PDE with Random Coefficients As A Problem in High-Dimensional Numerical Integration By Prof. Ian H. Sloan (University of New South Wales, Australia)
This talk is concerned with the use of quasi-Monte Carlo methods combined with finite-element methods to handle an elliptic PDE with a random field as a coefficient. A number of groups have considered such problems (under headings such as polynomial chaos, stochastic Galerkin and stochastic collocation) by reformulating them as deterministic problems in a high dimensional parameter space, where the dimensionality comes from the number of random variables needed to characterize the random field.
Overview
Abstract
This talk is concerned with the use of quasi-Monte Carlo methods combined with finite-element methods to handle an elliptic PDE with a random field as a coefficient. A number of groups have considered such problems (under headings such as polynomial chaos, stochastic Galerkin and stochastic collocation) by reformulating them as deterministic problems in a high dimensional parameter space, where the dimensionality comes from the number of random variables needed to characterize the random field. In recent joint work with Christoph Schwab (ETH) and Frances Kuo (UNSW) we have treated a problem of this kind as one of (truncated) infinite-dimensional integration - where integration arises because a multi-variable expected value is a multidimensional integral - together with finite-element approximation in the physical space. We use recent developments in the theory and practice of quasi-Monte Carlo integration rules in weighted Hilbert spaces, through which rules with optimal properties can be constructed once the weights are known. The novel feature of this work is that for the first time we are able to design weights for the weighted Hilbert space that achieve what is believed to be the best possible rate of convergence, under conditions on the random field that are exactly the same as in a recent paper by Cohen, DeVore and Schwab on best N-term approximation for the same problem.
Brief Biography
Ian Sloan completed physics and mathematics degrees at Melbourne University, a Master's degree in mathematical physics at Adelaide University, and a PhD in theoretical atomic physics at the University of London. After a decade of research on few-body collision problems in nuclear physics, and publishing some 35 papers in the physics literature, his main research interests shifted to computational mathematics . Since making that change he has published 200 papers on the numerical solution of integral equations, numerical integration and interpolation, boundary integral equations, approximation theory, multiple integration, continuous complexity theory and other parts of numerical analysis and approximation theory. He is a Fellow of the Australian Academy of Science, and of both SIAM and the American Mathematical Society. In 2001 was awarded the Lyle Medal of the Australian Academy of Science, and in 2005 was awarded the Information Based Complexity Prize. In 2008 he was appointed an Officer of the Order of Australia (AO).
He is a member of the editorial board SIAM Journal of Numerical Analysis, Numerische Mathematik, Advances in Computational Mathematics, Journal of Integral Equations and Applications, and the International Journal of Geomathematics, and is a Senior Editor of the Journal of Complexity.
From 2003 to 2007 he was President of the International Council for Industrial and Applied Mathematics.
