Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands

TL;DR

This paper introduces a new randomized quasi-Monte Carlo (RQMC) method using higher order scrambled digital nets that achieves the optimal convergence rate of the root mean square error for smooth integrands, surpassing previous methods.

Contribution

The paper proposes a novel RQMC algorithm based on higher order scrambled digital nets that attains the optimal RMSE convergence rate for integrands with high smoothness, matching known lower bounds.

Findings

Achieves RMSE convergence rate of N^{-eta} with eta > 1/2 for smooth functions.

Numerical examples show RMSE convergence rates of approximately N^{-5/2} and N^{-7/2}.

Theoretical proof confirms the optimality of the convergence rate for the proposed method.

Abstract

We study a random sampling technique to approximate integrals $\int_{[0,1]^s}f(\mathbf{x})\,\mathrm{d}\mathbf{x}$ by averaging the function at some sampling points. We focus on cases where the integrand is smooth, which is a problem which occurs in statistics. The convergence rate of the approximation error depends on the smoothness of the function $f$ and the sampling technique. For instance, Monte Carlo (MC) sampling yields a convergence of the root mean square error (RMSE) of order $N^{-1/2}$ (where $N$ is the number of samples) for functions $f$ with finite variance. Randomized QMC (RQMC), a combination of MC and quasi-Monte Carlo (QMC), achieves a RMSE of order $N^{-3/2+\varepsilon}$ under the stronger assumption that the integrand has bounded variation. A combination of RQMC with local antithetic sampling achieves a convergence of the RMSE of order $N^{-3/2-1/s+\varepsilon}$ (where $s\ge1$ is the dimension) for functions with mixed partial derivatives up to order two. Additional smoothness of the integrand does not improve the rate of convergence of these algorithms in general. On the other hand, it is known that without additional smoothness of the integrand it is not possible to improve the convergence rate. This paper introduces a new RQMC algorithm, for which we prove that it achieves a convergence of the root mean square error (RMSE) of order $N^{-\alpha-1/2+\varepsilon}$ provided the integrand satisfies the strong assumption that it has square integrable partial mixed derivatives up to order $\alpha>1$ in each variable. Known lower bounds on the RMSE show that this rate of convergence cannot be improved in general for integrands with this smoothness. We provide numerical examples for which the RMSE converges approximately with order $N^{-5/2}$ and $N^{-7/2}$, in accordance with the theoretical upper bound.