# Best way to generate Gaussian Field

I have to generate a homogeneous Gaussian Field with given correlation function of each points on a three dimension grid (500 x 500 x 500). A Cholesky decomposition method fails because of huge number of points so I need to find another method. Could you suggest me the best way to generation this Gaussian Field?

Thanks,

• You can look into spectral methods. Since your field is stationary, you can apply a Fourier transform on the autocorrelation to get the spectrum. Then, you scale complex white noise by the square root of the the spectrum (which is real) and voila. Equations: $S(f)=\mathcal F R(s)$, $H(f)=\sqrt{S(f)}$, $X = \mathcal F ^{-1} (N\cdot H ) = n * \mathcal F ^{-1} H$, $R_X(s)=\mathcal F ^{-1} S_X(f) = \mathcal F ^{-1} |H(f)|^2 = R(s)$, where $R$, $S$ denote autocorrelation and spectrum, $n$ and $N$ are white noise and its Fourier transform. You would need to figure out extending to more dimensions.
– yoki
Jun 10 '16 at 22:55