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Description
In this work, we present a parallel MPI software, called GENFIELD, to generate stationary Gaussian random fields, based on the circulant embedding method [7, 1, 4].
The advantage of the circulant embedding method over existing methods (Cholesky factorization of the covariance matrix, Karhunen Loève series expansion) is its computational efficiency. It relies on Discrete Fourier Transform that can be computed efficiently with a Fast Fourier Transform algorithm. In GENFIELD, we use the FFTW3 library [8], which performs MPI parallel one-dimensional transforms. To ensure reproducibility and independence between the generated fields, we take benefit of the streams and substreams structure offered by the Rngstream library [5].
GENFIELD is used in hydrogeology to model natural fields, like hydraulic conductivity or porosity fields [3, 6]. Such fields are a pre-requisite for uncertainty quantification studies [2].
References
[1] Graham, I. G. and Kuo, F. Y. and Nuyens, D. and Scheichl, R. and Sloan, I. H.. Analysis of circulant embedding methods for sampling stationary random fields, to appear in SIAM Journal on Numerical Analysis, 2018.
[2] Graham, I. G. and Kuo, F. Y. and Nuyens, D. and Scheichl, R. and Sloan, I. H., Circulant embedding with QMC – analysis for elliptic PDE with lognormal coefficients, ArXiv e-prints, 2017.
[3] Beaudoin, A. and de Dreuzy, J.-R. and Erhel, J. and Pichot, G., Convergence analysis of macro spreading in 3D heterogeneous, ESAIM: Proc., pages 59-76, EDP Sci., Les Ulis, 2013.
[4] Pichot, G., Algorithms for stationary Gaussian random field generation, Technical Report RT-484, INRIA Paris, Decembre 2016.
[5] L’Ecuyer, P. and Simard, R. and Chen, E. J. and Kelton, W. D., An Objected-Oriented Random-Number Package with Many Long Streams and Substreams, Operations Research, 50(6):1073–1075, 2002.
[6] , De Dreuzy, J.-R. and Pichot, G. and Poirriez, B. and Erhel, J., Synthetic benchmark for modeling flow in 3D fractured media, Comput. Geosci., 50:59–71, 2013.
[7] Dietrich, C. R. and Newsam, G. N., A fast and exact method for multidimensional Gaussian stochastic simulations, Water Resources Research, 29(8):28612869, 1993.
[8] Frigo, M. and Johnson, S. G., The design and implementation of FFTW3, Proc. IEEE 93, 2:216231, 2005
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