# Fast Update of Conditional Simulation Ensembles

## Abstract

Gaussian random field (GRF) conditional simulation is a key ingredient in many spatial statistics problems for computing Monte-Carlo estimators and quantifying uncertainties on non-linear functionals of GRFs conditional on data. Conditional simulations are known to often be computer intensive, especially when appealing to matrix decomposition approaches with a large number of simulation points. This work studies settings where conditioning observations are assimilated batch sequentially, with one point or a batch of points at each stage. Assuming that conditional simulations have been performed at a previous stage, the goal is to take advantage of already available sample paths and by-products to produce updated conditional simulations at minimal cost. Explicit formulae are provided, which allow updating an ensemble of sample paths conditioned on \(n\ge 0\) observations to an ensemble conditioned on \(n+q\) observations, for arbitrary \(q\ge 1\). Compared to direct approaches, the proposed formulae prove to substantially reduce computational complexity. Moreover, these formulae explicitly exhibit how the \(q\) new observations are updating the old sample paths. Detailed complexity calculations highlighting the benefits of this approach with respect to state-of-the-art algorithms are provided and are complemented by numerical experiments.

### Keywords

Gaussian random fields Residual kriging algorithm Batch-sequential strategies Kriging update equations## Notes

### Acknowledgments

Part of this work has been conducted within the frame of the ReDice Consortium, gathering industrial (CEA, EDF, IFPEN, IRSN, Renault) and academic (Ecole des Mines de Saint-Etienne, INRIA, and the University of Bern) partners around advanced methods for Computer Experiments. Clément Chevalier warmly thanks Prof. Julien Bect for fruitful discussions on GRF simulation. David Ginsbourger acknowledges support from the Department of Mathematics and Statistics of the University of Bern and from the Integrated methods for stochastic ensemble aquifer modelling (ENSEMBLE) project funded by the Swiss National Science Foundation under the contract CRSI22_12249/1. The authors are indebted to two anonymous referees, an associate editor, and Yves Deville for a number of remarks having contributed to substantially improve the paper.

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