galerkin scheme for the stability of pipe flow: I - linear analysis and transient growth
A stopping criterion for the conjugate gradient algorithm in a finite element method framework
Abstract
We combine linear algebra techniques with finite element techniques to obtain a reliable stopping criterion for the Conjugate Gradient algorithm. The finite element method approximates the weak form of an elliptic partial differential equation defined within a Hilbert space by a linear system of equations A x = b, where A is a real N by N symmetric and positive definite matrix. The conjugate gradient method is a very effective iterative algorithm for solving such systems. Nevertheless, our experiments provide very good evidence that the usual stopping criterion based on the Euclidean norm of the residual b - Ax can be totally unsatisfactory and frequently misleading. Owing to the close relationship between the conjugate gradient behaviour and the variational properties of finite element methods, we shall first summarize the principal properties of the latter. Then, we will use the recent results of [1,2,3,4]. In particular, using the conjugate gradient, we will compute the information which is necessary to evaluate the energy norm of the difference between the solution of the continuous problem, and the approximate solution obtained when we stop the iterations by our criterion.
Finally, we will present the numerical experiments we performed on a selected ill-conditioned problem.
References
- [1] M. Arioli, E. Noulard, and A. Russo, Vector Stopping Criteria for Iterative Methods: Applications to PDE's, IAN Tech. Rep. N.967, 1995.
- [2] G.H. Golub and G. Meurant, Matrices, moments and quadrature II; how to compute the norm of the error in iterative methods, BIT., 37 (1997), pp.687-705.
- [3] G.H. Golub and Z. Strakos, Estimates in quadratic formulas, Numerical Algorithms, 8, (1994), pp.~241--268.
- [4] G. Meurant, The computation of bounds for the norm of the error in the conjugate gradient algorithm, Numerical Algorithms, 16, (1997), pp.~77--87.
Computational problems in Interactive Boundary Layer Theory
Abstract
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.
Incompressible flow modelling can be a dodgy business
Abstract
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.
\\
\\
This is joint work with David Griffiths from the University of Dundee.
Long time-step methods for Hamiltonian dynamics from molecular to geophysical fluid dynamics
A sharp interface model for martensitic single crystal thin films
Saddle point preconditioners for the Navier-Stokes equations
Abstract
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.