def test_evaluate_multivariate_orthonormal_polynomial(self):
num_vars = 2
alpha = 0.
beta = 0.
degree = 2
deriv_order = 1
probability_measure = True
ab = jacobi_recurrence(degree + 1,
alpha=alpha,
beta=beta,
probability=probability_measure)
x, w = np.polynomial.legendre.leggauss(degree)
samples = cartesian_product([x] * num_vars, 1)
weights = outer_product([w] * num_vars)
indices = compute_hyperbolic_indices(num_vars, degree, 1.0)
# sort lexographically to make testing easier
I = np.lexsort((indices[0, :], indices[1, :], indices.sum(axis=0)))
indices = indices[:, I]
basis_matrix = evaluate_multivariate_orthonormal_polynomial(
samples, indices, ab, deriv_order)
exact_basis_vals_1d = []
exact_basis_derivs_1d = []
for dd in range(num_vars):
x = samples[dd, :]
exact_basis_vals_1d.append(
np.asarray([1 + 0. * x, x, 0.5 * (3. * x**2 - 1)]).T)
exact_basis_derivs_1d.append(
np.asarray([0. * x, 1.0 + 0. * x, 3. * x]).T)
exact_basis_vals_1d[-1] /= np.sqrt(1. /
(2 * np.arange(degree + 1) + 1))
exact_basis_derivs_1d[-1] /= np.sqrt(
1. / (2 * np.arange(degree + 1) + 1))
exact_basis_matrix = np.asarray([
exact_basis_vals_1d[0][:, 0], exact_basis_vals_1d[0][:, 1],
exact_basis_vals_1d[1][:, 1], exact_basis_vals_1d[0][:, 2],
exact_basis_vals_1d[0][:, 1] * exact_basis_vals_1d[1][:, 1],
exact_basis_vals_1d[1][:, 2]
]).T
# x1 derivative
exact_basis_matrix = np.vstack(
(exact_basis_matrix,
np.asarray([
0. * x, exact_basis_derivs_1d[0][:, 1], 0. * x,
exact_basis_derivs_1d[0][:, 2],
exact_basis_derivs_1d[0][:, 1] * exact_basis_vals_1d[1][:, 1],
0. * x
]).T))
# x2 derivative
exact_basis_matrix = np.vstack(
(exact_basis_matrix,
np.asarray([
0. * x, 0. * x, exact_basis_derivs_1d[1][:, 1], 0. * x,
exact_basis_vals_1d[0][:, 1] * exact_basis_derivs_1d[1][:, 1],
exact_basis_derivs_1d[1][:, 2]
]).T))
def func(x):
return evaluate_multivariate_orthonormal_polynomial(
x, indices, ab, 0)
basis_matrix_derivs = basis_matrix[samples.shape[1]:]
basis_matrix_derivs_fd = np.empty_like(basis_matrix_derivs)
for ii in range(samples.shape[1]):
basis_matrix_derivs_fd[ii::samples.shape[1], :] = approx_fprime(
samples[:, ii:ii + 1], func, 1e-7)
assert np.allclose(exact_basis_matrix[samples.shape[1]:],
basis_matrix_derivs_fd)
assert np.allclose(exact_basis_matrix, basis_matrix)
def test_evaluate_multivariate_mixed_basis_pce(self):
degree = 2
deriv_order = 1
probability_measure = True
gauss_mean, gauss_var = -1, 4
univariate_variables = [
uniform(-1, 2),
norm(gauss_mean, np.sqrt(gauss_var)),
uniform(0, 3)
]
variable = IndependentMultivariateRandomVariable(univariate_variables)
var_trans = AffineRandomVariableTransformation(variable)
num_vars = len(univariate_variables)
poly = PolynomialChaosExpansion()
poly_opts = define_poly_options_from_variable_transformation(var_trans)
poly.configure(poly_opts)
univariate_quadrature_rules = [
partial(gauss_jacobi_pts_wts_1D, alpha_poly=0, beta_poly=0),
gauss_hermite_pts_wts_1D,
partial(gauss_jacobi_pts_wts_1D, alpha_poly=0, beta_poly=0)
]
samples, weights = get_tensor_product_quadrature_rule(
degree + 1, num_vars, univariate_quadrature_rules,
var_trans.map_from_canonical_space)
indices = compute_hyperbolic_indices(num_vars, degree, 1.0)
# sort lexographically to make testing easier
indices = sort_indices_lexiographically(indices)
poly.set_indices(indices)
