In quasi-Monte Carlo methods, generating high-dimensional low discrepancy sequences by generator matrices is a popular and efficient approach. Historically, constructing or finding such generator matrices has been a hard problem. In particular, it is challenging to take advantage of the intrinsic structure of a given numerical problem to design samplers of low discrepancy in certain subsets of dimensions. To address this issue, we devise a greedy algorithm allowing us to translate desired net properties into linear constraints on the generator matrix entries. Solving the resulting integer linear program yields generator matrices that satisfy the desired net properties. We demonstrate that our method finds generator matrices in challenging settings, offering low discrepancy sequences beyond the limitations of classic constructions.

翻译：在拟蒙特卡洛方法中，利用生成矩阵生成高维低差异序列是一种流行且高效的思路。历史上，构造或寻找此类生成矩阵一直是一个难题。特别是，如何利用给定数值问题的内在结构，在特定维度子集上设计具有低差异性的采样器，极具挑战性。针对这一问题，我们设计了一种贪心算法，能够将期望的网性质转化为生成矩阵元素上的线性约束。通过求解由此产生的整数线性规划，可获得满足期望网性质的生成矩阵。我们证明，该方法能在挑战性场景下找到生成矩阵，从而提供超越经典构造方法局限的低差异序列。