def __init__(self, beta_coeff, idx_set, jpdf):
self.beta_coeff = beta_coeff
self.idx_set = idx_set
self.jpdf = jpdf
self.N = jpdf.getDimension()
# get the distribution type of each random variable
dist_types = []
for i in range(self.N):
dist_type = self.jpdf.getMarginal(i).getName()
dist_types.append(dist_type)
# create orthogonal univariate bases
poly_collection = ot.PolynomialFamilyCollection(self.N)
for i in range(self.N):
pdf = jpdf.getDistributionCollection()[i]
algo = ot.AdaptiveStieltjesAlgorithm(pdf)
poly_collection[i] = ot.StandardDistributionPolynomialFactory(algo)
# create multivariate basis
multivariate_basis = ot.OrthogonalProductPolynomialFactory(
poly_collection, ot.EnumerateFunction(self.N))
# get enumerate function (multi-index handling)
enum_func = multivariate_basis.getEnumerateFunction()
# get epansion
self.expansion = multivariate_basis.getSubBasis(
transform_multi_index_set(idx_set, enum_func))
# create openturns surrogate model
sur_model = ot.FunctionCollection()
for i in range(len(self.expansion)):
multi = str(beta_coeff[i]) + '*x'
help_function = ot.SymbolicFunction(['x'], [multi])
sur_model.add(ot.ComposedFunction(help_function,
self.expansion[i]))
self.surrogate_model = np.sum(sur_model)
myCollection[11] = distribution_J4 myCollection[12] = distribution_P1 myCollection[13] = distribution_P2 myCollection[14] = distribution_P3 myCollection[15] = distribution_P4 # Création d'une distribution ? en fonction de la collection myDistribution = ot.ComposedDistribution(myCollection) # ??? vectX = ot.RandomVector(myDistribution) ######################## ### Chaos Polynomial ### ######################## polyColl = ot.PolynomialFamilyCollection(dim) for i in range(dim): polyColl[i] = ot.HermiteFactory() enumerateFunction = ot.LinearEnumerateFunction(dim) multivariateBasis = ot.OrthogonalProductPolynomialFactory(polyColl, enumerateFunction) basisSequenceFactory = ot.LARS() fittingAlgorithm = ot.CorrectedLeaveOneOut() approximationAlgorithm = ot.LeastSquaresMetaModelSelectionFactory(basisSequenceFactory, fittingAlgorithm) # Génération du plan d'expériences N = 200 ot.RandomGenerator.SetSeed(77) Liste_test = ot.LHSExperiment(myDistribution, N) InputSample = Liste_test.generate()
def dali_pce(func,
N,
jpdf_cp,
jpdf_ot,
tol=1e-12,
max_fcalls=1000,
verbose=True,
interp_dict={}):
if not interp_dict: # if dictionary is empty --> cold-start
idx_act = [] # M_activated x N
idx_adm = [] # M_admissible x N
fevals_act = [] # M_activated x 1
fevals_adm = [] # M_admissible x 1
coeff_act = [] # M_activated x 1
coeff_adm = [] # M_admissible x 1
# start with 0 multi-index
knot0 = []
for n in range(N):
# get knots per dimension based on maximum index
kk, ww = seq_lj_1d(order=0, dist=jpdf_cp[n])
knot0.append(kk[0])
feval = func(knot0)
# update activated sets
idx_act.append([0] * N)
coeff_act.append(feval)
fevals_act.append(feval)
# local error indicators
local_error_indicators = np.abs(coeff_act)
# get the OT distribution type of each random variable
dist_types = []
for i in range(N):
dist_type = jpdf_ot.getMarginal(i).getName()
dist_types.append(dist_type)
# create orthogonal univariate bases
poly_collection = ot.PolynomialFamilyCollection(N)
for i in range(N):
if dist_types[i] == 'Uniform':
poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily(
ot.LegendreFactory())
elif dist_types[i] == 'Normal':
poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily(
ot.HermiteFactory())
elif dist_types[i] == 'Beta':
poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily(
ot.JacobiFactory())
elif dist_types[i] == 'Gamma':
poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily(
ot.LaguerreFactory())
else:
pdf = jpdf_ot.getDistributionCollection()[i]
algo = ot.AdaptiveStieltjesAlgorithm(pdf)
