def candidate_based_christoffel_leja_rule_1d(
recursion_coeffs, generate_candidate_samples, num_candidate_samples,
level, initial_points=None, growth_rule=leja_growth_rule,
samples_filename=None, return_weights_for_all_levels=True):
num_vars = 1
num_leja_samples = growth_rule(level)
leja_sequence = get_candidate_based_christoffel_leja_sequence_1d(
num_leja_samples, recursion_coeffs, generate_candidate_samples,
num_candidate_samples, initial_points, samples_filename)
from pyapprox.polynomial_sampling import christoffel_weights
def generate_basis_matrix(x):
return evaluate_orthonormal_polynomial_deriv_1d(
x[0, :], num_leja_samples, recursion_coeffs, deriv_order=0)
def weight_function(x): return christoffel_weights(
generate_basis_matrix(x))
ordered_weights_1d = get_leja_sequence_quadrature_weights(
leja_sequence, growth_rule, generate_basis_matrix, weight_function,
level, return_weights_for_all_levels)
return leja_sequence[0, :], ordered_weights_1d
def beta_leja_quadrature_rule(alpha_stat, beta_stat, level,
growth_rule=leja_growth_rule,
samples_filename=None,
return_weights_for_all_levels=True,
initial_points=None):
"""
Return the samples and weights of the Leja quadrature rule for the beta
probability measure.
By construction these rules have polynomial ordering.
Parameters
----------
level : integer
The level of the isotropic sparse grid.
alpha_stat : integer
The alpha shape parameter of the Beta distribution
beta_stat : integer
The beta shape parameter of the Beta distribution
samples_filename : string
Name of file to save leja samples and weights to
Return
------
ordered_samples_1d : np.ndarray (num_samples_1d)
The reordered samples.
ordered_weights_1d : np.ndarray (num_samples_1d)
The reordered weights.
"""
from pyapprox.multivariate_polynomials import PolynomialChaosExpansion
from pyapprox.leja_sequences import get_leja_sequence_1d,\
get_quadrature_weights_from_samples
num_vars = 1
num_leja_samples = growth_rule(level)
# print(('num_leja_samples',num_leja_samples))
# freezing beta rv like below has huge overhead
# it creates a docstring each time which adds up to many seconds
# for repeated calls to pdf
# univariate_weight_function=lambda x: beta_rv(
# alpha_stat,beta_stat).pdf((x+1)/2)/2
# univariate_weight_function = lambda x: beta_rv.pdf(
# (x+1)/2,alpha_stat,beta_stat)/2
def univariate_weight_function(x): return beta_pdf(
alpha_stat, beta_stat, (x+1)/2)/2
def univariate_weight_function_deriv(x): return beta_pdf_derivative(
alpha_stat, beta_stat, (x+1)/2)/4
weight_function = partial(
evaluate_tensor_product_function,
[univariate_weight_function]*num_vars)
weight_function_deriv = partial(
gradient_of_tensor_product_function,
[univariate_weight_function]*num_vars,
[univariate_weight_function_deriv]*num_vars)
# assert np.allclose(
# (univariate_weight_function(0.5+1e-8)-
# univariate_weight_function(0.5))/1e-8,
# univariate_weight_function_deriv(0.5),atol=1e-6)
poly = PolynomialChaosExpansion()
# must be imported locally otherwise I have a circular dependency
from pyapprox.variable_transformations import \
define_iid_random_variable_transformation
from scipy.stats import uniform
var_trans = define_iid_random_variable_transformation(
uniform(-1, 2), num_vars)
poly_opts = {'poly_type': 'jacobi', 'alpha_poly': beta_stat-1,
'beta_poly': alpha_stat-1, 'var_trans': var_trans}
poly.configure(poly_opts)
if samples_filename is None or not os.path.exists(samples_filename):
ranges = [-1, 1]
from scipy.stats import beta as beta_rv
if initial_points is None:
initial_points = np.asarray(
[[2*beta_rv(alpha_stat, beta_stat).ppf(0.5)-1]]).T
leja_sequence = get_leja_sequence_1d(
num_leja_samples, initial_points, poly,
weight_function, weight_function_deriv, ranges)
if samples_filename is not None:
np.savez(samples_filename, samples=leja_sequence)
else:
leja_sequence = np.load(samples_filename)['samples']
#print (leja_sequence.shape[1],growth_rule(level),level)
assert leja_sequence.shape[1] >= growth_rule(level)
leja_sequence = leja_sequence[:, :growth_rule(level)]
indices = np.arange(growth_rule(level))[np.newaxis, :]
poly.set_indices(indices)
ordered_weights_1d = get_leja_sequence_quadrature_weights(
leja_sequence, growth_rule, poly.basis_matrix, weight_function, level,
return_weights_for_all_levels)
return leja_sequence[0, :], ordered_weights_1d
def gaussian_leja_quadrature_rule(level,
growth_rule=leja_growth_rule,
samples_filename=None,
return_weights_for_all_levels=True,
initial_points=None):
"""
Return the samples and weights of the Leja quadrature rule for the beta
probability measure.
