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

Author

Listed:
  • Gracia Yunruo Dong

    (University of Toronto
    University of Victoria)

  • Erik Hintz

    (University of Waterloo)

  • Marius Hofert

    (The University of Hong Kong)

  • Christiane Lemieux

    (University of Waterloo)

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.

Suggested Citation

  • Gracia Yunruo Dong & Erik Hintz & Marius Hofert & Christiane Lemieux, 2024. "Randomized Quasi-Monte Carlo Methods on Triangles: Extensible Lattices and Sequences," Methodology and Computing in Applied Probability, Springer, vol. 26(2), pages 1-31, June.
    • Handle: RePEc:spr:metcap:v:26:y:2024:i:2:d:10.1007_s11009-024-10084-z
    DOI: 10.1007/s11009-024-10084-z
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