basis_matrix = poly.basis_matrix(samples, {'deriv_order': 1})
vals_basis_matrix = basis_matrix[:samples.shape[1], :]
inner_products = (vals_basis_matrix.T * weights).dot(vals_basis_matrix)
assert np.allclose(inner_products, np.eye(basis_matrix.shape[1]))
exact_basis_vals_1d = []
exact_basis_derivs_1d = []
for dd in range(num_vars):
x = samples[dd, :].copy()
if dd == 0 or dd == 2:
if dd == 2:
# y = x/3
# z = 2*y-1=2*x/3-1=2/3*x-3/2*2/3=2/3*(x-3/2)=(x-3/2)/(3/2)
loc, scale = 3 / 2, 3 / 2
x = (x - loc) / scale
exact_basis_vals_1d.append(
np.asarray([1 + 0. * x, x, 0.5 * (3. * x**2 - 1)]).T)
exact_basis_derivs_1d.append(
np.asarray([0. * x, 1.0 + 0. * x, 3. * x]).T)
exact_basis_vals_1d[-1] /= np.sqrt(
1. / (2 * np.arange(degree + 1) + 1))
exact_basis_derivs_1d[-1] /= np.sqrt(
1. / (2 * np.arange(degree + 1) + 1))
# account for affine transformation in derivs
if dd == 2:
exact_basis_derivs_1d[-1] /= scale
if dd == 1:
loc, scale = gauss_mean, np.sqrt(gauss_var)
x = (x - loc) / scale
exact_basis_vals_1d.append(
np.asarray([1 + 0. * x, x, x**2 - 1]).T)
exact_basis_derivs_1d.append(
np.asarray([0. * x, 1.0 + 0. * x, 2. * x]).T)
exact_basis_vals_1d[-1] /= np.sqrt(
sp.factorial(np.arange(degree + 1)))
exact_basis_derivs_1d[-1] /= np.sqrt(
sp.factorial(np.arange(degree + 1)))
# account for affine transformation in derivs
exact_basis_derivs_1d[-1] /= scale
exact_basis_matrix = np.asarray([
exact_basis_vals_1d[0][:, 0], exact_basis_vals_1d[0][:, 1],
exact_basis_vals_1d[1][:, 1], exact_basis_vals_1d[2][:, 1],
exact_basis_vals_1d[0][:, 2],
exact_basis_vals_1d[0][:, 1] * exact_basis_vals_1d[1][:, 1],
exact_basis_vals_1d[1][:, 2],
exact_basis_vals_1d[0][:, 1] * exact_basis_vals_1d[2][:, 1],
exact_basis_vals_1d[1][:, 1] * exact_basis_vals_1d[2][:, 1],
exact_basis_vals_1d[2][:, 2]
]).T
# x1 derivative
exact_basis_matrix = np.vstack(
(exact_basis_matrix,
np.asarray([
0. * x, exact_basis_derivs_1d[0][:, 1], 0. * x, 0 * x,
exact_basis_derivs_1d[0][:, 2],
exact_basis_derivs_1d[0][:, 1] * exact_basis_vals_1d[1][:, 1],
0. * x,
exact_basis_derivs_1d[0][:, 1] * exact_basis_vals_1d[2][:, 1],
0. * x, 0. * x
]).T))
# x2 derivative
exact_basis_matrix = np.vstack(
(exact_basis_matrix,
np.asarray([
0. * x, 0. * x, exact_basis_derivs_1d[1][:, 1], 0. * x, 0 * x,
exact_basis_derivs_1d[1][:, 1] * exact_basis_vals_1d[0][:, 1],
exact_basis_derivs_1d[1][:, 2], 0. * x,
exact_basis_derivs_1d[1][:, 1] * exact_basis_vals_1d[2][:, 1],
0. * x
]).T))
# x3 derivative
exact_basis_matrix = np.vstack(
(exact_basis_matrix,
np.asarray([
0. * x, 0. * x, 0. * x, exact_basis_derivs_1d[2][:, 1], 0 * x,
0 * x, 0 * x,
exact_basis_derivs_1d[2][:, 1] * exact_basis_vals_1d[0][:, 1],
exact_basis_derivs_1d[2][:, 1] * exact_basis_vals_1d[1][:, 1],
exact_basis_derivs_1d[2][:, 2]
]).T))
func = poly.basis_matrix
exact_basis_matrix_derivs = exact_basis_matrix[samples.shape[1]:]
basis_matrix_derivs_fd = np.empty_like(exact_basis_matrix_derivs)
for ii in range(samples.shape[1]):
basis_matrix_derivs_fd[ii::samples.shape[1], :] = approx_fprime(
samples[:, ii:ii + 1], func)
# print(np.linalg.norm(
# exact_basis_matrix_derivs-basis_matrix_derivs_fd,
# ord=np.inf))
assert np.allclose(exact_basis_matrix_derivs,
basis_matrix_derivs_fd,
atol=1e-7,
rtol=1e-7)
assert np.allclose(exact_basis_matrix, basis_matrix)