# Fast Update of Conditional Simulation Ensembles

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## 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.

## References

- Barnes RJ, Watson A (1992) Efficient updating of kriging estimates and variances. Math Geol 24(1):129–133CrossRefGoogle Scholar
- Binois M, Ginsbourger D, Roustant O (2014) Quantifying uncertainty on Pareto fronts with Gaussian process conditional simulations. Eur J Oper Res. doi: 10.1016/j.ejor.2014.07.032
- Chevalier C (2013) Fast uncertainty reduction strategies relying on Gaussian process models. Ph.D. Thesis, University of Bern, BernGoogle Scholar
- Chevalier C, Ginsbourger D, Emery X (2014) Corrected kriging update formulae for batch-sequential data assimilation. In: Pardo-Igúzquiza E et al (eds) Mathematics of planet Earth. Springer, Berlin, pp 119–122Google Scholar
- Chilès JP, Allard D (2005) Stochastic simulation of soil variations. In: Grunwald S (ed) Environmental soil-landscape modeling: geographic information technologies and pedometrics. CRC Press, Boca Raton, pp 289–321Google Scholar
- Chilès JP, Delfiner P (2012) Geostatistics: modeling spatial uncertainty, 2nd edn. Wiley, New YorkGoogle Scholar
- Davis MW (1987) Production of conditional simulations via the LU triangular decomposition of the covariance matrix. Math Geol 19(2):91–98Google Scholar
- de Fouquet C (1994) Reminders on the conditioning kriging. In: Armstrong M, Dowd P (eds) Geostatistical simulations. Kluwer Academic Publishers, Dordrecht, pp 131–145Google Scholar
- Delhomme J (1979) Spatial variability and uncertainty in groundwater flow parameters: a geostatistical approach. Water Resour Res 15(2):269–280CrossRefGoogle Scholar
- Deutsch C (2002) Geostatistical reservoir modeling. Oxford University Press, New YorkGoogle Scholar
- Dimitrakopoulos R (2010) Advances in orebody modelling and strategic mine planning. Australasian Institute of Mining and Metallurgy, Spectrum Series 20, MelbourneGoogle Scholar
- Emery X (2009) The kriging update equations and their application to the selection of neighboring data. Comput Geosci 13(1):211–219Google Scholar
- Emery X, Lantuéjoul C (2006) Tbsim: a computer program for conditional simulation of three-dimensional Gaussian random fields via the turning bands method. Comput Geosci 32(10):1615–1628CrossRefGoogle Scholar
- Gao H, Wang J, Zhao P (1996) The updated kriging variance and optimal sample design. Math Geol 28:295–313CrossRefGoogle Scholar
- Ginsbourger D, Baccou J, Chevalier C, Perales F, Garland N, Monerie Y (2014) Bayesian adaptive reconstruction of profile optima and optimizers. SIAM J Uncertain Quantif 2(1):490–510Google Scholar
- Handcock MS, Stein ML (1993) A Bayesian analysis of kriging. Technometrics 35(4):403–410CrossRefGoogle Scholar
- Hernández J, Emery X (2009) A geostatistical approach to optimize sampling designs for local forest inventories. Can J For Res 39:1465–1474CrossRefGoogle Scholar
- Hoshiya M (1995) Kriging and conditional simulation of Gaussian field. J Eng Mech 121(2):181–186CrossRefGoogle Scholar
- Journel A, Huijbregts C (1978) Mining geostatistics. Academic Press, LondonGoogle Scholar
- Journel A, Kyriakidis P (2004) Evaluation of mineral reserves: a simulation approach. Oxford University Press, New YorkGoogle Scholar
- Lantuéjoul C (2002) Geostatistical simulation: models and algorithms. Springer, BerlinCrossRefGoogle Scholar
- Matheron G (1973) The intrinsic random functions and their applications. Adv Appl Probab 5(3):439–468Google Scholar
- O’Hagan A (1978) Curve fitting and optimal design for prediction. J R Stat Soc Ser B (Methodol) 40(1):1–42Google Scholar
- Omre H, Halvorsen K (1989) The Bayesian bridge between simple and universal kriging. Math Geol 22(7):767–786CrossRefGoogle Scholar
- Roustant O, Ginsbourger D, Deville Y (2012) Dicekriging, diceoptim: two R packages for the analysis of computer experiments by kriging-based metamodelling and optimization. J Stat Softw 51Google Scholar
- Santner TJ, Williams BJ, Notz W (2003) The design and analysis of computer experiments. Springer, BerlinGoogle Scholar
- Stein ML (1999) Interpolation of spatial data: some theory for kriging. Springer, New YorkCrossRefGoogle Scholar
- Villemonteix J, Vazquez E, Walter E (2009) An informational approach to the global optimization of expensive-to-evaluate functions. J Global Optim 44(4):509–534CrossRefGoogle Scholar