chaospy.quadrature.sparse_grid

chaospy.quadrature.sparse_grid(order, dist, growth=None, recurrence_algorithm='stieltjes', rule='gaussian', tolerance=1e-10, scaling=3, n_max=5000)

Smolyak sparse grid constructor.

Args:

order (int, numpy.ndarray):

The order of the grid. If numpy.ndarray, it overrides both dim and skew.

dist (chaospy.distributions.baseclass.Distribution):

The distribution which density will be used as weight function.

growth (bool, None):

If True sets the growth rule for the quadrature rule to only include orders that enhances nested samples. Defaults to the same value as sparse if omitted.

recurrence_algorithm (str):

Name of the algorithm used to generate abscissas and weights in case of Gaussian quadrature scheme. If omitted, analytical will be tried first, and stieltjes used if that fails.

rule (str):

Rule for generating abscissas and weights. Either done with quadrature rules, or with random samples with constant weights.

tolerance (float):

The allowed relative error in norm between two quadrature orders before method assumes convergence.

scaling (float):

A multiplier the adaptive order increases with for each step quadrature order is not converged. Use 0 to indicate unit increments.

n_max (int):

The allowed number of quadrature points to use in approximation.

Returns:

(numpy.ndarray, numpy.ndarray):

Abscissas and weights created from sparse grid rule. Flatten such that abscissas.shape == (len(dist), len(weights)).

Example:

>>> distribution = chaospy.J(chaospy.Normal(0, 1), chaospy.Uniform(-1, 1))
>>> abscissas, weights = chaospy.quadrature.sparse_grid(1, distribution)
>>> abscissas.round(4)
array([[-1.    ,  0.    ,  0.    ,  0.    ,  1.    ],
       [ 0.    , -0.5774,  0.    ,  0.5774,  0.    ]])
>>> weights.round(4)
array([ 0.5,  0.5, -1. ,  0.5,  0.5])
>>> abscissas, weights = chaospy.quadrature.sparse_grid([2, 1], distribution)
>>> abscissas.round(2)
array([[-1.73, -1.  , -1.  , -1.  ,  0.  ,  1.  ,  1.  ,  1.  ,  1.73],
       [ 0.  , -0.58,  0.  ,  0.58,  0.  , -0.58,  0.  ,  0.58,  0.  ]])
>>> weights.round(2)
array([ 0.17,  0.25, -0.5 ,  0.25,  0.67,  0.25, -0.5 ,  0.25,  0.17])