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arXiv:2609.05979

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

Ziyang Ye, Chaokun Zhu, Zexin Pan

Published Sep 5, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2609.05979

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Monte Carlo (MQ) method is a powerful tool to approximate high dimensional integrals. The disadvantage of ordinary MQ is the slow convergence rate by cause of the essential randomness of this method. Computations of this convergence can lead to pure time consuming. Quasi-Monte Carlo (QMC) method yields considerably better results since this method is a deterministic alternative which uses Low-Discrepancy sequences instead of random samples . There are a multiplicity of big open problems in QMC-methods, problems partly arising from applications and partly arising from theory. QMC are developed to integrate over unite cube, where it has much more accuracy than MC for integrands of bounded variation. Integration over more general spaces such as triangles, disks and Cartesian products of such spaces is more challenging for QMC. Nevertheless in real-world applications various problems are defined over such spaces. The aim of this thesis is to provide a survey of a solution of such problems of numerical integration defined over non-cubical spaces. We present QMC and randomised QMC (RQMC) constructions in the triangle with a vanishing discrepancy based on the recent work of Basu 2016. The QMC construction is a version of the Van der Corput-Halton sequences specially made to the unit triangle. The attraction of scrambled net is replication based error estimation for QMC with slightly the same accuracy as QMC, and for smooth enough integrands.
We study a random sampling technique to approximate integrals ∫[0, 1]s f(x) dx 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+ε 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+ε (where s ≥ 1 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−α−1/2+ε provided the integrand satisfies the strong assumption that it has square integrable partial mixed derivatives up to order α > 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.
Recent advances in quasi-Monte Carlo integration demonstrate that the median of linearly scrambled digital net estimators achieves near-optimal convergence rates for high-dimensional integrals without requiring a priori knowledge of the integrand’s smoothness. Building on this framework, we prove that the median estimator attains dimension-independent convergence, a property known as strong tractability in complexity theory, under tractability conditions characterized by low effective dimensionality. Using a probabilistic, integrand-specific error criterion, our analysis establishes both faster and dimension-independent convergence under weaker assumptions than previously possible in the worst-case setting.
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Quasi-Monte Carlo methods are a variant of ordinary Monte Carlo methods that employ highly uniform quasirandom numbers in place of Monte Carlo's pseudorandom numbers. Monte Carlo methods offer statistical error estimates; however, while quasi-Monte Carlo has a faster convergence rate than normal Monte Carlo, one cannot obtain error estimates from quasi-Monte Carlo sample values by any practical way. A recently proposed method, called randomized quasi-Monte Carlo methods, takes advantage of Monte Carlo and quasi-Monte Carlo methods. Randomness can be brought to bear on quasirandom sequences through scrambling and other related randomization techniques in randomized quasi-Monte Carlo methods, which provide an elegant approach to obtain error estimates for quasi-Monte Carlo based on treating each scrambled sequence as a different and independent random sample. The core of randomized quasi-Monte Carlo is to find an effective and fast algorithm to scramble (randomize) quasirandom sequences. This dissertation surveys research on algorithms and implementations of scrambled quasirandom sequences and proposes some new algorithms to improve the quality of scrambled quasirandom sequences. Besides obtaining error estimates for quasi-Monte Carlo, scrambling techniques provide a natural way to parallelize quasirandom sequences. This scheme is especially suitable for distributed or grid computing. By scrambling a quasirandom sequence we can produce a family of related quasirandom sequences. Finding one or a subset of optimal quasirandom sequences within this family is an interesting problem, as such optimal quasirandom sequences can be quite useful for quasi-Monte Carlo. The process of finding such optimal quasirandom sequences is called the derandomization of a randomized (scrambled) family. We summarize aspects of this technique and propose some new algorithms for finding optimal sequences from the Halton, Faure and Sobol sequences. Finally we explore applications of derandomization.