Ian Hugh Sloan

Brief Bio

Ian Sloan completed physics and mathematics degrees at Melbourne University, a Master's degree in mathematical physics at Adelaide, and a PhD in theoretical atomic physics at the University of London, finishing in 1964.

He joined the University of New South Wales (UNSWAustralia) in 1965, and was appointed to a Personal Chair in Mathematics in 1983.

After a decade of research on few-body collision problems in nuclear physics, his research interests shifted to computational mathematics. Since making that change he has published some 200 papers on the numerical solution of integral equations, numerical integration and interpolation, boundary integral equations, approximation theory, continuous complexity theory, high dimensional integration, and other parts of numerical analysis and approximation theory. He is an ISI Highly Cited Author in Mathematics, and has won prizes, including the Information Based Complexity Prize and the Lyle Medal of the Australian Academy of Science.

He is a member of the editorial boards of Numerische Mathematik, Advances in Computational Mathematics, Journal of Integral equations and Applications, International Journal of Geomathematics, Foundations of Computational Mathematics, Chinese Journal of Engineering Mathematics, Computational Methods in Applied Mathematics, and Arabian Journal of Mathematics, and is a Senior Editor of the Journal of Complexity.

He is a Fellow of the Australian Academy of Science and the Royal Society of New South Wales. He was appointed a Fellow of both SIAM and the American Mathematical Society in their inaugural rounds.

From 2003 to 2007 he was President of the International Council for Industrial and Applied Mathematics.

Current Research Field

Ian Sloan’s current research mainly spans two broad areas of computational mathematics. First, he is centrally involved with modern developments of high-dimensional integration and approximation. This field was once the preserve of number theorists, but in recent decadesis attracting wide research interest as toolsare developed that can be applied to practical problems. High-dimensional problems (that is problems with the number of continuous variables ranging from ten or twenty to the hundreds of thousands) are increasingly encountered in practice. Some fields in which high-dimensional problems arise include mathematical finance; modelling of disease presentations in a community; and modelling the flow of subsurface water through a complex material considered as a random medium. Many other high dimensional problems occur in molecular physics and chemistry, for example in the Schrödinger equation for a multi-electron system.

A general characteristic of high-dimensional problems is that they are hard!The workhorse for such problems is the Monte Carlo method, which relies on randomness, but while it is admirably robust and flexible, the Monte Carlo rate of convergence is often too slow. A central concern of this area of research might be said to be to devise methods that can be provedto be better than the Monte Carlo method for particular classes of problems.

Second, he is involved with many aspects of approximation and modelling on manifolds, including polynomial and radial basis function approximations on spheres (including multiscale variants). Problems of this kind often arise in geophysical contexts, given the (approximate) spherical nature of Earth’s surface. Special problems arise because many measurements of geophysical importance are made not on Earth’s surface but on satellite orbits, raising interesting mathematical questions about mapping the measured data from the orbit to ground level. There are also many interesting questions about numerical integration over spheres and manifolds, not least because for spheres, with a small number of exceptions (e.g. the vertices of Platonic solids) there are NO point distributions on spheres that are convincingly regular. One theme of Ian Sloan’s research is the deep relation between quadrature and energy of point distributions on spheres. For instance, is it true (as many surmise) that point sets that have minimal Coulomb energy on the sphere are good point sets for (egequal-weight) numerical integration? (The answer is in general no.)

Personal Homepage : maths.unsw.edu.au/~sloan