Convergence rates for the Array-RQMC method

We study the convergence rate of randomized quasi-Monte Carlo (RQMC) methods for integrands with different smoothness. The Koksma--Hlawka inequality fails to be informative for certain $L^2$ singular integrands. We provide convergence results based on spectral analysis.We study the convergence rate of randomized quasi-Monte Carlo (RQMC) methods for integrands with different smoothness. The Koksma--Hlawka inequality fails to be informative for certain $L^2$ singular integrands. We provide convergence results based on spectral analysis.

Jan 1, 20250 citations

Parameterization based on randomized quasi-Monte Carlo methods

Giray Ökten, Matthew Willyard

Parallel ComputingMar 31, 201010 citations

Walk on spheres and Array-RQMC

Valerie N. P. Ho, Art B. Owen

We use Array-RQMC sampling in a walk on spheres (WoS) algorithm for Dirichlet boundary value problems. On a collection of problems, we find that Array-RQMC-WoS reduces the Monte Carlo MSE or variance by factors ranging from $71$ -fold to $3087$ -fold at $n=2^{17}$ trajectories. The variance is known to be $o(1/n)$ but attains empirical rates between $n^{-1.4}$ and $n^{-1.8}$ in our examples. A simpler RQMC-WoS algorithm studied in Ho and Owen (2026) has more theoretical support but only reduced variance by 1.8 to 10.7-fold on the same set of examples. In order to explain this improvement, we introduce a column-wise mean dimension of the RQMC error based on Sobol' indices. It matches the usual mean dimension for Monte Carlo and the mean dimension of a dual lattice error for randomized lattices. We find for a gasket example from Crane et al. (2025) that the mean dimension of Array-RQMC-WoS errors is much higher than an analogous Array-MC-WoS algorithm has. v2 replaced v1's QMCPy lattice with Korobov lattices from LatNet Builder, but left the old abstract in the meta-data v3 corrects the v2 abstract in this meta dataWe use Array-RQMC sampling in a walk on spheres (WoS) algorithm for Dirichlet boundary value problems. On a collection of problems, we find that Array-RQMC-WoS reduces the Monte Carlo MSE or variance by factors ranging from $71$-fold to $3087$-fold at $n=2^{17}$ trajectories. The variance is known to be $o(1/n)$ but attains empirical rates between $n^{-1.4}$ and $n^{-1.8}$ in our examples. A simpler RQMC-WoS algorithm studied in Ho and Owen (2026) has more theoretical support but only reduced variance by 1.8 to 10.7-fold on the same set of examples. In order to explain this improvement, we introduce a column-wise mean dimension of the RQMC error based on Sobol' indices. It matches the usual mean dimension for Monte Carlo and the mean dimension of a dual lattice error for randomized lattices. We find for a gasket example from Crane et al. (2025) that the mean dimension of Array-RQMC-WoS errors is much higher than an analogous Array-MC-WoS algorithm has. v2 replaced v1's QMCPy lattice with Korobov lattices from LatNet Builder, but left the old abstract in the meta-data v3 corrects the v2 abstract in this meta data

arXiv:2605.12844May 13, 20260 citations

Cluster Monte Carlo: Extending the range

Henk W. J. Blöte, Jouke R. Heringa, Erik Luijten

Computer Physics CommunicationsAug 1, 200222 citations

We study the convergence rate of randomized quasi-Monte Carlo (RQMC) methods for integrands with different smoothness. The Koksma--Hlawka inequality fails to be informative for certain

L^2

singular integrands. We provide convergence results based on spectral analysis.

71

-fold to

3087

-fold at

n=2^{17}

trajectories. The variance is known to be

o(1/n)

but attains empirical rates between

n^{-1.4}

and

n^{-1.8}

in our examples. A simpler RQMC-WoS algorithm studied in Ho and Owen (2026) has more theoretical support but only reduced variance by 1.8 to 10.7-fold on the same set of examples. In order to explain this improvement, we introduce a column-wise mean dimension of the RQMC error based on Sobol' indices. It matches the usual mean dimension for Monte Carlo and the mean dimension of a dual lattice error for randomized lattices. We find for a gasket example from Crane et al. (2025) that the mean dimension of Array-RQMC-WoS errors is much higher than an analogous Array-MC-WoS algorithm has. v2 replaced v1's QMCPy lattice with Korobov lattices from LatNet Builder, but left the old abstract in the meta-data v3 corrects the v2 abstract in this meta data