Approximation of High-Dimensional Numerical Problems - Algorithms, Analysis and Applications (15w5047)
Christiane Lemieux (University of Waterloo)
Ian H. Sloan (University of New South Wales)
Henryk Wozniakowski (Columbia University)
With the rapid growth in computational power over the last few years, problems of larger and larger scale are more and more feasible in many application areas, but the bottleneck is often the 'curse of dimensionality' - the tendency for problem difficulty to grow exponentially in the number of variables. This meeting, held after a decade of rapid theoretical advance, will provide a timely opportunity for analysts, algorithm designers and application experts to gain a clearer vision of current challenges and opportunities. The intense interactions between researchers of different background should lead to substantial progress in our ability to tackle problems involving complex models, which in turn could have a significant impact on various scientific disciplines, given the extent by which some of the tools we propose to study (e.g., Monte Carlo, quasi-Monte Carlo, sparse-grids methods) are used.
More specifically, the objectives of this workshop are:
(1) To bring together researchers in a range of theoretical disciplines concerned with the numerical analysis of high-dimensional approximation, alongside researchers involved with applications of high-dimensional problems. The theoretical tools include quasi-Monte Carlo and sparse grid methods (both of which have seen a spectacular flowering in the last decade); Monte Carlo methods; and Information-Based Complexity (the science of establishing what is the best possible performance for any algorithm solving a given problem). The application areas include finance, flow of water or oil through a porous medium modelled as a random field, climate modelling; stochastic PDE's; and more generally the burgeoning field of uncertainty quantification, since uncertainty with many degrees of freedom is a major source of high dimensionality.
(2) To identify key challenges that prevent the theoretical advances in high dimensional algorithms from being applied in practice. As a concrete example, in the now popular use of 'weights' to characterize the importance of different variables or groups of variables in a high-dimensional space, how in practice should one choose the weights for particular applications? Very recently progress has been made for a problem of QMC methods applied to PDEs with random coefficients [KSS12]. How can this progress be extended to other applications? On an even more practical side, as part of this objective we plan to have a discussion of the currently available software for implementing the algorithms in question (including quasi-Monte Carlo methods), with the goal being to understand what is used in practice and whether or why not the newest developments in theory have filtered through to practical use.
(3) To expose young researchers from Canada and elsewhere to the latest developments in this rapidly moving field of high-dimensional problems. In addition, given that most of the regular conferences for researchers in Monte Carlo and quasi-Monte Carlo methods are quite large, this smaller workshop will provide a more appropriate setting for them to exchange their most recent ideas, while at the same time, allowing new ideas to be generated.