By construction these rules have polynomial ordering.
Parameters
----------
level : integer
The level of the isotropic sparse grid.
samples_filename : string
Name of file to save leja samples and weights to
Return
------
ordered_samples_1d : np.ndarray (num_samples_1d)
The reordered samples.
ordered_weights_1d : np.ndarray (num_samples_1d)
The reordered weights.
"""
from pyapprox.multivariate_polynomials import PolynomialChaosExpansion
from pyapprox.leja_sequences import get_leja_sequence_1d,\
get_quadrature_weights_from_samples
# freezing scipy gaussian rv like below has huge overhead
# it creates a docstring each time which adds up to many seconds
# for repeated calls to pdf
from pyapprox.utilities import gaussian_pdf, gaussian_pdf_derivative
univariate_weight_function = partial(gaussian_pdf, 0, 1)
univariate_weight_function_deriv = partial(gaussian_pdf_derivative, 0, 1)
num_vars = 1
num_leja_samples = growth_rule(level)
weight_function = partial(
evaluate_tensor_product_function,
[univariate_weight_function]*num_vars)
weight_function_deriv = partial(
gradient_of_tensor_product_function,
[univariate_weight_function]*num_vars,
[univariate_weight_function_deriv]*num_vars)
assert np.allclose(
(univariate_weight_function(0.5+1e-8) -
univariate_weight_function(0.5))/1e-8,
univariate_weight_function_deriv(0.5), atol=1e-6)
poly = PolynomialChaosExpansion()
# must be imported locally otherwise I have a circular dependency
from pyapprox.variable_transformations import \
define_iid_random_variable_transformation
from scipy.stats import norm
var_trans = define_iid_random_variable_transformation(
norm(), num_vars)
poly_opts = {'poly_type': 'hermite', 'var_trans': var_trans}
poly.configure(poly_opts)
if samples_filename is None or not os.path.exists(samples_filename):
ranges = [None, None]
if initial_points is None:
initial_points = np.asarray([[0.0]]).T
leja_sequence = get_leja_sequence_1d(
num_leja_samples, initial_points, poly,
weight_function, weight_function_deriv, ranges)
if samples_filename is not None:
np.savez(samples_filename, samples=leja_sequence)
else:
leja_sequence = np.load(samples_filename)['samples']
assert leja_sequence.shape[1] >= growth_rule(level)
leja_sequence = leja_sequence[:, :growth_rule(level)]
indices = np.arange(growth_rule(level))[np.newaxis, :]
poly.set_indices(indices)
ordered_weights_1d = get_leja_sequence_quadrature_weights(
leja_sequence, growth_rule, poly.basis_matrix, weight_function, level,
return_weights_for_all_levels)
return leja_sequence[0, :], ordered_weights_1d