MST-Direct at Scale: Multivariate and Conditional Geostatistical Simulation via Sinkhorn Optimal Transport

Abstract

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.