Fast Update of Conditional Simulation Ensembles

Abstract

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 ≥ 0 observations to an ensemble conditioned on n + q observations, for arbitrary q ≥ 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.

Sequential design strategies for mean response surface metamodeling via stochastic kriging with adaptive exploration and exploitation

“…First, it is important to study the impact of the initial design and the step size on the predictive performance of SK applied with the sequential strategies proposed, with an emphasis on the roles that the step size ∆n and the initial number of simulation replications n play. Second, a natural extension is to consider augmenting an initial design in a batch sequential manner, that is, to add several new design points to the existing design-point set on each iteration (see, e.g., Loeppky et al, 2010), which will facilitate exploiting the distributed computing power to enable a fast parallel update of SK metamodels (see Chevalier et al, 2015). Third, for high dimensional problems, it is imperative that we are able to construct an adequate candidate designpoint set and to update the set adaptively.…”

Quantifying uncertainty on Pareto fronts with Gaussian process conditional simulations

“…In addition, techniques for simulating efficiently over more points and updating simulations with new observations [40] should be considered, as well as optimization of simulation points locations with re-interpolation or re-simulation [41]. Another direction for future research includes the integration of the proposed uncertainty estimate in a Stepwise Uncertainty Reduction (SUR) strategy [42] as an infill criterion.…”

Real-time resource model updating for improved coal quality control using online data

“…A framework with a similar aim has been recently proposed by (Chevalier et al, 2014) to update the conditional simulations at minimal cost. The formulae offers significant computational savings when the number of conditioning observations is large, and quantifies the effect of the newly assimilated observations on already simulated sample paths.…”