SPDEs

GaussianMarkovRandomFields.SPDE Type

SPDE

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

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GaussianMarkovRandomFields.MaternSPDE Type

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

The Whittle-Matérn SPDE is given by

$$({\kappa }^2-\Delta {)}^{\frac{\alpha }{2}}u(x)=𝒲(x),(x\in {\mathbb{R}}^d,\alpha =\nu +\frac{d}{2}),$$

where Δ is the Laplacian operator, $\kappa >0$, $\nu >0$.

The stationary solutions to this SPDE are Matérn processes.

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GaussianMarkovRandomFields.AdvectionDiffusionSPDE Type

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

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

$$[\frac{\partial }{\partial t}+\frac{1}{c}{({\kappa }^2-\nabla \cdot H\nabla )}^{\alpha }+\frac{1}{c}\gamma \cdot \nabla ]X(t,s)=\frac{\tau }{\sqrt{c}}Z(t,s),$$

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

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