2026/07/19 by Takehito Yoshiki
#math.NA #cs.NA
Base-2 digital nets are practical high-dimensional integration rules: the sample budget N=2m 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 2p 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.