Past Event: Oden Institute Seminar

High dimensional approximation and the curse of dimensionality

Ian Sloan, Professor, Mathematics, University of New South Wales

Abstract

High dimensional approximation problems commonly arise from parametric PDE problems in which an input random field depends on very many univariate random variables.  Typically (for example, in the method of “generalized polynomial chaos”, or GPC) the dependence on these variables is modelled by multivariate polynomials, leading to exponentially increasing difficulty and cost  (often expressed as the “curse of dimensionality”) as the dimension increases.  This is why sparsity of coefficients is a major focus in any implementation of  GPC.

In this lecture we develop a different approach to one version of GPC, in which there is no need for sparsification, and no curse of dimensionality.    The method, proposed in a 2022 paper with Frances Kuo, Vesa Kaarnioja, Yoshihito Kazashi and Fabio Nobile uses as the approximation a linear combination of periodic kernels, with the kernels located at lattice points, as advocated long ago by Hickernell and colleagues. The advantage is that the cost grows merely quadratically with dimension, giving no cause to appeal to sparsity of coefficients.  Numerical experiments in that paper for a parametric diffusion problem showed that computations with 10 or 100 stochastic dimensions are feasible.  In more recent work with Kuo and Kaarnioja we have been able to reduce the quadratic cost to merely linear in the dimension,  so making calculations with even 1,000 dimensions feasible.

Biography

Ian Sloan, an Australian, began his long research career in theoretical physics, with a PhD from the University of London and a decade of research into the quantum physics of scattering of particles from atoms and nuclei.  Since then he has contributed to many areas of computational mathematics, including the numerical solution of integral equations, constructive approximation, and high dimensional problems.  In 2015 he gave an invited lecture at the ICIAM Congress in Beijing.  He is a foundation Fellow of both SIAM and the American Mathematical Society, a Fellow of the Australian Academy of Science, and a former President of the International Council of Industrial and Applied Mathematics.

He is a member of the editorial board of a number of international journals, including SIAM Journal of Numerical Analysis, Numerische Mathematik, Advances in Conmputational Mathematics, Journal of Integral Equations and Applications and the new International Journal of Geomathematics, and is a Senior Editor of the Journal of Complexity.

His current research interests are in boundary integral methods, finite element methods, high dimensional numerical integration and related issues of information-based complexity, multivariate approximation theory, and the time discretisation of evolution problems.