rSPDE (version 2.5.1)

rSPDE.A1d: Observation matrix for finite element discretization on R

Description

A finite element discretization on R can be written as \(u(s) = \sum_i^n u_i \varphi_i(s)\) where \(\varphi_i(s)\) is a piecewise linear "hat function" centered at location \(x_i\). This function computes an \(m\times n\) matrix \(A\) that links the basis function in the expansion to specified locations \(s = (s_1,\ldots, s_m)\) in the domain through \(A_ij = \varphi_j(s_i)\).

Usage

rSPDE.A1d(x, loc)

Value

The sparse matrix A.

Arguments

x

The locations of the nodes in the FEM discretization.

loc

The locations \((s_1,\ldots, s_m)\)

Author

David Bolin

See Also

rSPDE.fem1d()

Examples

# create mass and stiffness matrices for a FEM discretization on [0,1]
x <- seq(from = 0, to = 1, length.out = 101)
fem <- rSPDE.fem1d(x)

# create the observation matrix for some locations in the domain
obs.loc <- runif(n = 10, min = 0, max = 1)
A <- rSPDE.A1d(x, obs.loc)