Weighted-L2 coefficients for a given spectral interval

Builds a table of weighted-L2 rational coefficients for a specific lower end of the spectral interval, so that the approximation is optimal on the interval the discrete operator actually has rather than on \((0,1]\).

The mesh-free tables that matern.operators() uses by default do not depend on the mesh or on \(\kappa\) and are stored in the package, which is why they cost nothing at set-up. They are, however, fitted on all of \((0,1]\), while a discretised operator only ever sees \(x \in (x_{\min}, 1]\) with \(x_{\min}\) set by the mesh resolution and by \(\kappa\). Fitting on that shorter interval spends the same number of terms where they matter and is appreciably more accurate; the price is that the table depends on the mesh and therefore has to be computed, which takes of the order of a second.

With the cache turned on (see rspde.cache()) there is usually nothing to do by hand: a model asked for with kappa_ref computes its table once and finds it again in later sessions without being told where it is. The two routes share one store, so a table built here is what a later matern.operators(kappa_ref = ...) picks up, and the other way round.

This function is for what the cache does not cover: holding \(x_{\min}\) fixed across several meshes or models rather than letting each derive its own, inspecting the coefficients and the errors of the fit, or working without writing to a cache at all. Pass the result to matern.operators() as wl2_table; supplying it overrides x_min and kappa_ref. The result is an ordinary data frame with attributes, so saveRDS() is also an option if you would rather manage it yourself than enable the cache.

Since \(x_{\min}\) grows with \(\kappa\), a lower bound for \(\kappa\) gives the shortest interval that is certainly long enough, and hence a table that stays valid while \(\kappa\) is estimated. That is what kappa_ref is.

Usage

rspde.wl2.table(
  m,
  d = NULL,
  nu = NULL,
  alpha = NULL,
  kappa_ref = NULL,
  x_min = NULL,
  C = NULL,
  G = NULL,
  mesh = NULL,
  loc_mesh = NULL,
  type = c("covariance", "operator"),
  eigenvalue = c("bound", "exact"),
  s = 0,
  k_term = FALSE,
  ...
)

Arguments

m

The order of the rational approximation.

d

The dimension of the domain. Taken from mesh when possible.

nu, alpha

The smoothness. Give one of them; alpha = nu + d/2. Only the integer part of alpha is used, since one table covers a whole unit interval of alpha.

kappa_ref

A lower bound for kappa, from which x_min is computed.

x_min

The lower end of the spectral interval, if it is known; then neither kappa_ref nor the mesh is needed.

C, G, mesh, loc_mesh

The finite element matrices or the mesh, used with kappa_ref to find x_min. See rspde.xmin().

type

Either "covariance" or "operator", matching the model the table is for.

eigenvalue

Passed to rspde.xmin().

s, k_term

The weight exponent and the constant term of the fit; see rational.coefficients.wl2(). These change what is being approximated and are for studying the approximation rather than for building a model: a constant term has no block in the covariance-based construction, and a table carrying one is refused by matern.operators().

...

Passed to the fit, for instance n_starts or by.

Value

A table of coefficients, with the attributes type, d, m, m_alpha, x_min and kind, for use as the wl2_table argument of matern.operators().

See also

rspde.xmin(), matern.operators(), rational.coefficients.wl2()

Examples

# \donttest{
mesh <- fmesher::fm_mesh_1d(seq(0, 1, length.out = 101))
tab <- rspde.wl2.table(m = 2, d = 1, nu = 0.8, kappa_ref = 5, mesh = mesh)
attr(tab, "x_min")
#> [1] 0.0006246096
# }