intrinsic.matern.operators: Covariance-based approximations of intrinsic fields

In rSPDE: Rational Approximations of Fractional Stochastic Partial Differential Equations

Covariance-based approximations of intrinsic fields

Description

intrinsic.matern.operators is used for computing a covariance-based rational SPDE approximation of intrinsic fields on R^d defined through the SPDE

(-\Delta)^{\beta/2}(\kappa^2-\Delta)^{\alpha/2} (\tau u) = \mathcal{W}

Usage

intrinsic.matern.operators(
  kappa,
  tau,
  alpha,
  beta = 1,
  G = NULL,
  C = NULL,
  d = NULL,
  mesh = NULL,
  graph = NULL,
  loc_mesh = NULL,
  m_alpha = 2,
  m_beta = 2,
  compute_higher_order = FALSE,
  return_block_list = FALSE,
  type_rational_approximation = c("chebfun", "brasil", "chebfunLB"),
  fem_mesh_matrices = NULL,
  scaling = NULL
)

Arguments

kappa	range parameter
tau	precision parameter
alpha	Smoothness parameter
beta	Smoothness parameter
G	The stiffness matrix of a finite element discretization of the domain of interest.
C	The mass matrix of a finite element discretization of the domain of interest.
d	The dimension of the domain.
mesh	An inla mesh.
graph	An optional metric_graph object. Replaces d, C and G.
loc_mesh	locations for the mesh for d=1.
m_alpha	The order of the rational approximation for the Matérn part, which needs to be a positive integer. The default value is 2.
m_beta	The order of the rational approximation for the intrinsic part, which needs to be a positive integer. The default value is 2.
compute_higher_order	Logical. Should the higher order finite element matrices be computed?
return_block_list	Logical. For type = "covariance", should the block parts of the precision matrix be returned separately as a list?
type_rational_approximation	Which type of rational approximation should be used? The current types are "chebfun", "brasil" or "chebfunLB".
fem_mesh_matrices	A list containing FEM-related matrices. The list should contain elements c0, g1, g2, g3, etc.
scaling	second lowest eigenvalue of g1

Details

The covariance operator

\tau^{-2}(-\Delta)^{\beta}(\kappa^2-\Delta)^{\alpha}

is approximated based on rational approximations of the two fractional components. The Laplacians are equipped with homogeneous Neumann boundary conditions and a zero-mean constraint is additionally imposed to obtained a non-intrinsic model.

Value

intrinsic.matern.operators returns an object of class "intrinsicCBrSPDEobj". This object is a list containing the following quantities:

C	The mass lumped mass matrix.
Ci	The inverse of C.
GCi	The stiffness matrix G times Ci
Gk	The stiffness matrix G along with the higher-order FEM-related matrices G2, G3, etc.
fem_mesh_matrices	A list containing the mass lumped mass matrix, the stiffness matrix and the higher-order FEM-related matrices.
m_alpha	The order of the rational approximation for the Matérn part.
m_beta	The order of the rational approximation for the intrinsic part.
alpha	The fractional power of the Matérn part of the operator.
beta	The fractional power of the intrinsic part of the operator.
type	String indicating the type of approximation.
d	The dimension of the domain.
A	Matrix that sums the components in the approximation to the mesh nodes.
kappa	Range parameter of the covariance function
tau	Scale parameter of the covariance function.
type	String indicating the type of approximation.

Examples

if (requireNamespace("RSpectra", quietly = TRUE)){
 x <- seq(from = 0, to = 10, length.out = 201)
 beta <- 1
 alpha <- 1
 kappa <- 1
 op <- intrinsic.matern.operators(kappa = kappa, tau = 1, alpha = alpha, 
                                 beta = beta, loc_mesh = x, d=1) 
 # Compute and plot the variogram of the model
 Sigma <- op$A %*% solve(op$Q,t(op$A))
 One <- rep(1, times = ncol(Sigma))
 D <- diag(Sigma)
 Gamma <- 0.5*(One %*% t(D) + D %*% t(One) - 2 * Sigma)
 k <- 100
 plot(x, Gamma[k, ], type = "l")
 lines(x, 
      variogram.intrinsic.spde(x[k], x, kappa, alpha, beta, L = 10, d = 1),
      col=2, lty = 2)
}

rSPDE documentation built on Nov. 6, 2023, 1:06 a.m.