BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Talks.cam//talks.cam.ac.uk// X-WR-CALNAME:Talks.cam BEGIN:VEVENT SUMMARY:Explicit error bounds for randomized Smolyak algorithms and an app lication to infinite-dimensional integration - Michael Gnewuch (Christian- Albrechts-Universität zu Kiel) DTSTART:20190221T110000Z DTEND:20190221T113500Z UID:TALK120193@talks.cam.ac.uk CONTACT:INI IT DESCRIPTION:Smolyak'\;s method\, also known as hyperbolic cross a pproximation or sparse grid method\, is a powerful %black box tool to tack le multivariate tensor product problems just with the help of efficient al gorithms for the corresponding univariate problem. We provide upper and lo wer error bounds for randomized Smolyak algorithms with fully explicit dep endence on the number of variables and the number of information evaluatio ns used. The error criteria we consider are the worst-case root mean squar e error (the typical error criterion for randomized algorithms\, often ref erred to as ``randomized error'\;'\;) and the root mean square worst -case error (often referred to as ``worst-case error'\;'\;). Randomi zed Smolyak algorithms can be used as building blocks for efficient method s\, such as multilevel algorithms\, multivariate decomposition methods or dimension-wise quadrature methods\, to tackle successfully high-dimensiona l or even infinite-dimensional problems. As an example\, we provide a very general and sharp result on infinite-dimensional integration on weighted reproducing kernel Hilbert spaces and illustrate it for the special case o f weighted Korobov spaces. We explain how this result can be extended\, e. g.\, to spaces of functions whose smooth dependence on successive variable s increases (``spaces of increasing smoothness'\;'\;) and to the pro blem of L_2-approximation (function recovery).
 LOCATION:Seminar Room 1\, Newton Institute END:VEVENT END:VCALENDAR