Rational Approximation Coefficients for a Fractional SPDE

Returns the coefficient descriptor for an SPDE Matern field of operator order alpha = nu + 1 (in 2D). For integer nu the construction is exact and no rational approximation is needed. For fractional nu it returns the rational-SPDE roots from the BRASIL best-rational approximation (see Details).

Usage

rational_spde_coefficients(nu, m = 4L, lambda_range = c(1e-04, 10000))

Arguments

nu

Matern smoothness parameter. A positive number; may be fractional (e.g. 0.5, 1.5, 2.5).

m

Rational approximation order (number of numerator / denominator factors) used for fractional nu. Higher m lowers the approximation error of the field's spectral density. Default 4.

lambda_range

The generalized-eigenvalue spectrum c(l_min, l_max) of the FEM operator CiL = C^{-1}(kappa^2 C + G) over which the rational approximation is fitted. The approximation acts on the normalized interval [l_min / l_max, 1]. Used for fractional nu only.

Value

For integer nu, a list with is_integer = TRUE, the operator order alpha, beta = alpha / 2, m = 0, and empty poles / weights. For fractional nu, a list with is_integer = FALSE, alpha, beta, the rSPDE roots rb (denominator factors, drive Pl) and rc (numerator factors, drive Pr), the integer power m_beta, the remaining fractional exponent beta_rem, the scale constant scale, the rational order m, and the approximation error.

Details

For integer nu (1, 2, 3, ...) the operator order alpha = nu + 1 is an integer and the precision is assembled directly from integer powers of the FEM operator L = kappa^2 C + G – an exact construction.

For fractional nu the field uses the operator-based rational SPDE approximation (Bolin & Kirchner 2020). With beta = alpha / 2 and m_beta = max(1, floor(beta)), the field-mode variance of the assembled precision Q = Pl' C^{-1} Pl (field u = Pr x, x ~ N(0, Q^{-1})) tracks the Matern spectral density l^{-2 beta} when prod(1 - l rc) / (l^{m_beta - 1} prod(1 - l rb)) approximates l^{-beta_rem} on the scaled spectrum, beta_rem = beta - (m_beta - 1). The roots come from the degree-(m, m) best uniform (minimax) rational approximation of x^{-beta_rem}, computed by the BRASIL algorithm (Hofreither 2021); its numerator zeros map to rc = 1 / zero and denominator poles to rb = 1 / pole. The field assembly from these roots is .spde_rational_assemble(); it is validated against the Matern spectral density in test-spde-rational.R.

References

Bolin, D. & Kirchner, K. (2020). The rational SPDE approach for Gaussian random fields with general smoothness. Journal of Computational and Graphical Statistics, 29(2), 274-285.

Hofreither, C. (2021). An algorithm for best rational approximation based on barycentric rational interpolation. Numerical Algorithms, 88, 365-388.