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A universal median quasi-Monte Carlo integration

Takashi Goda, Kosuke Suzuki, Makoto Matsumoto

TL;DR

The paper addresses universal quasi-Monte Carlo integration that does not require prior knowledge of target function smoothness or weights by employing a median of several randomized QMC estimates from linearly scrambled digital nets or polynomial lattice point sets. It provides probabilistic bounds on the worst-case error across three weighted Sobolev spaces with increasing smoothness, showing that the failure probability decays exponentially with the number of estimates and that convergence is nearly optimal for finite smoothness and dimension-independent super-polynomial for infinitely differentiable functions. The authors develop duality-based analyses (via NRT and Dick weights) to bound the dual-net contributions and establish universality results applicable across spaces and weight regimes. Numerical experiments corroborate the theory, illustrating robust, space- and dimension-agnostic performance of the median-QMC rule and highlighting its stability against poorly conditioned point sets.

Abstract

We study quasi-Monte Carlo (QMC) integration over the multi-dimensional unit cube in several weighted function spaces with different smoothness classes. We consider approximating the integral of a function by the median of several integral estimates under independent and random choices of the underlying QMC point sets (either linearly scrambled digital nets or infinite-precision polynomial lattice point sets). Even though our approach does not require any information on the smoothness and weights of a target function space as an input, we can prove a probabilistic upper bound on the worst-case error for the respective weighted function space, where the failure probability converges to 0 exponentially fast as the number of estimates increases. Our obtained rates of convergence are nearly optimal for function spaces with finite smoothness, and we can attain a dimension-independent super-polynomial convergence for a class of infinitely differentiable functions. This implies that our median-based QMC rule is universal in the sense that it does not need to be adjusted to the smoothness and the weights of the function spaces and yet exhibits the nearly optimal rate of convergence. Numerical experiments support our theoretical results.

A universal median quasi-Monte Carlo integration

TL;DR

The paper addresses universal quasi-Monte Carlo integration that does not require prior knowledge of target function smoothness or weights by employing a median of several randomized QMC estimates from linearly scrambled digital nets or polynomial lattice point sets. It provides probabilistic bounds on the worst-case error across three weighted Sobolev spaces with increasing smoothness, showing that the failure probability decays exponentially with the number of estimates and that convergence is nearly optimal for finite smoothness and dimension-independent super-polynomial for infinitely differentiable functions. The authors develop duality-based analyses (via NRT and Dick weights) to bound the dual-net contributions and establish universality results applicable across spaces and weight regimes. Numerical experiments corroborate the theory, illustrating robust, space- and dimension-agnostic performance of the median-QMC rule and highlighting its stability against poorly conditioned point sets.

Abstract

We study quasi-Monte Carlo (QMC) integration over the multi-dimensional unit cube in several weighted function spaces with different smoothness classes. We consider approximating the integral of a function by the median of several integral estimates under independent and random choices of the underlying QMC point sets (either linearly scrambled digital nets or infinite-precision polynomial lattice point sets). Even though our approach does not require any information on the smoothness and weights of a target function space as an input, we can prove a probabilistic upper bound on the worst-case error for the respective weighted function space, where the failure probability converges to 0 exponentially fast as the number of estimates increases. Our obtained rates of convergence are nearly optimal for function spaces with finite smoothness, and we can attain a dimension-independent super-polynomial convergence for a class of infinitely differentiable functions. This implies that our median-based QMC rule is universal in the sense that it does not need to be adjusted to the smoothness and the weights of the function spaces and yet exhibits the nearly optimal rate of convergence. Numerical experiments support our theoretical results.
Paper Structure (15 sections, 22 theorems, 119 equations, 4 figures)

This paper contains 15 sections, 22 theorems, 119 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. QMC point sets
  4. Digital $(t,m,s)$-nets and linear scrambling
  5. Polynomial lattice point sets
  6. Duality and some new results
  7. Linear scrambled digital nets
  8. Polynomial lattice point sets
  9. Median QMC integration
  10. Basic result
  11. Universality
  12. Weighted Sobolev space of first order
  13. Weighted Sobolev space of high order
  14. Weighted space of infinitely smooth functions
  15. Numerical experiments

Key Result

Lemma 2.3

\newlabellem:t-value0 Let $m,n \in \mathbb{N}$ and $C_1, \dots, C_s \in \mathbb{F}_b^{n \times m}$. The point set $P(C_1,\ldots,C_s)$ is a digital $(t,m,s)$-net in base $b$ with $t$ being the smallest non-negative integer such that, for any $a_1,\ldots,a_s\in \mathbb{N}_0$ with $a_1+\cdots+a_s=m-t

Figures (4)

  • Figure 1: Histograms of $\log_2 \mathcal{S}_{\alpha,\boldsymbol{\gamma}}$ for randomly generated polynomial lattices (left panels) and those of the median of $\log_2 \mathcal{S}_{\alpha,\boldsymbol{\gamma}}$ for randomly generated sets of $r=15$ polynomial lattices (right panels). The upper panels correspond to the case $\alpha=2$, while the lower panels correspond to $\alpha=3$.
  • Figure 2: Comparison of the one-dimensional integration error by QMC rule using Sobol' points (left) and two median QMC rules using linearly scrambled Sobol' points (middle) and randomly chosen polynomial lattice point sets (right) for the test functions $f_1$ (blue), $f_2$ (orange) and $f_3$ (yellow), respectively.
  • Figure 3: Comparison of the 20-dimensional integration error by QMC rule using Sobol' points (left) and two median QMC rules using linearly scrambled Sobol' points (middle) and randomly chosen polynomial lattice point sets (right) for the finitely smooth function $f_4$ with $c=0.5$ (blue), $c=1.5$ (orange) and $c=2.5$ (yellow), respectively.
  • Figure 4: Comparison of the 5-dimensional integration error by QMC rule using Sobol' points (left) and two median QMC rules using linearly scrambled Sobol' points (middle) and randomly chosen polynomial lattice point sets (right) for the infinitely smooth function $f_5$ with $c=0$ (blue), $c=1$ (orange) and $c=2$ (yellow), respectively.

Theorems & Definitions (53)

  • Definition 2.1: digital net
  • Definition 2.2: elementary interval and $(t,m,s)$-net
  • Lemma 2.3
  • Remark 2.4
  • Definition 2.5: linearly scrambled digital net
  • Remark 2.6
  • Remark 2.7
  • Definition 2.8: polynomial lattice point set
  • Lemma 2.9
  • Remark 2.10
  • ...and 43 more