We introduce an efficient numerical implementation of a Markov Chain Monte Carlo method to sample a probability distribution on a manifold (introduced theoretically in Zappa, Holmes-Cerfon, Goodman (2018)), where the manifold is defined by the level set of constraint functions, and the probability distribution may involve the pseudodeterminant of the Jacobian of the constraints, as arises in physical sampling problems. The algorithm is easy to implement and scales well to problems with thousands of dimensions and with complex sets of constraints provided their Jacobian retains sparsity. The algorithm uses direct linear algebra and requires a single matrix factorization per proposal point, which enhances its efficiency over previously proposed methods but becomes the computational bottleneck of the algorithm in high dimensions. We test the algorithm on several examples inspired by soft-matter physics and materials science to study its complexity and properties.

翻译：我们提出了一种高效的马尔可夫链蒙特卡洛方法数值实现，用于在流形上对概率分布进行采样（该方法由Zappa、Holmes-Cerfon、Goodman（2018）在理论上提出）。该流形由约束函数的水平集定义，概率分布可能涉及约束雅可比矩阵的伪行列式——这类问题常见于物理采样场景中。该算法易于实现，且能够良好扩展到数千维问题以及约束条件复杂但雅可比矩阵保持稀疏性的场景。算法采用直接线性代数方法，每个提议点仅需一次矩阵分解，这相较于已有方法提升了计算效率，但在高维情况下该矩阵分解会成为算法计算瓶颈。我们通过软物质物理和材料科学领域的多个算例对算法复杂度及特性进行了测试。