Randomized Quasi-Monte Carlo Methods on Triangles: Extensible Lattices and Sequences

Gracia Yunruo Dong (), Erik Hintz (), Marius Hofert () and Christiane Lemieux ()
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Gracia Yunruo Dong: University of Toronto
Erik Hintz: University of Waterloo
Marius Hofert: The University of Hong Kong
Christiane Lemieux: University of Waterloo

Methodology and Computing in Applied Probability, 2024, vol. 26, issue 2, 1-31

Abstract: Abstract Two constructions were recently proposed for constructing low-discrepancy point sets on triangles. One is based on a finite lattice, the other is a triangular van der Corput sequence. We give a continuation and improvement of these methods. We first provide an extensible lattice construction for points in the triangle that can be randomized using a simple shift. We then examine the one-dimensional projections of the deterministic triangular van der Corput sequence and quantify their sub-optimality compared to the lattice construction. Rather than using scrambling to address this issue, we show how to use the triangular van der Corput sequence to construct a stratified sampling scheme. We show how stratified sampling can be used as a more efficient implementation of nested scrambling, and that nested scrambling is a way to implement an extensible stratified sampling estimator. We also provide a test suite of functions and a numerical study for comparing the different constructions.

Keywords: Quasi-Monte Carlo; Triangular van der Corput sequence; Lattice methods; Stratified sampling; 11K31; 11K36; 11K45; 65D30; 65D32 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s11009-024-10084-z

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