Coarse Scrambling for Sobol’ and Niederreiter Sequences

Abstract

SIAM Journal on Numerical Analysis, Volume 64, Issue 4, Page 1342-1363, August 2026.
Abstract. We introduce 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 [math]-sequence property of the underlying points. For sufficiently smooth integrands, we prove that this method achieves the canonical [math] variance decay rate, matching that of standard Owen’s scrambling. Crucially, we show that its maximal gain coefficient grows only logarithmically with dimension [math], 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.