sobolev_kernel_generation: Definition of the kernel for the Sobolev Space

In ardeeshany/AFSSEN: Adaptive Function on Scalar Elastic Net

Definition of the kernel for the Sobolev Space

Description

Definition of the eigenfunctions and eigenvalues of the kernel for the Sobolev Space H^1((a,b)).

Usage

sobolev_kernel_generation(a, b, m, sigma, plot.eigen = FALSE)

Arguments

a	scalar. left end point of the domain.
b	scalar. right end point of the domain.
m	scalar, integer. number of points of the interval domain.
sigma	scalar. weight σ associated to the derivative in the norm associated to the kernel. See Details for the explicit definition of the norm.
plot.eigen	bool. if TRUE the cumulative sum of the eigenvalues of the kernel is plotted. Default is FALSE.

Details

The norm associated to the Sobolev kernel, dependent on the smoothing parameter σ is

\|f\|^2 = \|f\|^2_{L^2} + 1/σ \|f^{\prime}\|^2_{L^2}

The function sobolev_kernel_generation is implicitly called in the generation_kernel function when type parameter is 'sobolev'. See the Vignette for the explicit definition of the kernel.

Value

list containing

• vectors matrix. m \times m matrix containing the eigenvectors of the kernel. Each column contains the evaluation of an eigenfunction on the domain seq(a, b, length = m).

• values vector. m length vector containing the eigenvalues of the kernel

Examples

sobolev_kernel_generation(a = 0, b = 1, m = 100,
               sigma = 1, plot.eigen = FALSE)

ardeeshany/AFSSEN documentation built on Aug. 28, 2022, 2:22 p.m.