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Lower end of the spectral interval of a rational approximation

Source: R/rational_wl2.R
rspde.xmin.Rd

Computes the value \(x_{\min} = 1/(1 + \mu_{\max}/\kappa_{\mathrm{lo}}^2)\) used by the weighted-L2 rational coefficients, where \(\mu_{\max}\) is the largest generalised eigenvalue of \((G, C)\) and \(\kappa_{\mathrm{lo}}\) is a lower bound for the values that \(\kappa\) can take.

Usage

rspde.xmin(
  C = NULL,
  G = NULL,
  mesh = NULL,
  kappa_ref = NULL,
  nu = NULL,
  diameter = NULL,
  loc_mesh = NULL,
  eigenvalue = c("bound", "exact")
)

Arguments

C

The mass matrix of the finite element discretisation.

G

The stiffness matrix of the finite element discretisation.

mesh

An optional mesh; C and G are computed from it if they are not given, and it is used for the diameter of the domain.

kappa_ref

The reference value \(\kappa_{\mathrm{lo}}\). If NULL, it is taken to be sqrt(8 * nu) / diameter.

nu

The smoothness parameter, used for the default kappa_ref.

diameter

The diameter of the domain, used for the default kappa_ref. Computed from mesh if not given.

loc_mesh

Mesh locations, an alternative to mesh for the diameter.

eigenvalue

Either "bound" (the default), which uses the Gershgorin bound max(rowSums(abs(G)) / diag(C)) for \(\mu_{\max}\), or "exact", which computes the eigenvalue by Lanczos iteration. The bound is an over-estimate of \(\mu_{\max}\), and therefore gives a conservative (too small) \(x_{\min}\), which is the safe direction.

Value

The value of \(x_{\min}\).

Details

The reference \(\kappa_{\mathrm{lo}}\) must be a lower bound: a reference above the true \(\kappa\) leaves part of the spectrum outside the fitted interval, and the error then grows by one to two orders of magnitude. A reference below the true \(\kappa\) is safe; the approximation then degrades gracefully towards the mesh-free fit. The default, \(\kappa_{\mathrm{lo}} = \sqrt{8\nu}/\mathrm{diam}\), corresponds to a range equal to the diameter of the domain.

See also

rational.coefficients.wl2()

Examples

x <- seq(from = 0, to = 1, length.out = 201)
fem <- rSPDE.fem1d(x)
# reference kappa corresponding to a range equal to the domain diameter
rspde.xmin(C = fem$C, G = fem$G, loc_mesh = x, nu = 0.5)
#> [1] 2.499938e-05
# a user-supplied lower bound for kappa
rspde.xmin(C = fem$C, G = fem$G, kappa_ref = 10)
#> [1] 0.0006246096