Abstract: Since the mid-twentieth century invention of Quasi-Monte Carlo integration by number theorists, there have always been two versions of QMC: the periodic, associated with Korobov, Bahvalov, Zaremba, Hua and others; and the non-periodic, associated with Halton, Hlawka, Sobol’, and more recently Niederreiter, Dick and others. The periodic setting offers the profound benefits of Fourier series, but has been lacking in serious high-dimensional applications. After a review of the periodic setting, I will describe the recent development of a periodic QMC method for high-dimensional approximation in the context of uncertainty quantification. In this problem an elliptic partial differential equation with a random input field is modelled in a non-standard way with periodic random variables. The claim to high-dimensional applicability is based on a computational cost that grows only linearly with dimensionality. Collaborators in this work are Vesa Kaarnioja, Yoshihito Kazashi, Frances Kuo and Fabio Nobile. The work rests, of course, on the contributions of numerous others, including Wozniakowski, Hickernell, Cools, Nuyens, …
After schooling in Ballarat in Victoria, Australia, 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 (under the supervision of HSW Massey) at the University of London, finishing in 1964. 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 was employed by Australia's CSR Company from 1961 to 1965, before joining the University of New South Wales as a Lecturer. After several promotions, he was appointed to a Personal Chair in Mathematics in 1983. He was Head of the School of Mathematics from 1986 to 1990 and from 1992 to 1993. He completed a term as Chair of the Chemistry, Mathematics and Physics Panel of the Australian Research Council and member of the ARC's Research Grants Committee, and is a former President of the Australian Mathematical Society .
He was elected a Fellow of the Australian Academy of Science in 1993. In 1997 he was awarded the ANZIAM Medal by Australian and New Zealand Industrial and Applied Mathematics(ANZIAM), and in 2001 was awarded the Thomas Ranken Lyle Medal of the Australian Academy of Science. In 2002 he was awarded the Szekeres Medal of the Australian Mathematical Society, 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 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.
From 2003 to 2007 he was President of the International Council for Industrial and Applied Mathematics, Before that he was the Chair of the International Program Committee for ICIAM 2003, the fifth International Congress on Industrial and Applied Mathematics, held in Sydney in 2003. He is currently Deputy Director of MASCOS, the ARC Centre of Excellence for Mathematics and Statistics of Complex Sysytems.
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