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SUMMARY:Optimal Confidence for Monte Carlo Integration of Smooth Functions
  - Robert J. Kunsch (Universität Osnabrück)
DTSTART:20190221T134000Z
DTEND:20190221T141500Z
UID:TALK120199@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:We study the complexity $n(\\varepsilon\,\\delta)$ of approxim
 ating the integral of smooth functions at absolute precision $\\varepsilon
  &gt\; 0$ with confidence level $1 - \\delta \\in (0\,1)$ using function e
 valuations as information within randomized algorithms. Methods that achie
 ve optimal rates in terms of the root mean square error (RMSE) are not alw
 ays optimal in terms of error at confidence\, usually we need some non-lin
 earity in order to suppress outliers. Besides\, there are numerical proble
 ms which can be solved in terms of error at confidence but no algorithm ca
 n guarantee a finite RMSE\, see [1]. Hence\, the new error criterion seems
  to be more general than the classical RMSE. The sharp order for multivari
 ate functions from classical isotropic Sobolev spaces $W_p^r([0\,1]^d)$ ca
 n be achieved via control variates\, as long as the space is embedded in t
 he space of continuous functions $C([0\,1]^d)$. It turns out that the inte
 grability index $p$ has an effect on the influence of the uncertainty $\\d
 elta$ to the complexity\, with the limiting case $p = 1$ where determinist
 ic methods cannot be improved by randomization. In general\, higher smooth
 ness reduces the effort we need to take in order to increase the confidenc
 e level. Determining the complexity $n(\\varepsilon\,\\delta)$ is much mor
 e challenging for mixed smoothness spaces $\\mathbf{W}_p^r([0\,1]^d)$. Whi
 le optimal rates are known for the classical RMSE (as long as $\\mathbf{W}
 _p^r([0\,1]^d)$ is embedded in $L_2([0\,1]^d)$)\, see [2]\, basic modifica
 tions of the corresponding algorithms fail to match the theoretical lower 
 bounds for approximating the integral with prescribed confidence. <br><br>
 <i>Joint work with Daniel Rudolf&nbsp\;</i><br><br>[1]&nbsp\; R.J. Kunsch\
 , E. Novak\, D. Rudolf.&nbsp\;Solvable integration problems and optimal sa
 mple size selection.&nbsp\;<span>To appear in Journal of Complexity.<br></
 span>[2]&nbsp\; M. Ullrich.&nbsp\;A Monte Carlo method for integration of 
 multivariate smooth functions.&nbsp\;SIAM Journal on Numerical Analysis\, 
 55(3):1188-1200\, 2017.<br><br><br>
LOCATION:Seminar Room 1\, Newton Institute
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