GMRF: Preliminary Write-up Kelvin Loh April 2014 1 Theory 1.1 General Information A Gaussian Field with a Matern covariance (1) can be approximated by the so- lution to the stochastic partial differential equation (SPDE) (2) [2]. Instead of having to do a Karhunen-Loeve expansion (identical to Singular Value Decom- position for the case of zero mean fields [1]) on the dense covariance matrix for the generation of a sample, one can just solve the SPDE (2), taking advantage of sparsity in the discretization matrix to generate the approximate Gaussian field instead. Cov(a(x 1 ),a(x 2 )) = σ 2 2 ν-1 Γ(ν ) (κ||x 1 − x 2 ||) ν K ν (κ||x 1 − x 2 ||) (1) (κ 2 − Δ) α/2 (τa(x)) = W (x) (2) where x, x 1 ,x 2 ∈ Ω d , W (x) ∼ MVN(0, 1) The solution, a(x), of the SPDE (2) has a covariance which approximates Eq.(1) by the following relations, and open boundary condition ∇a(x) · ˆ n =0 x ∈ ∂ Ω (3) τ 2 = Γ(ν ) Γ(ν )(4π) d/2 κ 2ν σ 2 (4) κ = √ 8ν λ (5) ν = α − d 2 (6) 1.2 Problem Description - Level 1 The variable poisson equation, Eq. (7) is to be solved in the square domain, Ω = [0, 1] × [0, 1], with the exponential Gaussian field e a(x) as the coefficients. 1