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:在半蒙特卡洛方法中,通过发电机矩阵生成高维低差异序列是一种流行和高效的方法。从历史上看,建造或发现这类发电机矩阵是一个棘手的问题。特别是,利用特定数字问题的内在结构设计某些维度子集差异小的取样员是具有挑战性的。为了解决这一问题,我们设计了贪婪的算法,使我们能够将理想的净属性转化为发电机矩阵条目的线性限制。解决由此产生的整数线性程序生成发电机矩阵,从而满足理想的净属性。我们证明,我们的方法是在具有挑战性的环境中找到发电机矩阵,提供了超出典型构造局限的低差异序列。</s>