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SUMMARY:Approximate and integrate: Variance reduction in Monte Carlo integ
 ration via function approximation
DTSTART:20180706T111500
DTEND:20180706T121500
DTSTAMP:20260407T010625Z
UID:996fe9160a869cb770c6bc58f462462ed027dccce7e777bdede80d5f
CATEGORIES:Conferences - Seminars
DESCRIPTION:Prof. Yuji Nakatsukasa (National Institute of Informatics\, To
 kyo\, Japan)\nClassical algorithms in numerical analysis for numerical int
 egration\n(quadrature/cubature) follow the principle of approximate and in
 tegrate:\nthe integrand is approximated by a simple function (e.g. a polyn
 omial)\, which is then integrated exactly. In high-dimensional integration
 \, such methods quickly become infeasible due to the curse of dimensionali
 ty. A common alternative is the Monte Carlo method (MC)\, which simply tak
 es the average of random samples\, improving the estimate as more and more
  samples are taken. The main issue with MC is its slow (though\ndimension-
 independent) convergence\, and various techniques have been proposed to re
 duce the variance. In this talk we suggest a numerical analyst's interpret
 ation of MC: it approximates the integrand with a constant function\, and 
 integrates that constant exactly. This observation leads naturally to MC-l
 ike methods where the approximant is a non-constant function\, for example
  low-degree polynomials\, sparse grids or low-rank functions. We show that
  these methods have the same O(1/sqrt(N)) asymptotic convergence as in MC\
 , but with reduced variance\, equal to the quality of the underlying funct
 ion approximation. We also discuss methods that improve the approximation 
 quality as more samples are taken\, and thus can converge faster than O(1/
 sqrt(N)). The main message is that techniques in high-dimensional approxim
 ation theory can be combined with Monte Carlo integration to accelerate co
 nvergence.
LOCATION:MA A1 12 https://plan.epfl.ch/theme/generalite_thm_plan_public?re
 quest_locale=en&room=MA%2BA1%2B12&domain=places&dim_floor=1&lang=fr&dim_la
 ng=fr&tree_groups=centres_nevralgiques%2Cacces%2Cmobilite_reduite%2Cen
STATUS:CONFIRMED
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