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Coarse scrambling for Sobol' and Niederreiter sequences

Kosuke Suzuki

TL;DR

The paper introduces coarse scrambling, a block-wise digit permutation method for digital sequences that preserves the $(0,\boldsymbol{e},d)$-sequence structure. It establishes that coarse scrambling achieves the canonical $O(n^{-3+\epsilon})$ variance decay for sufficiently smooth integrands, matching Owen's scrambling, while its maximal gain coefficient grows only as $O(\log d)$, mitigating the dimensionality curse. The authors provide a unified equidistribution framework and derive gain-coefficient bounds for scrambled $(0,\boldsymbol{e},d)$-sequences, showing a logarithmic scaling in dimension and comparing coarse with usual scrambling. Numerical experiments corroborate the theory, revealing that usual scrambling excels for low-dimensional projections, whereas coarse scrambling is competitive for functions with low effective truncation dimension. This work offers a robust, dimension-aware alternative for randomized QMC with practical implications for high-dimensional integration tasks.

Abstract

We introduce \emph{coarse scrambling}, a novel randomization for digital sequences that permutes blocks of digits in a mixed-radix representation. This construction is designed to preserve the powerful $(0,\boldsymbol{e},d)$-sequence property of the underlying points. For sufficiently smooth integrands, we prove that this method achieves the canonical $O(n^{-3+ε})$ variance decay rate, matching that of standard Owen's scrambling. Crucially, we show that its maximal gain coefficient grows only logarithmically with dimension, $O(\log d)$, thus providing theoretical robustness against the curse of dimensionality affecting scrambled Sobol' sequences. Numerical experiments validate these findings and illustrate a practical trade-off: while Owen's scrambling is superior for integrands sensitive to low-dimensional projections, coarse scrambling is competitive for functions with low effective truncation dimension.

Coarse scrambling for Sobol' and Niederreiter sequences

TL;DR

The paper introduces coarse scrambling, a block-wise digit permutation method for digital sequences that preserves the -sequence structure. It establishes that coarse scrambling achieves the canonical variance decay for sufficiently smooth integrands, matching Owen's scrambling, while its maximal gain coefficient grows only as , mitigating the dimensionality curse. The authors provide a unified equidistribution framework and derive gain-coefficient bounds for scrambled -sequences, showing a logarithmic scaling in dimension and comparing coarse with usual scrambling. Numerical experiments corroborate the theory, revealing that usual scrambling excels for low-dimensional projections, whereas coarse scrambling is competitive for functions with low effective truncation dimension. This work offers a robust, dimension-aware alternative for randomized QMC with practical implications for high-dimensional integration tasks.

Abstract

We introduce \emph{coarse scrambling}, a novel randomization for digital sequences that permutes blocks of digits in a mixed-radix representation. This construction is designed to preserve the powerful -sequence property of the underlying points. For sufficiently smooth integrands, we prove that this method achieves the canonical variance decay rate, matching that of standard Owen's scrambling. Crucially, we show that its maximal gain coefficient grows only logarithmically with dimension, , thus providing theoretical robustness against the curse of dimensionality affecting scrambled Sobol' sequences. Numerical experiments validate these findings and illustrate a practical trade-off: while Owen's scrambling is superior for integrands sensitive to low-dimensional projections, coarse scrambling is competitive for functions with low effective truncation dimension.

Paper Structure

This paper contains 17 sections, 15 theorems, 66 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Digital Constructions and Equidistribution
  4. Generalized Niederreiter Sequences
  5. Stronger Equidistribution and Further Generalization
  6. Scrambling
  7. Nested ANOVA decomposition and gain coefficients
  8. Gain coefficients of scrambled points
  9. General results
  10. Gain coefficients of scrambled (0,e,d)-sequences
  11. Comparison of coarse and usual scrambling
  12. Numerical Experiments
  13. Experimental Setup
  14. Results and Discussion
  15. Linear Function
  16. ...and 2 more sections

Key Result

Theorem 2.6

Any generalized Niederreiter sequence in prime power base $b$ with the base polynomials $p_1(x),\dots,p_d(x)$ is a $(t,d)$-sequence with $t = \sum_{j=1}^d (\deg(p_j) - 1)$.

Figures (1)

  • Figure 1: Log-log plot of the number of points vs. the root mean square error for coarse (block) and usual affine scrambling. Reference lines for $O(n^{-1})$ and $O(n^{-1.5})$ are included.

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Definition 2.7
  • Theorem 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 24 more