rSPDE (version 2.5.1)

spacetime.operators: Space-time random fields

Description

spacetime.operators is used for computing a FEM approximation of a Gaussian random field defined as a solution to the SPDE $$d u + \gamma(\kappa^2 + \kappa^{d/2}\rho \cdot\nabla - \Delta)^\alpha u = \sigma dW_C.$$ where C is a Whittle-Matern covariance operator with smoothness parameter \(\beta\) and range parameter \(\kappa\)

Usage

spacetime.operators(
  mesh_space = NULL,
  mesh_time = NULL,
  space_loc = NULL,
  time_loc = NULL,
  graph = NULL,
  kappa = NULL,
  sigma = NULL,
  gamma = NULL,
  rho = NULL,
  alpha = NULL,
  beta = NULL,
  graph_dirichlet = TRUE,
  bounded_rho = TRUE
)

Value

An object of type spacetimeobj.

Arguments

mesh_space

Spatial mesh for FEM approximation

mesh_time

Temporal mesh for FEM approximation

space_loc

Locations of mesh nodes for spatial mesh for 1d models.

time_loc

Locations of temporal mesh nodes.

graph

An optional metric_graph object. Replaces mesh for models on metric graphs.

kappa

Positive spatial range parameter

sigma

Positive variance parameter

gamma

Temporal range parameter.

rho

Drift parameter. Real number on metric graphs and one-dimensional spatial domains, a vector with two number on 2d domains.

alpha

Integer smoothness parameter alpha.

beta

Integer smoothness parameter beta.

graph_dirichlet

For models on metric graphs, use Dirichlet vertex conditions at vertices of degree 1? When bounded_rho = TRUE, the rspde_lme models enforce bounded rho for consistency. If the estimated value of rho approaches the upper bound too closely, we recommend refitting the model with bounded_rho = FALSE. However, this should be done with caution, as it may lead to instability in some cases, though it can also result in a better model fit. The actual bound used for rho can be accessed from the bound_rho element of the returned object.

bounded_rho

Logical. Specifies whether rho should be bounded to ensure the existence, uniqueness, and well-posedness of the solution. Defaults to TRUE. Note that this bounding is not a strict condition; there may exist values of rho beyond the upper bound that still satisfy these properties.

Examples

s <- seq(from = 0, to = 20, length.out = 101)
t <- seq(from = 0, to = 20, length.out = 31)

op_cov <- spacetime.operators(space_loc = s, time_loc = t,
                             kappa = 5, sigma = 10, alpha = 1,
                             beta = 2, rho = 1, gamma = 0.05)
Q <- op_cov$Q
v <- rep(0,dim(Q)[1])
v[1565] <- 1
Sigma <- solve(Q,v)

image(matrix(Sigma, nrow=length(s), ncol = length(t)))