基于求积节点的数字网：在低维投影上继承求积精度
Digital Nets on Cubature Nodes: Inheriting Cubature Accuracy on Low-Dimensional Projections
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中文总结 AI 辅助
研究能否将低维求积精度插入高维数字网规则，通过简单坐标嵌入方法，当投影形成完整\(p\)位网格时变换规则与乘积求积规则一致，实验表明该方法对特定测试函数有有限预算改进。
中文摘要 AI 辅助
基数为2的数字网是实用的高维积分规则，样本预算\(N = 2^m\)可独立于环境维度选择，生成矩阵能控制投影和沃尔什对偶权重。但对于低维投影分量，具有可比节点数的低维求积规则可能更精确。本文通过简单坐标嵌入回答能否将低维求积精度插入高维数字网规则的问题。当投影形成完整\(p\)位网格时，变换后的规则与相应乘积求积规则在该投影上一致。实验表明该机制对光滑低阶和坐标衰减测试函数有有限预算改进。
英文摘要
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.
