Fast Update of Conditional Simulation Ensembles

Clément Chevalier (1, 2) , Xavier Emery (3) , David Ginsbourger (1, 2)

1 IMSV - Institute of Mathematical Statistics and Actuarial Science [Bern]
2 GdR MASCOT-NUM - Méthodes d'Analyse Stochastique des Codes et Traitements Numériques
3 Department of Department of Mining Engineering

Gaussian random fields (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. Here we study the settings where conditioning observations are assimilated batch-sequentially, i.e. one point or batch of points at each stage. Assuming that conditional simulations have been performed at a previous stage, we aim at taking advantage of already available sample paths and by-products in order to produce updated conditional simulations at minimal cost. We provide explicit formulas allowing to update an ensemble of sample paths conditioned on $n\geq 0$ observations to an ensemble conditioned on $n+q$ observations, for arbitrary $q\geq 1$. Compared to direct approaches, the proposed formulas prove to substantially reduce computational complexity. Moreover, these formulas enable explicitly exhibiting how the $q$ ''new'' observations are updating the ''old'' sample paths. Detailed complexity calculations highlighting the benefits of our approach with respect to state-of-the-art algorithms are provided and are complemented by numerical experiments.

Mots clés: Gaussian random fields; kriging residual algorithm; batch-sequential strategies; kriging update equations

Domaines: Modélisation et simulation; Applications [stat.AP]; Méthodologie [stat.ME]; Statistiques [math.ST]; Théorie [stat.TH]; Géophysique [physics.geo-ph]; Géophysique [physics.geo-ph]; Milieux et Changements globaux; Machine Learning [stat.ML]

Clément Chevalier, Xavier Emery, David Ginsbourger. Fast Update of Conditional Simulation Ensembles. 2014. ⟨hal-00984515⟩