BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//talks.cam.ac.uk//v3//EN
BEGIN:VTIMEZONE
TZID:Europe/London
BEGIN:DAYLIGHT
TZOFFSETFROM:+0000
TZOFFSETTO:+0100
TZNAME:BST
DTSTART:19700329T010000
RRULE:FREQ=YEARLY;BYMONTH=3;BYDAY=-1SU
END:DAYLIGHT
BEGIN:STANDARD
TZOFFSETFROM:+0100
TZOFFSETTO:+0000
TZNAME:GMT
DTSTART:19701025T020000
RRULE:FREQ=YEARLY;BYMONTH=10;BYDAY=-1SU
END:STANDARD
END:VTIMEZONE
BEGIN:VEVENT
CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Optimal Confidence for Monte Carlo Integration of
Smooth Functions - Robert J. Kunsch (UniversitÃ¤t O
snabrÃ¼ck)
DTSTART;TZID=Europe/London:20190221T134000
DTEND;TZID=Europe/London:20190221T141500
UID:TALK120199AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/120199
DESCRIPTION:We study the complexity $n(\\varepsilon\,\\delta)$
of approximating the integral of smooth functions
at absolute precision $\\varepsilon >\; 0$ with
confidence level $1 - \\delta \\in (0\,1)$ using
function evaluations as information within randomi
zed algorithms. Methods that achieve optimal rates
in terms of the root mean square error (RMSE) are
not always optimal in terms of error at confidenc
e\, usually we need some non-linearity in order to
suppress outliers. Besides\, there are numerical
problems which can be solved in terms of error at
confidence but no algorithm can guarantee a finite
RMSE\, see [1]. Hence\, the new error criterion s
eems to be more general than the classical RMSE. T
he sharp order for multivariate functions from cla
ssical isotropic Sobolev spaces $W_p^r([0\,1]^d)$
can be achieved via control variates\, as long as
the space is embedded in the space of continuous f
unctions $C([0\,1]^d)$. It turns out that the inte
grability index $p$ has an effect on the influence
of the uncertainty $\\delta$ to the complexity\,
with the limiting case $p = 1$ where deterministic
methods cannot be improved by randomization. In g
eneral\, higher smoothness reduces the effort we n
eed to take in order to increase the confidence le
vel. Determining the complexity $n(\\varepsilon\,\
\delta)$ is much more challenging for mixed smooth
ness spaces $\\mathbf{W}_p^r([0\,1]^d)$. While opt
imal rates are known for the classical RMSE (as lo
ng as $\\mathbf{W}_p^r([0\,1]^d)$ is embedded in $
L_2([0\,1]^d)$)\, see [2]\, basic modifications of
the corresponding algorithms fail to match the th
eoretical lower bounds for approximating the integ
ral with prescribed confidence.

*Joint w
ork with Daniel Rudolf \;*

[1] \
; R.J. Kunsch\, E. Novak\, D. Rudolf. \;Solvab
le integration problems and optimal sample size se
lection. \;To appear in Journal of Compl
exity.

[2] \; M. Ullrich. \;A Mo
nte Carlo method for integration of multivariate s
mooth functions. \;SIAM Journal on Numerical A
nalysis\, 55(3):1188-1200\, 2017.

LOCATION:Seminar Room 1\, Newton Institute
CONTACT:INI IT
END:VEVENT
END:VCALENDAR