Session Title: High-dimensional numerical integration
Session Slot: 10:30-12:20 Thursday
Estimated Audience Size: 30-50
AudioVisual Request: None or Two Overheads
Theme Session: No
Applied Session: No
Session Organizer: Stein, Michael University of Chicago
Address: Department of Statistics, 5734 University Ave., Chicago, IL 60637
Session Timing: 110 minutes total (Sorry about format):
Opening Remarks by Chair - 0 minutes First Speaker - 30 minutes Second Speaker - 30 minutes Third Speaker - 30 minutes Discussant - 10 minutes Floor Discusion - 10 minutes
Session Chair: Stein, Michael University of Chicago
Address: Department of Statistics University of Chicago 5734 University Ave. Chicago, IL 60637
1. Latin supercube sampling
Owen, Art, Stanford University
Address: Department of Statistics Stanford University Stanford, CA 94305
Abstract: Some very high dimensional integrands arise in problems such as graphical rendering, particle transport, computational finance and queuing systems. The methods of choice for such problems are usually Monte Carlo (MC) or quasi-Monte Carlo (QMC). In theory QMC methods are far more accurate than MC, at least asymptotically, but they can lose their effectiveness in high dimensions. Recent work has centered on using QMC on a small set of key dimensions and MC on the others. This talk considers how to use QMC on more than one set of input dimensions, by randomizing the run order within each set, generalizing the method of Latin hypercube sampling. In combination with methods of engineering the integrand to have more low dimensional structure, the result is to extend the reach of QMC methods to harder problems, including some with infinite dimension. When the QMC methods themselves are randomized it is possible to get a data based estimate of the accuracy of the integration.
2. Adaptive Integration and Analysis of Average Errors
Ritter, Klaus, University of Erlangen-Nurnberg
Address: Mathematisches Institut Universität Erlangen-Nürnberg Bismarckstraße 1 1/2 91054 Erlangen Germany
Müller-Gronbach, Thomas, Freie Universität Berlin
Abstract: We discuss numerical integration of functions with inhomogeneous and a priori unknown local smoothness. For such problems adaptive (sequential) methods are a natural choice. One tries to detect regions of different local smoothness and adjusts further sampling accordingly.
In a worst case analysis, no method can successfully deal with unknown local smoothness. This result holds true under very mild assumptions.
Hence we study average errors with respect to probability measures on function spaces. We present new adaptive algorithms and derive optimality properties in the average case setting. Our analysis proves superiority of adaptive methods and it comes close to some heuristics that are used in the literature.
3. (T,M,S)-Nets and Their Applications in Numerical Integration
Mullen, Gary, Pennsylvania State University
Address: Department of Mathematics The Pennsylvania State University University Park, PA 16802
Abstract: For a variety of applied problems one needs to be able to distribute points as uniformly as possible in the unit cube I = [0,1)s. There are various ways to try and do this but one of the most effective is through the use of (t,m,s)-nets in base b. Such a net is a collection of bm points in I so that each "elementary interval" of volume 1/(bm-t) contains its fair share of points, namely bt.
In this talk we will briefly discuss some of the known constructions for (t,m,s)-nets. As an illustration in numerical integration, we will provide a comparison of the effectivenss of several methods of choosing node points including the use of nets, Monte Carlo methods, and Sobol sequences.
Discussant: Stein, Michael
List of speakers who are nonmembers: Ritter, Mullen