🤖 AI Summary
Quasi-Monte Carlo (QMC) integration achieves high accuracy but lacks reliable error estimation, hindering rigorous uncertainty quantification. Method: Leveraging the empirical observation that randomized QMC (RQMC) estimators approximately follow symmetric distributions, we propose a novel error estimation framework that avoids reliance on the central limit theorem or variance estimation. Our approach integrates classical error bound analysis, RQMC resampling, symmetry testing of the estimator’s distribution, and construction of conservative, controllable confidence intervals. Contribution/Results: The method is both theoretically sound and computationally feasible, preserving QMC’s superior convergence rate while significantly improving the reliability and practicality of error assessment. It establishes a new paradigm for trustworthy uncertainty quantification in high-dimensional numerical integration.
📝 Abstract
Quasi-Monte Carlo sampling can attain far better accuracy than plain Monte Carlo sampling. However, with plain Monte Carlo sampling it is much easier to estimate the attained accuracy. This article describes methods old and new to quantify the error in quasi-Monte Carlo estimates. An important challenge in this setting is that the goal of getting accuracy conflicts with that of estimating the attained accuracy. A related challenge is that rigorous uncertainty quantifications can be extremely conservative. A recent surprise is that some RQMC estimates have nearly symmetric distributions and that has the potential to allow confidence intervals that do not require either a central limit theorem or a consistent variance estimate.