MST-Direct Extended to Large-Scale Multivariate Geostatistical Simulation Using Optimal Transport
A research team has extended MST-Direct, a geostatistical simulation framework based on Sinkhorn optimal transport, to handle multivariate, conditional, and large-grid scenarios that the original method could not address. The updated approach introduces a sparse, candidate-restricted matcher to reduce memory demands, supports simultaneous simulation of multiple variables, and incorporates hard-data conditioning via kriging. The work is significant because it offers exact reproduction of multivariate joint distributions and hard-data honoring, outperforming the widely used Projection Pursuit Multivariate Transform (PPMT) benchmark on a six-variate nonlinear reference dataset.
The paper, posted to arXiv in May 2026, presents a substantially upgraded version of MST-Direct, a method that uses the Sinkhorn algorithm from optimal transport theory to simulate spatially correlated, multivariate geological variables. The original method was limited to bivariate, unconditional simulations on small grids of a few thousand nodes; the new formulation scales to grids of 200×200 nodes and beyond through a sparse, candidate-restricted matching strategy with O(nC) memory complexity. Multiple variables are handled simultaneously by matching target value tuples onto an independent FFT-MA Gaussian backbone that is constructed to reproduce a prescribed variogram model. Hard-data conditioning — fixing simulated values at locations where measurements exist — is achieved by anchoring observed data tuples in place and conditioning the Gaussian backbone through kriging. Because the transport plan is structured as a permutation of target tuples, the multivariate joint distribution is preserved exactly rather than approximated. Validation against a six-variate, heteroscedastic, strongly nonlinear reference distribution demonstrates zero histogram error and accurate spatial correlation reproduction, while the competing PPMT method introduces approximation errors. The authors benchmark both unconditional (200×200 grid) and conditional (100×100 grid, 200 hard-data samples) scenarios.
What's missing
The paper has not yet undergone peer review, as it is a preprint. Generalization to three-dimensional grids and to datasets with more than six variables is not demonstrated.
What different sources said
- arXiv cs.LGCenter
MST-Direct at Scale: Multivariate and Conditional Geostatistical Simulation via Sinkhorn Optimal Transport
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