Randomized quasi-Monte Carlo and Owen's boundary growth conditions: A spectral analysis

Overview

Abstract

In this work, we analyze the convergence rate of randomized quasi-Monte Carlo (RQMC) methods under Owen's boundary growth condition [Owen, 2006] via spectral analysis. Specifically, we examine the RQMC estimator variance for the two commonly studied low-discrepancy sequences: the lattice rule and the Sobol' sequence, applying the Fourier transform and Walsh-Fourier transform, respectively, for this analysis. With certain regularity assumptions, our findings reveal that the asymptotic convergence rate of the RQMC estimator's variance closely aligns with the exponent specified in Owen's boundary growth condition for both sequence types. We also provide guidance on choosing the importance of sampling density to minimize RQMC estimator variance.

Brief Biography

Yang Liu is a Ph.D. candidate at Stochastic Numerics Research Group (STOCHNUM) under the supervision of Professor Raul F. Tempone at King Abdullah University of Science and Technology (KAUST). His primary research interests involve uncertainty quantification, Monte Carlo methods, and finite element methods. He obtained a Master's Degree in Applied Mathematics and Computational Science from KAUST in 2019.