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

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

2010-07-06

arxiv.org/abs/1007.0842v3

Mathematics

Numerical Analysis

Scientific paper

We study numerical approximations of integrals $\int_{[0,1]^s} f(\boldsymbol{x}) \,\mathrm{d} \boldsymbol{x}$ by averaging the function at some sampling points. 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). Quasi-Monte Carlo (QMC) sampling on the other hand achieves a convergence of order $N^{-1+\varepsilon}$, for any $\varepsilon >0$. Randomized QMC (RQMC), a combination of MC and QMC, achieves a RMSE of order $N^{-3/2+\varepsilon}$. A combination of RQMC with local antithetic sampling achieves a convergence of the RMSE of order $N^{-3/2-1/s+\varepsilon}$ (where $s \ge 1$ is the dimension). QMC, RQMC and RQMC with local antithetic sampling require that the integrand has some smoothness (for instance, bounded variation). Stronger smoothness assumptions on the integrand do not improve the convergence of the above algorithms further. This paper introduces a new RQMC algorithm, for which we prove that it achieves a convergence of the RMSE of order $N^{-\alpha-1/2+\varepsilon}$ if the integrand has square integrable partial mixed derivatives up to order $\alpha$ in each variable. Known lower bounds show that this rate of convergence cannot be improved. 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.

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