We study the worst case integration error of combinations of quadrature rules in a reproducing kernel Hilbert space. We show that the error, with respect to the total number N of function evaluations used, cannot decrease faster than O(N^{-1}) in the case where several quasi-Monte Carlo rules are combined to a compound quasi-Monte Carlo rule. However, if the errors of the quadrature rules constituting the compound rule have an order of convergence O(N^{-a}) for a > 1 then, by introducing weights, this order of convergence can be shown to be recovered for the compound rule. We apply our results to the case of lattice sequences.We study the worst case integration error of combinations of quadrature rules in a reproducing kernel Hilbert space. We show that the error, with respect to the total number N of function evaluations used, cannot decrease faster than O(N^{-1}) in the case where several quasi-Monte Carlo rules are combined to a compound quasi-Monte Carlo rule. However, if the errors of the quadrature rules constituting the compound rule have an order of convergence O(N^{-a}) for a > 1 then, by introducing weights, this order of convergence can be shown to be recovered for the compound rule. We apply our results to the case of lattice sequences.

Feb 1, 20090 citations

Weighted QMC rules - Stop anywhere

Dirk Nuyens

We study the worst case integration error of combinations of quadrature rules in a reproducing kernel Hilbert space. We show that the error, with respect to the total number $N$ of function evaluations used, cannot decrease faster than $O(N^{-1})$ for an equal-weight rule. However, if the errors of the quadrature rules constituting a compound rule have an order of convergence $O(N^{-\alpha})$ for $\alpha>1$ then, by introducing weights, this order of convergence can be shown to be recovered for the compound rule. The theory applies to lattice sequences as well as digital sequences.We study the worst case integration error of combinations of quadrature rules in a reproducing kernel Hilbert space. We show that the error, with respect to the total number $N$ of function evaluations used, cannot decrease faster than $O(N^{-1})$ for an equal-weight rule. However, if the errors of the quadrature rules constituting a compound rule have an order of convergence $O(N^{-\alpha})$ for $\alpha>1$ then, by introducing weights, this order of convergence can be shown to be recovered for the compound rule. The theory applies to lattice sequences as well as digital sequences.

Sep 1, 20090 citations

New averaging technique for approximating weighted integrals

Leszek Plaskota, G.W. Wasilkowski, Yunge Zhao

Journal of ComplexityFeb 14, 200912 citations

Finite-order Integration Weights Can be Dangerous

Ian H. Sloan

Abstract Finite-order weights have been introduced in recent years to describe the often occurring situation that multivariate integrands can be approximated by a sum of functions each depending only on a small subset of the variables. The aim of this paper is to demonstrate the danger of relying on this structure when designing lattice integration rules, if the true integrand has components lying outside the assumed finiteorder function space. It does this by proving, for weights of order two, the existence of 3-dimensional lattice integration rules for which the worst case error is of order O(N¯½), where N is the number of points, yet for which there exists a smooth 3- dimensional integrand for which the integration rule does not converge.Abstract Finite-order weights have been introduced in recent years to describe the often occurring situation that multivariate integrands can be approximated by a sum of functions each depending only on a small subset of the variables. The aim of this paper is to demonstrate the danger of relying on this structure when designing lattice integration rules, if the true integrand has components lying outside the assumed finiteorder function space. It does this by proving, for weights of order two, the existence of 3-dimensional lattice integration rules for which the worst case error is of order O(N¯½), where N is the number of points, yet for which there exists a smooth 3- dimensional integrand for which the integration rule does not converge.

Computational Methods in Applied MathematicsJan 1, 20076 citations

We study the worst case integration error of combinations of quadrature rules in a reproducing kernel Hilbert space. We show that the error, with respect to the total number

N

of function evaluations used, cannot decrease faster than

O(N^{-1})

for an equal-weight rule. However, if the errors of the quadrature rules constituting a compound rule have an order of convergence

O(N^{-\alpha})

for

\alpha>1

then, by introducing weights, this order of convergence can be shown to be recovered for the compound rule. The theory applies to lattice sequences as well as digital sequences.