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On Integration Methods Based on Scrambled Nets of Arbitrary Size

Abstract

We consider the problem of evaluating I(φ)=\uisφ(\bx)\dxI(\varphi)=\int_{\ui^s}\varphi(\bx) \dx for a function φL2[0,1)s\varphi \in L_{2}[0,1)^{s}. In situations where I(φ)I(\varphi) can be approximated by an estimate of the form N1n=0N1φ(\bxn)N^{-1}\sum_{n=0}^{N-1}\varphi(\bx^n), with {\bxn}n=0N1\{\bx^n\}_{n=0}^{N-1} a point set in \uis\ui^s, \citet{Owen1995, Owen1997a,Owen1997b,Owen1998} shows that the \bigOP(N1/2)\bigO_P(N^{-1/2}) Monte Carlo convergence rate can be improved by taking for {\bxn}n=0N1\{\bx^n\}_{n=0}^{N-1} the first N=λbmN=\lambda b^m points, 1λ<b1\leq\lambda<b, of a scrambled (t,s)(t,s)-sequence in base b2b\geq 2. For more complex integration problems, \citet{Gerber2014} have recently developed a sequential quasi-Monte Carlo (SQMC) algorithm which has an error of size \smalloP(N1/2)\smallo_P(N^{-1/2}) for bounded and continuous functions φ\varphi, when it uses the first N=λbmN=\lambda b^m points of scrambled (t,s)(t,s)-sequences as inputs. In this paper we extend these results by relaxing the constraint on NN. Our main contribution is to provide a bound for the variance of scrambled net quadratures which is of order \smallo(N1)\smallo(N^{-1}), without any restriction on NN. This bound allows us to provide simple conditions to get an integration error of size \smalloP(N1/2)\smallo_P(N^{-1/2}) for functions that depend on the quadrature size NN and, as a corollary, to establish that SQMC reaches the \smalloP(N1/2)\smallo_P(N^{-1/2}) convergence rate for any patterns of NN. Finally, we show in a numerical study that for scrambled net quadrature rules we can relax the constraint on NN without any loss of efficiency when the integrand φ\varphi is a discontinuous function while, for the univariate version of SQMC, taking N=λbmN=\lambda b^m may only provide moderate gains.

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