KLExpansion_Full

class cuqi.geometry.KLExpansion_Full(grid, std=1.0, cor_len=0.2, nu=3.0, axis_labels=None, **kwargs)

Class representation of the random field in the sine basis

\[ \begin{align}\begin{aligned}f = \frac{\text{std}^2}{\pi}\sum_{i=0}^{N-2} \left(\frac{\tau^\gamma}{(\tau+i^2)^\gamma}\right) p_i \, \text{sin}\left(\frac{\pi}{N}(i+1)(K+\frac{1}{2})\right)\\+ \frac{\text{std}^2}{\pi}\frac{(-1)^K}{2}\left(\frac{\tau^\gamma}{\left(\tau+(N-1)^2\right)^\gamma}\right) p_{N-1}\end{aligned}\end{align} \]

where:

\[\tau = \frac{1}{\text{cor_len}^2},\]

\[\gamma = \text{nu}+1,\]

\(K=\{0, 1, 2, 3, ..., N-1\}\), \(N\) is the number of nodes in the grid, and \(p_i\) are the expansion coefficients.

The above transformation is the inverse of DST-II (see https://en.wikipedia.org/wiki/Discrete_sine_transform)

Parameters:

• grid (array-like) – One dimensional regular grid on which the random field is defined.

• cor_len (float, default 1.0) – The correlation length of the random field.

• nu (float, default 2.5) – Smoothness parameter of the random field.

• std (float, default 1.0) – Standard deviation of the random field.

__init__(grid, std=1.0, cor_len=0.2, nu=3.0, axis_labels=None, **kwargs)

Methods

__init__(grid[, std, cor_len, nu, axis_labels])
fun2par(funvals)	The function to parameter map used to map function values back to parameters, if available.
fun2vec(funvals)	Maps function values to a vector representation of the function values, if available.
par2fun(p)	The parameter to function map used to map parameters to function values in e.g. plotting.
plot(values[, is_par, plot_par])	Plots a function over the set defined by the geometry object.
plot_envelope(lo_values, hi_values[, ...])	Plots an envelope from lower and upper bounds over the set defined by the geometry object.
vec2fun(funvec)	Maps function vector representation, if available, to function values.

Attributes

coefs
fun_dim	The dimension of the geometry (function space).
fun_is_array	Flag to indicate whether the function value is an array.
fun_shape	The shape of the geometry (function space).
funvec_dim	The dimension of the geometry (dimension of the vector representation of the function value).
funvec_shape	The shape of the geometry (shape of the vector representation of the function value).
grid
par_dim	The dimension of the geometry (parameter space).
par_shape	The shape of the parameter space
variables