Frances Kuo | University of New South Wales | Australia
Quasi-Monte Carlo (QMC) methods offer tailored point constructions for solving high dimensional integration and approximation problems by sampling. In recent years the modern QMC theory has been successfully applied to computational models in the field of Uncertainty Quantification (UQ) involving partial differential equations (PDEs) with random coefficients.
This talk will showcase new and ongoing works where we take QMC methods to new territories in UQ, including (i) modeling random fields by periodic random variables, (ii) moving QMC from integration to function approximation with kernel methods, and (iii) applying QMC to PDE-constrained optimization under uncertainty. Periodic random variables are naturally connected with a family of QMC methods called ``lattice rules'', which are very easy to implement and proven to achieve higher order convergence rates with dimension- independent error bounds. Lattice points combined with a periodic reproducing kernel means that the associated linear system in a kernel method involves a circulant matrix so can be solved very efficiently using the fast Fourier transform, making this strategy rather competitive compared to other approximation methods. One advantage of using QMC in optimal control over other methods such as sparse grid quadrature is that they have positive quadrature weights, preserving convexity of the optimization problem. The need to work with coupled PDE systems (the ``KKT system'') leads to novel and challenging analysis.
The talk is based on joint works with Vesa Kaarnioja and Ian H. Sloan (UNSW Sydney), Ronald Cools and Dirk Nuyens (KU Leuven), Yoshihito Kazashi and Fabio Nobile (EPFL), and Philipp Guth and Claudia Schillings (U Mannheim).