477 research outputs found
On Integration Methods Based on Scrambled Nets of Arbitrary Size
We consider the problem of evaluating for a function . In situations where
can be approximated by an estimate of the form
, with a point set in
, it is now well known that the Monte Carlo
convergence rate can be improved by taking for the first
points, , of a scrambled
-sequence in base . In this paper we derive a bound for the
variance of scrambled net quadrature rules which is of order
without any restriction on . As a corollary, this bound allows us to provide
simple conditions to get, for any pattern of , an integration error of size
for functions that depend on the quadrature size . Notably,
we establish that sequential quasi-Monte Carlo (M. Gerber and N. Chopin, 2015,
\emph{J. R. Statist. Soc. B, to appear.}) reaches the
convergence rate for any values of . In a numerical study, we show that for
scrambled net quadrature rules we can relax the constraint on without any
loss of efficiency when the integrand is a discontinuous function
while, for sequential quasi-Monte Carlo, taking may only
provide moderate gains.Comment: 27 pages, 2 figures (final version, to appear in The Journal of
Complexity
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