Iterative algorithms for non-conditional and conditional simulation of Gaussian random vectors
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ORIGINAL PAPER
Iterative algorithms for non-conditional and conditional simulation of Gaussian random vectors Daisy Arroyo1 • Xavier Emery2,3 Accepted: 5 September 2020 / Published online: 18 September 2020 Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The conditional simulation of Gaussian random vectors is widely used in geostatistical applications to quantify uncertainty in regionalized phenomena that have been observed at finitely many sampling locations. Two iterative algorithms are presented to deal with such a simulation. The first one is a variation of the propagative version of the Gibbs sampler aimed at simulating the random vector without any conditioning data. The novelty of the presented algorithm stems from the introduction of a relaxation parameter that, if adequately chosen, allows quickening the rates of convergence and mixing of the sampler. The second algorithm is meant to convert the non-conditional simulation into a conditional one, based on the successive over-relaxation method. Again, a relaxation parameter allows quickening the convergence in distribution to the desired conditional random vector. Both algorithms are applicable in a very general setting and avoid the pivoting, inversion, square rooting or decomposition of the variance-covariance matrix of the vector to be simulated, thus reduce the computation costs and memory requirements with respect to other discrete geostatistical simulation approaches. Keywords Gaussian random fields Gibbs sampler Mixing Gauss-Seidel method Successive over-relaxation method
1 Introduction The simulation of random fields is widespread in geostatistics to quantify the uncertainty in regionalized phenomena that have been observed at a limited number of sampling locations. Applications in the earth sciences, among other disciplines, include the modeling of mineral deposits, hydrocarbon reservoirs, aquifers, forests, bedrocks, soils, lands and agricultural fields, see for instance Delfiner and Chile`s (1977), Journel and Huijbregts (1978), Delhomme (1979), Mate´rn (1986), Shive et al. (1990), Chile`s and Allard (2005) and Webster and Oliver (2007). & Daisy Arroyo [email protected] Xavier Emery [email protected] 1
Department of Statistics, University of Concepcio´n, Concepcio´n, Chile
2
Department of Mining Engineering, University of Chile, Santiago, Chile
3
Advanced Mining Technology Center, University of Chile, Santiago, Chile
When restricting to Gaussian random fields (i.e., random fields whose finite-dimensional distributions are multivariate normal) and to finitely many locations in space, the problem boils down to simulating a n-dimensional Gaussian random vector Y with a pre-specified mean m and variance-covariance matrix C. Numerous algorithms have been proposed in the past decades to perform such a simulation, see, for instance, Lantue´joul (2002) and Chile`s and Delfiner (2012) for a general overview. A few of them, such as the covariance matrix decomposition (Davis 1987; Alabert 1987; Rue 2001), circulant-emb
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