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A randomized lattice rule without component-by-component construction

Takashi Goda

Abstract

We study the multivariate integration problem for periodic functions from the weighted Korobov space in the randomized setting. We introduce a new randomized rank-1 lattice rule with a randomly chosen number of points, which avoids the need for component-by-component construction in the search for good generating vectors while still achieving nearly the optimal rate of the randomized error. Our idea is to exploit the fact that at least half of the possible generating vectors yield nearly the optimal rate of the worst-case error in the deterministic setting. By randomly choosing generating vectors $r$ times and comparing their corresponding worst-case errors, one can find one generating vector with a desired worst-case error bound with a very high probability, and the (small) failure probability can be controlled by increasing $r$ logarithmically as a function of the number of points. Numerical experiments are conducted to support our theoretical findings.

A randomized lattice rule without component-by-component construction

Abstract

We study the multivariate integration problem for periodic functions from the weighted Korobov space in the randomized setting. We introduce a new randomized rank-1 lattice rule with a randomly chosen number of points, which avoids the need for component-by-component construction in the search for good generating vectors while still achieving nearly the optimal rate of the randomized error. Our idea is to exploit the fact that at least half of the possible generating vectors yield nearly the optimal rate of the worst-case error in the deterministic setting. By randomly choosing generating vectors times and comparing their corresponding worst-case errors, one can find one generating vector with a desired worst-case error bound with a very high probability, and the (small) failure probability can be controlled by increasing logarithmically as a function of the number of points. Numerical experiments are conducted to support our theoretical findings.
Paper Structure (11 sections, 9 theorems, 60 equations, 3 figures, 1 algorithm)

This paper contains 11 sections, 9 theorems, 60 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weighted Korobov space
  4. Rank-1 lattice rule
  5. A randomized lattice rule
  6. Algorithm and basic properties
  7. Error bounds
  8. Additional remarks
  9. Numerical experiments
  10. Distribution of the worst-case error
  11. Performance for test functions

Key Result

Lemma 2.3

Let $N\in \mathbb{N}$ with $N\geq 2$ and $\boldsymbol{z}\in \{1,\ldots,N-1\}^d$. For any $\boldsymbol{k}\in \mathbb{Z}^d$, it holds that

Figures (3)

  • Figure 1: Relative frequency histograms of the base-10 logarithm of the worst-case error in the weighted Korobov space with $d=20$, $\alpha=2$, and $\gamma_j=1/j^3$. The left column displays the results for $N=251$, while the right column shows those for $N=2039$. The top row displays the results for rank-1 lattice points with randomly chosen $\boldsymbol{z}\in \{1,\ldots,N-1\}^d$, the middle row for those obtained by Algorithm \ref{['alg:without']} with fixed $N$, $\eta=1/2$, and $r$ given by \ref{['eq:number_trials']} in which $M$ is replaced by $N$, and the bottom row for those generated by the randomized CBC construction from dick2022component with fixed $N$ and $\eta=1/2$ (note that the symbol $\tau$ was used instead of $\eta$ in dick2022component).
  • Figure 2: Convergence behavior of randomized lattice rules for test functions in $d=2$. Each panel corresponds to a different test function: $f_1$ (top left), $f_2$ (top right), $f_3$ (bottom left), and $f_4$ (bottom right). The horizontal axis represents the maximum number of points $M$, while the vertical axis does the sample variance (logarithmically scaled with base-$10$). The results of the standard Monte Carlo method are represented by blue $\ast$ for reference. The result of Algorithm \ref{['alg:without']} is represented by orange $\square$, while that of the randomized CBC approach is by yellow $\triangledown$.
  • Figure 3: Convergence behavior of randomized lattice rules for test functions in higher dimension $d=20$. Other descriptions are the same as Figure \ref{['fig:test_d2']}.

Theorems & Definitions (20)

  • Definition 2.1: Rank-1 lattice point set
  • Definition 2.2: Dual lattice
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • ...and 10 more