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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Explicit error bounds for randomized Smolyak algor
ithms and an application to infinite-dimensional i
ntegration - Michael Gnewuch (Christian-Albrechts-
Universität zu Kiel)
DTSTART;TZID=Europe/London:20190221T110000
DTEND;TZID=Europe/London:20190221T113500
UID:TALK120193AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/120193
DESCRIPTION:Smolyak'\;s method\, also known as hyperb
olic cross approximation or sparse grid method\, i
s a powerful %black box tool to tackle multivariat
e tensor product problems just with the help of ef
ficient algorithms for the corresponding univariat
e problem. We provide upper and lower error bounds
for randomized Smolyak algorithms with fully expl
icit dependence on the number of variables and the
number of information evaluations used. The error
criteria we consider are the worst-case root mean
square error (the typical error criterion for ran
domized algorithms\, often referred to as ``random
ized error'\;'\;) and the root mean square w
orst-case error (often referred to as ``worst-case
error'\;'\;). Randomized Smolyak algorithms
can be used as building blocks for efficient meth
ods\, such as multilevel algorithms\, multivariate
decomposition methods or dimension-wise quadratur
e methods\, to tackle successfully high-dimensiona
l or even infinite-dimensional problems. As an exa
mple\, we provide a very general and sharp result
on infinite-dimensional integration on weighted re
producing kernel Hilbert spaces and illustrate it
for the special case of weighted Korobov spaces. W
e explain how this result can be extended\, e.g.\,
to spaces of functions whose smooth dependence on
successive variables increases (``spaces of incre
asing smoothness'\;'\;) and to the problem o
f L_2-approximation (function recovery).

LOCATION:Seminar Room 1\, Newton Institute
CONTACT:info@newton.ac.uk
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