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

By Josef Dick. 2010

52 citations recorded by the source corpus.

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.

Subfields: Computer Vision and Pattern Recognition, Numerical Analysis, Statistics and Probability

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Citations in the corpus

Convergence guarantees for kernel-based quadrature rules in misspecified settings (2016, background): “We also note that there are certain quadratures which are adaptive to less smooth integrands [8, 9, 10].”
Effectively Unbiased FID and Inception Score and Where to Find Them (2019, background): “However, there has been a substantial amount of work in the area of estimating high dimensional integrals such as sparse grids [34], higher order scrambled digital nets [10], randomized lattice rules [17] which we have yet to touch upon.”
Control Functionals for Quasi-Monte Carlo Integration (2015, background): “Our focus was on Sobolev spaces, but it is known that a faster rate O(N−α+ ) is possible in the subspace SHα([0, 1]d), for any > 0, and explicit point sets are available (for integer α) [6].”
Acta Numerica: High dimensional integration - the Quasi-Monte Carlo way (2014, unknown): “A construction of digital nets constructed in R s has been studied by Dick (2011b).”
A constraint on extensible quadrature rules (2014, methodology): “Also, there are quasi-Monte Carlo methods such as higher order digital nets (Dick, 2011), that are simultaneously rate optimal for more than one class of functions, each with its own rate.”