The new millennium sees both the drawing to close of many of the current projects in Advanced Computation and the possibility of significant new initiatives. The departure of Professor Brent (RSISE) to Oxford, the retirement of Professor Osborne, the winding up of the successful Fujitsu Area 4 project and the departure of the highly skilled and productive staff, and the doubt over the scale of activity of the ACSys data mining group are events which signal the end of an era in which the ANU has been one of the most important international centres of computational mathematics. It is pleasant to reflect on such superb achievements as the recent resolution of the Painleve paradox by Stewart (Texas A & M) who has spent much of his professional career at ANU. It is less pleasant to reflect on the scattering of productive researchers to the four quarters of the globe that has occurred in the last few years, and, in particular, in 1999. The proposed APAC Centre and associated initiatives in education offer significant hope for new developments in Advanced Computation. One may question the apparent slow pace of developments and perhaps the lack of real experience (maybe even opportunism rather than professional involvement) of the apparent decision makers. It is surprising that the experience accumulated by the Area 4 personnel should be allowed to dissipate so easily. However, it has meant that the time scale for developments is not incompatible with the making of a new senior appointment in SMS.
Despite the impending changes, 1999 was a year of significant activity. In no small part this was built around the SMS Special Year Initiative in Computational Mathematics. In particular, this provided funds which were used to bring international participants to the CTAC99 (Computational Techniques and Applications Conference) meeting of the Computational Mathematics Group of ANZIAM. This very successful meeting was also able to take advantage of a German-Australia exchange which brought a number of leading researchers in boundary integral methods to Sydney for an ANU supported National Symposium. Keynote speakers supported by Special Year funds were Professor Alistair Watson of (Dundee) (Professor Watson was the First PhD in Computational Mathematics from the ANU), and Professor Rainer Kress from Gottingen. Other visitors supported by special year funds included Professor Andreas Griewank (Dresden), and Professor John Reid (Rutherford Appleton Laboratory).
Numerical linear algebra --
Work in this area was largely directed to
producing subroutines for the Fujitsu VPP300 vector parallel
computer under the Area 4 contract. Grosz and Zhou have worked on
methods for solving large scale linear systems with special
reference to the particular problems of data distribution and load
balancing that arise from the conjunction of vector facilities in a
parallel machine architecture. Grosz has continued to develop
multi-level and ILUM-type preconditioning techniques for algebraic
multigrid iterative schemes, including clever vectorization schemes
to take account of the parallel vector structure of the VPP. Zhou
has completed an implementation of the Choleski method for the
direct solution of large sparse positive definite matrices. Harrar
and Osborne have worked on the development, analysis, and
implementation of techniques for the solution of large-scale
eigenvalue problems and this work has been applied very
successfully to a range of practical problems. The methods
implemented have involved rational Arnoldi and Newton based schemes
using efficient existing routines, including the wrap-around
partitioning software developed under the Area 4 contract.
Numerical solution of differential and integral equations --
Grosz developed a new and
flexible a-posteriori error estimate technique for finite element
methods. Tran is concluding the numerical studies necessary to
completed the work on tolerant qualocation methods for
pseudo-differential equations with variable coefficients. He has
studied preconditioners for a fully-discrete and symmetric boundary
element methods, and inverse scattering problems for a locally
perturbed half-plane. Work has commenced on the study of fast
solution methods for three-dimensional many-particle problems in
linear elasticity.
