THE REES DISTINGUISHED LECTURER SERIES
Prof. Ian H. Sloan
University of New South Wales
Sydney, Australia
Approximating and Designing on the Sphere
Thursday September 29, 2005
006 Kirkbride Hall
3:40 - 5:00
The sphere is an important setting for applied mathematics, with the applications ranging from geodesy through climate change to scattering in the ionosphere. Yet there are many surprising gaps in our understanding of mathematics on the sphere, and even simple questions often go unanswered. Many questions revolve around nding good distributions of points on the sphere. Questions of this kind arise in coding theory, numerical integration, and minimal energy configurations.
This talk, intended for non-specialists, reviews recent work on interpolation, approximation and numerical integration on the sphere S2, and related questions of how to choose 'good' point distributions on the sphere. Special focus will be on extremal systems, extremal spherical designs, and optimal cubature.
Refreshments served immediately after talk in Ewing 436.
Friday September 30, 2005
104 Gore Hall
3:45
- 5:00
Numerical integration in high dimensions confronts us with the curse of dimensionality -- the number of function values needed to obtain an acceptable approximation can grow exponentially in the number of dimensions d. The exponential increase is clearly inevitable with any form of product integration rule, and for many theoretical settings is now known to be unavoidable no matter how the integration rule is chosen.
It has been known since 1998 that the curse of dimensionality can in principle be overcome within the "weighted Sobolev space" setting introduced by Sloan and Wózniakowki, if the "weights" that describe the behaviour with respect to different variables satisfy a certain (necessary and sufficient) condition. In that work it was show that, under the appropriate condition on the weights, there exist integration rules for which the "worst-case error" is bounded independently of d. That 1998 result was non-constructive, giving no clue as to how we might construct "good" integration rules. More recently it has been shown that "good" rules can be found within the much smaller class of (shifted) lattice rules, and even more recently that good rules can be constructed one component at a time.
This talk will review these developments, from early existence proofs and non-constructive methods to recent fast constructions of good integration rules in hundreds of dimensions, that may use hundreds of thousands of sample points.
Click here for more information on the Rees Lecture Series and Prof Sloan.