用最优传输方法实现大规模多变量地质模拟，精确保留分布与空间相关性。
MST-Direct at Scale: Multivariate and Conditional Geostatistical Simulation via Sinkhorn Optimal Transport
- 基于稀疏候选匹配的Sinkhorn算法，支持上万节点的大规模计算。
- 在200x200网格上零直方图误差复现六变量联合分布。
- 适合需要精确硬数据约束和复杂空间结构的地质建模任务。
本文将MST-Direct方法从原始的双变量、无条件、小网格设置扩展至多变量、条件化及大网格场景。针对原方法三大局限：(i) 通过稀疏候选限制的Sinkhorn匹配器实现可扩展性，内存复杂度降至O(nC)；(ii) 采用独立的FFT-MA高斯基底匹配多变量值元组，复现指定变差函数；(iii) 通过克里金法对基底进行硬数据条件化，固定观测值位置。由于传输计划保持为目标元组的置换，多变量联合分布被精确保留。在相同的六变量、异方差、强非线性参考分布下，分别在200x200无条件与100x100含200个硬数据样本的条件下验证，对比投影追逐多变量变换（PPMT）方法。结果表明，MST-Direct零直方图误差复现联合分布，完全满足硬数据约束，并准确再现空间相关结构，而PPMT仅为近似解。
原文摘要 · Abstract (English)
This paper extends MST-Direct, a Matching-via-Sinkhorn-Transport approach for multivariate geostatistical simulation, from the original bivariate, unconditional, small-grid formulation to multivariate, conditional, and large-grid settings. We address the three main limitations identified in the original work: (i) scalability beyond a few thousand nodes through a sparse, candidate-restricted Sinkhorn matcher with O(nC) memory complexity; (ii) extension to multiple variables by matching target value tuples onto an independent FFT-MA Gaussian backbone that reproduces a prescribed variogram; and (iii) hard-data conditioning by fixing observed data tuples at their spatial locations while conditioning the backbone through kriging. Because the transport plan remains a permutation of the target tuples, the multivariate joint distribution is preserved exactly. The method is validated using the same six-variate, heteroscedastic, strongly nonlinear reference distribution employed in Direct Multivariate Simulation (DMS), under both unconditional (200x200) and conditional (100x100, 200 hard-data samples) scenarios, and is benchmarked against the Projection Pursuit Multivariate Transform (PPMT). Results show that MST-Direct reproduces the joint distribution with zero histogram error, exactly honours hard data, and accurately reproduces the prescribed spatial correlation structure, whereas PPMT remains an approximation. Index Terms-Optimal transport, Sinkhorn algorithm, geostatistical simulation, multivariate simulation.
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