Friday, May 10, 2019 3:00 pm - 3:00 pm EDT
Jaspar Wiart, RICAM, Austrian Academy of Sciences
"An Unexpected Appearance of the Shift"
Monte Carlo integration is a method of estimating integrals particularly well-suited to high dimensions. Simply pick a number of random points, evaluate the function at those points, and average the results. Randomized quasi-Monte Carlo seeks to improve upon this method by choosing the points in a clever way. We pick the points so that each point is uniformly distributed but with a dependence structure that prevents clustering.
Scrambled (0,m,s)-nets are a class of point sets with very nice equidistribution properties. We recently found that these point sets are negatively lower orthant dependent. This result is equivalent to showing that the restriction of a certain continuous linear functional on l1 to a particular subset has norm one. The norm calculation involved a convexity argument and a convenient but unexpected appearance of the shift. Knowing that these nets satisfy this dependence condition allowed us to gain a deeper understanding of when these point sets preform better than Monte Carlo.
With the help of plenty of pictures I will introduce scrambled (0,m,s)-nets and negative lower orthant dependence as well as explain the absurd appearance of the shift in this project.