Digital Nets on Cubature Nodes: Inheriting Cubature Accuracy on Low-Dimensional Projections

Abstract

Base-2 digital nets are practical high-dimensional integration rules: the sample budget $N=2^m$ can be chosen independently of the ambient dimension, and the generating matrices provide algebraic control of projections and Walsh-dual weights. They are therefore well suited to problems whose error is governed by weighted or low-dimensional projection structure. However, when one restricts attention to a smooth low-dimensional projected component, a low-dimensional cubature rule with a comparable number of nodes can be substantially more accurate than the projected digital-net points. This raises the question of whether low-dimensional cubature accuracy can be inserted into a high-dimensional digital-net rule without forming the full tensor product. We answer this question by a simple coordinate embedding: read the leading $p$ binary digits of each coordinate as an index into $2^p$ equal-weight cubature nodes, and replace the coordinate by the indexed node. When a projection forms the full $p$-bit grid, the transformed rule coincides on that projection with the corresponding product cubature rule; small projected $t$-values provide sufficient conditions for such full-grid recovery. For general integrands, the error separates into the corresponding product cubature error and a residual digital-net term. Experiments with scrambled Sobol' nets in dimension $50$ illustrate this mechanism and show finite-budget improvements for the smooth low-order and coordinate-decaying test functions considered here.