Fast and Exact Simulation of Large Gaussian Lattice Systems in ...

2048alpha = 0.52048alpha = 1.010241024512512256256128128n64n6432321616880.01 0.05 0.1 0.2 0.3 0.7 1 3 5 10thetaIECESE0.01 0.05 0.1 0.2 0.3 0.7 1 3 5 10thetaIECESE2048alpha = 1.52048alpha = 1.7510241024512512256256128128n64n6432321616880.01 0.05 0.1 0.2 0.3 0.7 1 3 5 10thetaIECESE0.01 0.05 0.1 0.2 0.3 0.7 1 3 5 10thetaIECESEFigure 5: Current Computational Limits of Exact Simulation. We consider the simulation of intrinsicallystationary Gaussian surfaces with powered exponential variogram, γ(h) = 1 − exp(−|θh| α ),on a square lattice in [0, 1] 2 with spacing 1/n along each coordinate, using the standard circulantembedding (SE), cut-off embedding (CE) and intrinsic embedding (IE) techniques. For eachmethod and for α = 0.50, 1.00, 1.50 and 1.75, the associated graph divides the (n, θ) plane into alower area, for which exact simulation is feasible, and an upper area, for which the method fails.4.2 Current computational limits of exact simulationThe cut-off embedding and intrinsic embedding approaches push the computational limits of exactsimulation. Figure 5 illustrates the current limits for random fields with powered exponential dependencestructure. Specifically, we consider the simulation of an intrinsically stationary Gaussianrandom field with variogram γ(h) = 1 − exp(−|θh| α ) on a square lattice in [0, 1] 2 with spacing 1/nalong each coordinate. Assuming that we can apply the fast Fourier transform to square matricesof size at most 4096 × 4096, the graphs show the largest value of n for which exact simulation withthe standard circulant embedding technique (SE), the cut-off embedding approach (CE) and theintrinsic embedding approach (IE) is feasible. For each technique and for α = 0.50, 1.00, 1.50 and1.75, respectively, the associated graph divides the (n, θ) plane into a lower area, for which exactsimulation is feasible, and an upper area, for which the method fails. The intrinsic embedding approachapplies very generally and allows for exact simulation on grids with spacing 1/512 or largerfor all parameter combinations that we considered. The cut-off embedding method outperforms14