4.6.16 pde.tools.spectral module

Functions making use of spectral decompositions.

make_colored_noise

Return a function creating an array of random values that obey.

make_correlated_noise

Return a function creating random values with specified spatial correlations.
make_colored_noise(shape, dx=1.0, exponent=0, scale=1, rng=None)[source]

Return a function creating an array of random values that obey.

\[\langle c(\boldsymbol k) c(\boldsymbol k’) \rangle = \Gamma^2 |\boldsymbol k|^\nu \delta(\boldsymbol k-\boldsymbol k’)\]

in spectral space on a Cartesian grid. The special case \(\nu = 0\) corresponds to white noise. For simplicity, the correlations respect periodic boundary conditions.

Parameters:

  • shape (tuple of ints) – Number of supports points in each spatial dimension. The number of the list defines the spatial dimension.

  • dx (float or list of floats) – Discretization along each dimension. A uniform discretization in each direction can be indicated by a single number.

  • exponent (float) – Exponent \(\nu\) of the power spectrum

  • scale (float) – Scaling factor \(\Gamma\) determining noise strength

  • rng (Generator) – Random number generator (default: default_rng())

Returns:

a function returning a random realization

Return type:

callable

make_correlated_noise(shape, correlation, *, discretization=1.0, rng=None, **kwargs)[source]

Return a function creating random values with specified spatial correlations.

The returned field \(f\) generally obeys a selected correlation function \(C(k)\). In Fourier space, we thus have

\[\langle f(\boldsymbol k) f(\boldsymbol k’) \rangle = C(|\boldsymbol k|) \delta(\boldsymbol k-\boldsymbol k’)\]

For simplicity, the correlations respect periodic boundary conditions.

Parameters:

  • shape (tuple of ints) – Number of supports points in each spatial dimension. The number of the list defines the spatial dimension.

  • correlation (str) – Selects the correlation function used to make the correlated noise. Many of the options (described below) support additional parameters that can be supplied as keyword arguments.

  • discretization (float or list of floats) – Discretization along each dimension. A uniform discretization in each direction can be indicated by a single number.

  • rng (Generator) – Random number generator (default: default_rng())

  • **kwargs – Additional parameters can affect details of the correlation function

Return type:

Callable[[], ndarray]

Supported correlation functions

Identifier

Correlation function

none

No correlation, \(C(k) = \delta(k)\)

gaussian

\(C(k) = \exp(\frac12 k^2 \lambda^2)\) with the length scale \(\lambda\) set by argument length_scale.

power law

\(C(k) = k^{\nu/2}\) with exponent \(\nu\) set by argument exponent.

cosine

\(C(k) = \exp\bigl(-s^2(\lambda k - 1)^2\bigr)\) with the length scale \(\lambda\) set by argument length_scale, whereas the sharpness parameter \(s\) is set by sharpness and defaults to 10.
Returns:

a function returning a random realization

Return type:

callable

Parameters: