This study investigates the performance of median-of-means sampling compared to traditional mean-of-means sampling for computing the Keister function integral using Randomized Quasi-Monte Carlo (RQMC) methods. The research tests both lattice points and digital nets as point distributions across dimensions 2, 3, 5, and 8, with sample sizes ranging from 2^8 to 2^19 points. Results demonstrate that median-of-means sampling consistently outperforms mean-of-means for sample sizes larger than 10^3 points, while mean-of-means shows better accuracy with smaller sample sizes, particularly for digital nets. The study also confirms previous theoretical predictions about median-of-means' superior performance with larger sample sizes and reflects the known challenges of maintaining accuracy in higher-dimensional integration. These findings support recent research suggesting median-of-means as a promising alternative to traditional sampling methods in numerical integration, though limitations in sample size and dimensionality warrant further investigation with different test functions and larger parameter spaces.
We study the sample median of independently generated quasi-Monte Carlo estimators based on randomized digital nets and prove it approximates the target integral value at almost the optimal convergence rate for various function spaces. In contrast to previous methods, the algorithm does not require a priori knowledge of underlying function spaces or even an input of pre-designed ( t , m , s ) (t,m,s) -digital nets, and is therefore easier to implement. This study provides further evidence that some types of randomized quasi-Monte Carlo estimators are heavy-tailed when applied to smooth integrands and taking the median can significantly improve the error by filtering out the outliers.
We survey basic ideas and results on randomized quasi-Monte Carlo (RQMC) methods, discuss their practical aspects, and give numerical illustrations. RQMC can improve accuracy compared with standard Monte Carlo (MC) when estimating an integral interpreted as a mathematical expectation. RQMC estimators are unbiased and their variance converges at a faster rate (under certain conditions) than MC estimators, as a function of the sample size. Variants of RQMC also work for the simulation of Markov chains, for function approximation and optimization, for solving partial differential equations, etc. In this introductory survey, we look at how RQMC point sets and sequences are constructed, how we measure their uniformity, why they can work for high-dimensional integrals, and how can they work when simulating Markov chains over a large number of steps.
Abstract. We study quasi-Monte Carlo (QMC) integration over the multidimensional unit cube in several weighted function spaces with different smoothness classes. We consider approximating the integral of a function by the median of several integral estimates under independent and random choices of the underlying QMC point sets (either linearly scrambled digital nets or infinite-precision polynomial lattice point sets). Even though our approach does not require any information on the smoothness and weights of a target function space as an input, we can prove a probabilistic upper bound on the worst-case error for the respective weighted function space, where the failure probability converges to 0 exponentially fast as the number of estimates increases. Our obtained rates of convergence are nearly optimal for function spaces with finite smoothness, and we can attain a dimension-independent super-polynomial convergence for a class of infinitely differentiable functions. This implies that our median-based QMC rule is universal in the sense that it does not need to be adjusted to the smoothness and the weights of the function spaces and yet exhibits the nearly optimal rate of convergence. Numerical experiments support our theoretical results.
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.
The Monte Carlo method uses a random sample of size n to approximate the value of an s-dimensional integral and has a convergence rate of O(n−1/2). By using lowdiscrepancy sampling, the quasi-Monte Carlo (QMC) method achieves a convergence rate of O(n−1(log n)), which is asymptotically faster than that of the Monte Carlo method. However, due to the dependence of the QMC error bound on the dimension s, this result doesn’t hold at practical values of n for even relatively small values of s. Still, many empirical studies have shown that QMC can outperform Monte Carlo integration in highdimensional problems at practical values of n. Empirical studies are often conducted by researchers to investigate the behaviour of QMC methods since the QMC error bound suffers from several practical limitations. This paper presents a Matlab program that implements some of the more commonly used numerical tests and provides a simple graphical summary of the results.
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
L2 integrands with a dimension-independent constant. The median trick then enables near-optimal convergence for function classes beyond
L2: for
Lp integrands with
p∈(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
Hwav,α,s,p,q with
αp:=α−(1/p−1/2)+>0, we prove a high-probability error bound of
O(N−αp−1/2+ε) for any
ε>0; the bound improves to
O(N−r−αp−1/2+ε) when the integrand has dominating mixed derivatives of order
r≥1 that belong to
Hwav,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.