SPDEs

SPDE

An abstract type for a stochastic partial differential equation (SPDE).

MaternSPDE{D}(κ::Real, ν::Union{Integer, Rational}) where D

The Whittle-Matérn SPDE is given by

\[(κ^2 - Δ)^{\frac{α}{2}} u(x) = 𝒲(x), \quad \left( x \in \mathbb{R}^d, α = ν + \frac{d}{2} \right),\]

where Δ is the Laplacian operator, $κ > 0$, $ν > 0$.

The stationary solutions to this SPDE are Matérn processes.

AdvectionDiffusionSPDE{D}(κ::Real, α::Rational, H::AbstractMatrix,
γ::AbstractVector, c::Real, τ::Real) where {D}

Spatiotemporal advection-diffusion SPDE as proposed in [2]:

\[\left[ \frac{∂}{∂t} + \frac{1}{c} \left( κ^2 - ∇ ⋅ H ∇ \right)^\alpha + \frac{1}{c} γ ⋅ ∇ \right] X(t, s) = \frac{τ}{\sqrt{c}} Z(t, s),\]

where Z(t, s) is spatiotemporal noise which may be colored.