Robust high-dimensional integration using medians of coarsely scrambled Sobol' sequences

Abstract

We study the numerical approximation of high-dimensional integrals using randomized quasi-Monte Carlo (RQMC) methods, with a focus on scrambled Sobol' sequences. While asymptotically faster than Monte Carlo, classical RQMC suffers from error bounds that grow exponentially in the dimension $s$, and its root mean squared error (RMSE) convergence rate is generally no better than $O(N^{-3/2})$. To overcome these limitations, we combine two recent developments: Suzuki's coarse scrambling and the median trick. Coarse scrambling randomizes Sobol' sequences according to their generating base polynomials and significantly reduces the maximal gain coefficient. By appropriately choosing the base polynomials, we show that the coefficient can be made uniformly bounded in $s$, yielding an $O(N^{-1/2})$ RMSE for $L^2$ integrands with a dimension-independent constant. The median trick then enables near-optimal convergence for function classes beyond $L^2$: for $L^p$ integrands with $p\in(1,2)$, the median of independent coarsely scrambled estimates achieves an error of $O(N^{-1+1/p})$ with high probability; for integrands in the Haar wavelet space $\mathcal H_{\mathrm{wav},α,s,p,q}$ with $α_p:=α-(1/p-1/2)_+>0$, we prove a high-probability error bound of $O(N^{-α_p-1/2+\varepsilon})$ for any $\varepsilon>0$; the bound improves to $O(N^{-r-α_p-1/2+\varepsilon})$ when the integrand has dominating mixed derivatives of order $r\ge1$ that belong to $\mathcal H_{\mathrm{wav},α,s,p,q}$. The latter two bounds are uniform in $s$ under suitable conditions on the ANOVA components of the integrands. Numerical experiments confirm the predicted convergence rates and demonstrate the robustness of the proposed approach.