Optimization --
Osborne has continued work with Presnell (ANU and Florida) and
Turlach (ANU and Adelaide) on a simplicial algorithm for a new
variable selection technique named the Lasso. In this, the
estimation procedure is constrained by a parametric constraint
based on the l1 norm. The effect of this constraint can be
studied conveniently using homotopy methods. This work has been
generalised to a study of optimisation problems subject to a
polyhedral constraint. Osborne and Watson have considered also
Lagrangian vesions of this problem. It turns out that an important
range of problems in data analysis and data mining are posed in
this form. A variant generalised to include block effects as well
as single variables has been developed by Bakin in his PhD thesis
submitted during the year. It is interesting that the spectrum of
applications that connects with his work includes both the Lasso
and ridge regression. Li with Osborne and Prvan (Canberra) have
studied methods for relaxing the requirements of Fisher scoring
that restrict it to large data sets and exact models. Promising
algorithms using variants of sequential quadratic programming have
been developed for both constrained and unconstrained problems. Li
has developed new algorithms for estimating differential equations
based on this work. The book by Osborne on the minimization of
polyhedral functions will be published by Cambridge University
Press.
Modelling applications --
During 1999, Dr Barnes
continued work with Marchant (Wollongong), Zoppou (CSIRO) and
Roberts on Rapidly Varying Flows, using money from an ARC grant.
This work will contribute towards more accuracy in the modelling of
dam breaks and subsequent wave fronts. Dr Barnes was approached
during 1999 by the Australian Entertainment Industry Association
(AEIA), to build a voting system for the Helpmann awards. This
system will come into existence in 2001 The book on mathematical
modelling by Barnes and Fulford (Massey) is in press. Harrar and
Osborne have applied a variety of their eigenvalue techniques to
solve a broad range of applications problems arising in
hydrodynamic stability, molecular dynamics, chemical reactions,
acoustics, computational quantum chemistry, optical physics, and
magneto-hydrodynamics. The availability of very large scale
computing facilities has meant that some of these problems have
been considered in much greater detail than was possible
previously. This has led both to some revision of known results,
and to the detection of new phenomena. Grosz has continued his
active interest in the implementation of simulation software on
parallel computers involving the application of his package for
solving partial differential equations (VECFEM). This includes work
with Ehlers (Karlsruhe) on immersion cooling of tempering steel,
and Kalyanasundaram (FEIT, ANU) on elasto-plastic stress analysis.
Petersons obtained an improved estimate of the depth of a possible
sharp rise in the electrical conductivity of the Earth beneath
Europe. The study of the very long period 11 year response of the
Earth as a whole is continuing.
Projects --
Fujitsu Area 4 project - The program has made an
important contribution to this project which supports 4 staff (two
fractional) and a PhD student and is jointly directly by the
Program and the Computer Systems Laboratory in RSISE.
Unfortunately, the project has been terminated by Fujitsu, and
final deliveries have been made. These have included final
versions of the algebraic multigrid software developed by Grosz,
eigenvalue software including Arnoldi and block Arnoldi software
developed by Harrar, wrap-around partitioning software developed
by Osborne, including driver software for mapping band
matrices to block bidiagonal form, and software for the direct
solution of large sparse linear systems developed by Zhou. Advanced
Computational Systems CRC AC-1 project - This project aims to give
theoretical underpinning to other activities of the CRC especially
in the areas of high performance computing and portable software
development. A major project in support of these activities is
being undertaken by the program in the area of data mining in
conjunction with the program in Statistical Science. Direct
benefits have been a Research Fellow post in Statistical Science,
and supplementation of scholarship and other support for Bakin who
completed his PhD studies in 1999. The aim is to seek parsimonious
representations of very large data sets to permit rapid resolution
of queries, and the detection of unusual events. Bakin has
developed his PLASM methodology for block selection of variables in
large scale problems. Roberts has been attached to the project for
two years to concentrate on the further development of the finite
element surface smoothing techniques.
Other activities --
The program provided the major organisational
effort in support of the CTAC99 Conference. Workshops were
organised by Grosz on high performance computing, and Osborne on
recent developments in the numerical solution of ordinary
differential equations. Harrar, together with Ryan (Auckland) and
Weber (ADFA) organised a meeting in conjunction with CTAC99 to
celebrate the contributions of M R Osborne to computational
mathematics. The annual one-day seminar organised by the program
and the local ANZIAM branch considered probabilistic and
deterministic models of the spead of bushfires, epidemics, and
rumours.