Two constructions were recently proposed for constructing low-discrepancy point sets on triangles. One is based on a finite lattice, the other is a triangular van der Corput sequence. We give a continuation and improvement of these methods. We first provide an extensible lattice construction for points in the triangle that can be randomized using a simple shift. We then examine the one-dimensional projections of the deterministic triangular van der Corput sequence and quantify their sub-optimality compared to the lattice construction. Rather than using scrambling to address this issue, we show how to use the triangular van der Corput sequence to construct a stratified sampling scheme. We show how stratified sampling can be used as a more efficient implementation of nested scrambling, and that nested scrambling is a way to implement an extensible stratified sampling estimator. We also provide a test suite of functions and a numerical study for comparing the different constructions.

最近提出了两种用于在三角形上构造低差异点集的构造。一种是基于有限格，另一种是三角范德科普特序列。我们对这些方法进行了延续和改进。我们首先为三角形中的点提供可扩展的点阵结构，可以使用简单的移位将其随机化。然后，我们检查确定性三角范德科普特序列的一维投影，并与格结构相比量化它们的次优性。我们没有使用置乱来解决这个问题，而是展示了如何使用三角范德科普特序列来构建分层采样方案。我们展示了如何使用分层采样作为嵌套置乱的更有效实现，并且嵌套置乱是实现可扩展分层采样估计器的一种方法。我们还提供了功能测试套件和数值研究来比较不同的结构。