The transparent barrier model

This model considers an SPDE over a domain Ω which is partitioned into k subdomains Ω_d, d ∈ {1, …, k}, where ∪_d = 1^kΩ_d = Ω. A common marginal variance is assumed but the range can be particular to each Ω_d, r_d.

From Bakka et al. (2019), the precision matrix is $$\mathbf{Q}=\frac{1}{{\sigma }^2}\mathbf{R}{\stackrel{\mathbf{~}}{\mathbf{C}}}^{-1}\mathbf{R}\text{ for }{\mathbf{R}}_r=\mathbf{C}+\frac{1}{8}\sum _{d=1}^k{r}_d^2{\mathbf{G}}_d,{\stackrel{\mathbf{~}}{\mathbf{C}}}_r=\frac{\pi }{2}\sum _{d=1}^k{r}_d^2{\stackrel{\mathbf{~}}{\mathbf{C}}}_d$$ where σ² is the marginal variance. The Finite Element Method - FEM matrices: C, defined as C_i, j = ⟨ψ_i, ψ_j⟩ = ∫_Ωψ_i(s)ψ_j(s)∂s, computed over the whole domain, while G_d and C̃_d are defined as a pair of matrices for each subdomain (G_d)_i, j = ⟨1_Ωd∇ψ_i, ∇ψ_j⟩ = ∫_Ωd∇ψ_i(s)∇ψ_j(s)∂s  and  (C̃_d)_i, i = ⟨1_Ωdψ_i, 1⟩ = ∫_Ωdψ_i(s)∂s.

In the case when r = r₁ = r₂ = … = r_k we have $\mathbf{R}_r = \mathbf{C}+\frac{r^2}{8}\mathbf{G}$ and $\mathbf{\tilde{C}}_r = \frac{\pi r^2}{2}\mathbf{\tilde{C}}$ giving $$\mathbf{Q}=\frac{2}{\pi {\sigma }^2}(\frac{1}{r^2}\mathbf{C}{\stackrel{\mathbf{~}}{\mathbf{C}}}^{-1}\mathbf{C}+\frac{1}{8}\mathbf{C}{\stackrel{\mathbf{~}}{\mathbf{C}}}^{-1}\mathbf{G}+\frac{1}{8}\mathbf{G}{\stackrel{\mathbf{~}}{\mathbf{C}}}^{-1}\mathbf{C}+\frac{r^2}{64}\mathbf{G}{\stackrel{\mathbf{~}}{\mathbf{C}}}^{-1}\mathbf{G})$$ which coincides with the stationary case in Lindgren and Rue (2015), when using $\tilde{\mathbf{C}}$ in place of C.

Implementation

In practice we define r_d as r_d = p_dr, for known p₁, …, p_k constants. This gives $${\stackrel{\mathbf{~}}{\mathbf{C}}}_r=\frac{\pi r^2}{2}\sum _{d=1}^k{p}_d^2{\stackrel{\mathbf{~}}{\mathbf{C}}}_d=\frac{\pi r^2}{2}{\stackrel{\mathbf{~}}{\mathbf{C}}}_{p_1,\dots ,p_k}\text{ and }\frac{1}{8}\sum _{d=1}^k{r}_d^2{\mathbf{G}}_d=\frac{r^2}{8}\sum _{d=1}^k{p}_d^2{\stackrel{\mathbf{~}}{\mathbf{G}}}_d=\frac{r^2}{8}{\stackrel{\mathbf{~}}{\mathbf{G}}}_{p_1,\dots ,p_k}$$ where C̃_{p1, …, pk} and G̃_{p1, …, pk} are pre-computed.