Scalable parallel scheme for sampling of Gaussian random fields over very large domains

Abstract

Summary This paper proposes a new scheme for the generation of Gaussian random fields over large domains (domain size much larger than the correlation length). The scheme decomposes the simulation domain into overlapping subdomains and essentially generates independent random fields over each of them before merging them on the overlaps. It is naturally suited for simulation over clusters of computers. With this approach, the number of operations for each processor depends only on the number of local degrees of freedom and not on the total number over all processors. Hence, weak scalability is perfectly met. This paper describes the general scheme and introduces two error estimates for comparison with classical sampling schemes. Improvements in terms of scalability are demonstrated both theoretically and through numerical examples. The behavior in the overlap is studied in detail. Simulations using the localized approach were performed in up to 512 processors and allowed to generate in 41 seconds a realization of a random field over a cube of side 300 correlation lengths (close to 2‐billion sampling points), much more efficiently and rapidly than with classical methods.

SEM3D: A 3D High-Fidelity Numerical Earthquake Simulator for Broadband (0–10 Hz) Seismic Response Prediction at a Regional Scale

“…To generate random samples, an open-source library randomField (https://github. com/cottereau/randomField.git, accessed on 24 January 2022) was developed by de Carvalho Paludo et al [39] in the framework of BBS2S construction in order to be linked to SEM3D and feed it with large scale geological configurations with small random fluctuations. randomField is a part of SEM3D and exploits the spectral quadrature formula proposed by Shinozuka and Deodatis [89,90], with a variation suited for isotropic media.…”

“…When dealing with large domains (L C−θ ), it applies a generative strategy to overcome the scalability issue and efficiently reduce the generation time (see Figure 7). It generates realizations of G(x) over the entire domain as superpositions of the i th smaller independent realizations G i (x) supported on overlapping subdomains Ω i of Ω [39]:…”

“…Essentially, this means that the scheme is O(1) when we consider a constant number of GLL nodes per processor is considered. The overlapping Equation ( 6) involves an approximation that does not alter the average and variance of the resulting field G(x) [39]. The influence on the correlation structure depends on the overlap, relative to the correlation length, and has been studied by de Carvalho Paludo et al [39].…”

“…To describe the natural heterogeneity of the Earth's crust and soil properties at different scales (i.e., regional geology, local basin-type structures, and the heterogeneity of granular materials) [10,39]. Particular attention should be paid to the remarkable amplification of surface waves due to basin-edge effects and shallow basin deposits [24]; 4.…”

Influence of local cubic anisotropy on the transition towards an equipartition regime in a 3D texture-less random elastic medium

“…One of the advantage of this technique is that its computational efficiency can be drastically improved with Fast Fourier Transform algorithms [46]. For very large domains, the algorithm can also be parallelized [47]. A typical realization of the shear modulus of the randomly heterogeneous medium characterized with the aforementioned parameters is depicted in Fig.…”