## UK and Republic of Ireland SIAM Section Holds Fifth Annual Meeting in Glasgow

**March 9, 2001**

Mark Davis of Imperial College, recently returned to academia from the finance industry, gave an invited presentation on option pricing and hedging strategies.

**Andy Wathen**

The fifth annual meeting of the UK and Republic of Ireland (UKIE) SIAM Section---the first SIAM meeting of the millennium (depending on how you count!)---attracted over 40 applied and industrial mathematicians to the University of Strathclyde in Glasgow, Scotland, on Friday, January 5. Participants heard an interesting and varied collection of presentations.

Section officers Trevor Stuart (president, Imperial College), Des Higham (vice president, University of Strathclyde), and Peter Jimack (secretary and treasurer, University of Leeds) should be congratulated for arranging speakers across such a varied cross section of the discipline. This aspect of generality and nonspecialism has quickly become a hallmark of this event and is much appreciated by the many section members whose main other exposure to the broad spectrum of research in industrial and applied maths is through the SIAM annual meeting.

The venue for the event was the rather grand Court/Senate Suite at the University of Strathclyde, which provided an ideal small lecture theatre for the presentations. After introductory comments by the section president, Tim Goodman (University of Dundee) got the program under way with a discussion of results in approximation theory that relate to refinable functions. It is well known that B-splines, when suitably scaled, converge to a Gaussian; of interest are the conditions under which a general refinable function has this property of "asymptotic normality" and the rate at which the convergence takes place. Goodman gave a range of results, some of which had been proved in weaker form by researchers in probability theory. He also explained how the results relate to Heisenberg's uncertainty principle and mentioned applications to signal processing.

Howard Elman (University of Maryland) described joint work with Alison Ramage (University of Strathclyde) on the numerical solution of advection-diffusion partial differential equation problems in two dimensions. The oscillatory nature of Galerkin finite-element or centered finite-difference approximations is well understood in a one-dimensional (ODE) setting, with the onset of oscillations being triggered when the characteristic grid Peclet number is greater than unity. Using discrete Fourier analysis techniques, Elman showed that in the two-dimensional case, oscillations can exist perpendicular to the flow direction even when the associated mesh Peclet number is less than one.

*Invited speaker Howard Elman of the University of Maryland sparked lively discussion with his presentation on the optimal stabilisation parameter for streamline diffusion discretisations in multidimensional settings. Elsewhere in this issue, Elman---as editor-in-chief of *SIAM Journal on Scientific Computing*---looks into the high level of interest in iterative methods for nonsymmetric linear systems revealed by a recent study of scientific journals.*

Elman also made use of this analysis to shed light on the thorny issue of the optimal stabilisation parameter for streamline diffusion discretisations in multidimensional settings. In particular, he identified two parameter choices, corresponding to (a) the most accurate discrete solution and (b) the smallest value precluding the onset of oscillations. The merits of these alternative philosophies, and the presence of other international experts in the audience, provided for an interesting discussion.

During the business session, there was general approval of the way funds had been dispersed to support applied and industrial mathematics in the UK and RoI, and in the ensuing discussion it was agreed to look further at the small amount of the budget spent on student prizes. Since the current section president happens to be the president of the London Mathematical Society as well, it was perhaps forseeable that he would be questioned about possible reciprocity between the two societies; we await further development. . . .

The two-year terms of the president and vice president end in March 2001, and the secretary announced the results of the e-mail election for new officers. David Parker (University of Edinburgh) will be the next section president, and Ivan Graham (University of Bath) will serve as the next vice president, each for a two-year term beginning in April 2001.

After lunch, Valerie Isham (University College, London) discussed spatiotemporal processes in hydrology. Based on radar and rain gauge data, both stochastic, point process models and empirical multivariate statistical models were developed. She illustrated the techniques with an interesting application: the prediction of rainfall patterns, and consequent flood damage, for housing in Galway, Ireland.

Mark Davis (Imperial College), who had recently returned to academia from the financial sector, demonstrated the relevance of mathematical analysis even in the hectic world of option pricing. The work he described arises from a practical constraint: Although it is often impossible to trade the asset on which an option is written, a highly correlated commodity can be traded. If the correlation is not perfect, he asked, what is a good hedging strategy?

Davis defined the problem as maximizing a utility function. By using a dual formulation, he was able to express the problem in terms of stochastic control under a quadratic penalty function; an elegant approach to computing a numerical solution based on an asymptotic expansion followed.

In the final talk of the day, Alex Craik (University of St. Andrews) described a system of three nonlinear ordinary differential equations arising in work on exact solutions of the Navier-Stokes equations. A special case of the system, which arises in the context of a rotating rigid body, had been dealt with in a masterly manner by Euler, using Jacobian elliptic functions. The more general system studied by Craik can give rise to finite-time blowup and spring-like solutions. A complicated four-leaf manifold of periodic solutions was illustrated numerically; in conjunction with two saddle points, this manifold was seen to organise the behaviour of the spring-like solutions.

About half of the attendees stayed for the lively evening dinner---very much in keeping with the New Year spirit! Overall, this was a friendly and thought-provoking meeting---a good way to see some of the active research being pursued in the UK and RoI, to catch up with colleagues, and to make new aquaintances. The next meeting will be held in January 2002, venue to be decided.

*Further details about the UKIE SIAM section can be found at **http://www.ma.hw.ac.uk/~gabriel/ukiesiam/**.*

*Andy Wathen, a reader in the Oxford University Computing Lab, thanks Des Higham and David Silvester for much help in the writing of this report.*