SIAM's UK/Ireland Section Holds Third Annual Meeting at Imperial College

April 22, 1999

A model for groups interested in formng SIAM sections worldwide, the section has held a series of lively meetings covering applied mathematics across a broad spectrum.

Des Higham

The UK and Republic of Ireland (UKIE) SIAM Section held its third annual meeting at Imperial College, London, on January 8. The one-day meeting, which attracted more than 60 attendees, was organized by the section officers---Bill Morton (president), John Dold (vice president), and David Silvester (secretary)---along with local contact John Barrett. The five invited speakers covered a range of topics in applied analysis, numerical analysis, and mathematical modeling.

In the first invited talk, Charlie Elliott (University of Sussex) discussed mean field vortex density models of type II superconductors. After an overview of the hierarchy of mathematical models in the area, he focused on two special two-dimensional mean field models: an elliptic equation coupled to a first-order equation of either the conservation law or the Hamilton-Jacobi type. He presented a range of results for "viscosity" or "order-preserving" solutions and used comparison techniques to describe a convergence theory for a class of numerical methods.

Recent work on partly random graphs was the subject of a lively talk by SIAM president Gil Strang (MIT). Motivated by a paper of Strogatz and Watts in the June 1998 issue of Nature, this work is related to the widely held belief that just about any two people in the world can be joined by at most six links, where each link involves two people who have met. Strang mentioned, by way of example, that because he knows an acquaintance of Bill Clinton's, everybody in the audience had a Monica Lewinsky index of at most four.

Following Strogatz and Watts, Strang and his co-workers have investigated whether highly structured networks with added random links offer a route to this phenomenon. The idea can be tested by three lines of MATLAB code, he explained, and he and his co-workers have also performed some analysis with the help of Maple, Mathematica, and the sequence-fitting "superseeker" package (available on the Internet). A literature search, though, revealed that many of the relevant results are available from a more low-tech source---a 1981 journal article by Brendan McKay.

Section president Bill Morton, who is also vice president of the Institute of Mathematics and its Applications in the UK, considered the role of the section at the business session. Very positive about the benefits of SIAM membership, he also spoke of the opportunities offered by the section to local members. Strang, as SIAM president, also addressed the audience, explaining that he is keen to tap into the strong overseas support enjoyed by SIAM and to make it a truly international society. UKIE SIAM, he said, is an excellent model for a local section; an East Asian section was formed recently, and similar developments are being discussed in other parts of the world. Both Morton and Strang stressed that SIAM sections should be seen as complementing existing national mathematical societies.

Concluding the business meeting, section secretary David Silvester announced the results of the section's recent election: Trevor Stuart (Imperial College) and Des Higham (University of Strathclyde) were elected president and vice president, respectively; their two-year terms begin in April 1999.

After lunch, Martin Stynes (National University of Ireland, Cork) surveyed the numerical solution of convection-dominated convection-diffusion problems. These mathematical problems, which arise, for example, in linearized Navier-Stokes models, offer many challenges to the numerical analyst. In particular, the effective resolution of boundary and interior layers poses serious difficulties.

Stynes explained how standard finite differences and finite elements on uniform grids can come unstuck, and discussed two remedies---Shiskin grids and the streamline-diffusion finite-element method---with numerical examples and analytical error bounds attesting to the benefits of these ideas. Looking to the future, Stynes suggested a third technique---fully adaptive grids---as the most promising approach, despite the current lack of a sound theoretical basis.

Colin Please (University of Southampton) discussed the modeling of avascular, in vitro, spheroidal tumor growth, with experimental observations used to drive the modeling process. A key challenge in this area, he explained, is to account for the development of a necrotic region in the center of a tumor; his model uses force balance laws as a mechanism for introducing this phenomenon. The final model includes a diffusion equation, a porous-medium law, and a conservation law. In assessing the model, Please explained the difficulty of obtaining and interpreting precise experimental data. Future directions, he said, include the numerical solution of the model and the incorporation of realistic cell-life-cycle effects.

In the final talk of the day, Chris Budd (University of Bath) combined modeling, analysis, and numerical simulation to give a mathematical description of single-layer rock folding. Such folding arises, for example, when continents move together and push up the intervening sea bed to form mountains. The physical process involves enormous pressures and temperatures and extremely long time scales.

Working with colleagues in the geosciences, Budd arrived at a final model in the form of a linear fourth-order parabolic equation, with a destabilizing second-order term and a stabilizing fourth-order term, coupled to a nonlinear constraint. The model admits folding at all length scales. Although numerical experiments suggested that the model produces self-similar solutions, a careful analysis revealed the story to be more complicated: Approximate self-similarity, Budd explained, can be established by incorporating a logarithmic correction. Overall, this was a well-organized and well-attended meeting in a series that continues to showcase high-quality applied mathematics across a broad spectrum.

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

Des Higham is a reader in the Department of Mathematics at the University of Strathclyde.


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