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

Yang Liu

TL;DR

Analysis of the convergence rate of randomized quasi-Monte Carlo methods under Owen's boundary growth condition via spectral analysis reveals that the asymptotic convergence rate of the RQMC estimator's variance closely aligns with the exponent specified in Owen's boundary growth condition.

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 sequences: the lattice rule and the Sobol' sequence, applying the Fourier transform and Walsh--Fourier transform, respectively, for this analysis. Assuming certain regularity conditions, 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 analysis for certain discontinuous integrands.

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

TL;DR

Analysis of the convergence rate of randomized quasi-Monte Carlo methods under Owen's boundary growth condition via spectral analysis reveals that the asymptotic convergence rate of the RQMC estimator's variance closely aligns with the exponent specified in Owen's boundary growth condition.

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 sequences: the lattice rule and the Sobol' sequence, applying the Fourier transform and Walsh--Fourier transform, respectively, for this analysis. Assuming certain regularity conditions, 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 analysis for certain discontinuous integrands.
Paper Structure (20 sections, 16 theorems, 137 equations, 6 figures)

This paper contains 20 sections, 16 theorems, 137 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Notations
  4. Lattice rule
  5. Sobol' sequence
  6. Owen's boundary growth condition
  7. Spectral Analysis of RQMC and Owen's boundary growth condition
  8. Lattice Rule
  9. Sobol' sequence
  10. Unbounded integrands with interior discontinuities
  11. Axis-parallel sets
  12. General discontinuous integrand
  13. Partially axis-parallel sets
  14. Numerical Examples
  15. Conclusions
  16. ...and 5 more sections

Key Result

Lemma 1

The variance of the RSLR integration estimator, ${\mathrm{Var}\mspace{-2mu}\left[I_N^{\mathrm{lat}}\right]}$, is given by where $c_{\bm{n}}$ are the Fourier coefficients of $f$, given by and the set $\{ \bm{n} \in \mathbb{Z}^s: \bm{n} \cdot \bm{z} \equiv 0 \mod N \}$ is known as the dual lattice corresponding to the $N$-point rank-1 lattice rule generated by $\bm{z}$. To summarize: the variance

Figures (6)

  • Figure 1: Ex1: Convergence of RQMC estimator variances. Top: RSLR; bottom: Sobol' sequence. The columns from left to right correspond to $s = 2, 3, 5$ respectively. $M = 0.1$. Each boxplot displays the 1st to 99th percentile of 2,048 samples.
  • Figure 2: Ex1: Convergence of RQMC estimator variances. Top: RSLR; bottom: Sobol' sequence. The columns from left to right correspond to $s = 2, 3, 5$, with $M = 0.2$ when $s = 2, 3$, and $M = 0.225$ when $s = 5$. Each boxplot displays the 1st to 99th percentile of 2,048 samples.
  • Figure 3: Ex2: Convergence of RQMC estimator variances. Top: RSLR; bottom: Sobol' sequence. The columns from left to right correspond to $s = 2, 3, 5$, respectively. $M = 0.1$. Each boxplot displays the 1st to 99th percentile of 2,048 samples.
  • Figure 4: Ex2: Convergence of RQMC estimator variances. Top: RSLR; bottom: Sobol' sequence. The columns from left to right correspond to $s = 2, 3, 5$, with $M = 0.2$ when $s = 2, 3$, and $M = 0.225$ when $s = 5$. Each boxplot displays the 1st to 99th percentile of 2,048 samples.
  • Figure 5: Ex3: Convergence of RQMC estimator variances. Top: RSLR, bottom: Sobol' sequence. The columns from left to right correspond to $s = 2, 3, 5$, respectively. $M = 0.1$. Each boxplot displays the 1st to 99th percentile of 2,048 samples.
  • ...and 1 more figures

Theorems & Definitions (33)

  • Lemma 1: Variance of RSLR estimator lecuyer2000variance
  • Remark 1: Monte Carlo estimator variance
  • Remark 2: Fourier coefficients and integration error
  • Definition 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • Remark 3: Comparison with Zaremba's condition
  • Lemma 5
  • ...and 23 more