poly_collection[i] = ot.StandardDistributionPolynomialFactory(
algo)
# create multivariate basis
mv_basis = ot.OrthogonalProductPolynomialFactory(
poly_collection, ot.EnumerateFunction(N))
# get enumerate function (multi-index handling)
enum_func = mv_basis.getEnumerateFunction()
else:
idx_act = interp_dict['idx_act']
idx_adm = interp_dict['idx_adm']
coeff_act = interp_dict['coeff_act']
coeff_adm = interp_dict['coeff_adm']
fevals_act = interp_dict['fevals_act']
fevals_adm = interp_dict['fevals_adm']
mv_basis = interp_dict['mv_basis']
enum_func = interp_dict['enum_func']
# local error indicators
local_error_indicators = np.abs(coeff_adm)
# compute global error indicator
global_error_indicator = local_error_indicators.sum() # max or sum
# fcalls / M approx. terms up to now
fcalls = len(idx_act) + len(idx_adm) # fcalls = M --> approx. terms
# maximum index per dimension
max_idx_per_dim = np.max(idx_act + idx_adm, axis=0)
# univariate knots and polynomials per dimension
knots_per_dim = {}
for n in range(N):
kk, ww = seq_lj_1d(order=max_idx_per_dim[n], dist=jpdf_cp[n])
knots_per_dim[n] = kk
# start iterations
while global_error_indicator > tol and fcalls < max_fcalls:
if verbose:
print(fcalls)
print(global_error_indicator)
# the index added last to the activated set is the one to be refined
last_act_idx = idx_act[-1][:]
# compute the knot corresponding to the lastly added index
last_knot = [
knots_per_dim[n][i] for n, i in zip(range(N), last_act_idx)
]
# get admissible neighbors of the lastly added index
adm_neighbors = admissible_neighbors(last_act_idx, idx_act)
for an in adm_neighbors:
# update admissible index set
idx_adm.append(an)
# find which parameter/direction n (n=1,2,...,N) gets refined
n_ref = np.argmin(
[idx1 == idx2 for idx1, idx2 in zip(an, last_act_idx)])
# sequence of 1d Leja nodes/weights for the given refinement
knots_n, weights_n = seq_lj_1d(an[n_ref], jpdf_cp[int(n_ref)])
# update max_idx_per_dim, knots_per_dim, if necessary
if an[n_ref] > max_idx_per_dim[n_ref]:
max_idx_per_dim[n_ref] = an[n_ref]
knots_per_dim[n_ref] = knots_n
# find new_knot and compute function on new_knot
new_knot = last_knot[:]
new_knot[n_ref] = knots_n[-1]
feval = func(new_knot)
fevals_adm.append(feval)
fcalls += 1 # update function calls
# create PCE basis
idx_system = idx_act + idx_adm
idx_system_single = transform_multi_index_set(idx_system, enum_func)
system_basis = mv_basis.getSubBasis(idx_system_single)
# get corresponding evaluations
fevals_system = fevals_act + fevals_adm
# multi-dimensional knots
M = len(idx_system) # equations terms
knots_md = [[knots_per_dim[n][idx_system[m][n]] for m in range(M)]
for n in range(N)]
knots_md = np.array(knots_md).T
# design matrix
D = get_design_matrix(system_basis, knots_md)
# solve system of equaations
Q, R = scl.qr(D, mode='economic')
c = Q.T.dot(fevals_system)
coeff_system = scl.solve_triangular(R, c)
# find the multi-index with the largest contribution, add it to idx_act
# and delete it from idx_adm
coeff_act = coeff_system[:len(idx_act)].tolist()
coeff_adm = coeff_system[-len(idx_adm):].tolist()
help_idx = np.argmax(np.abs(coeff_adm))
idx_add = idx_adm.pop(help_idx)
pce_coeff_add = coeff_adm.pop(help_idx)
fevals_add = fevals_adm.pop(help_idx)
idx_act.append(idx_add)
coeff_act.append(pce_coeff_add)
fevals_act.append(fevals_add)
# re-compute coefficients of admissible multi-indices
# local error indicators
local_error_indicators = np.abs(coeff_adm)
# compute global error indicator
global_error_indicator = local_error_indicators.sum() # max or sum
# store expansion data in dictionary
interp_dict = {}
interp_dict['idx_act'] = idx_act
interp_dict['idx_adm'] = idx_adm
interp_dict['coeff_act'] = coeff_act
interp_dict['coeff_adm'] = coeff_adm
interp_dict['fevals_act'] = fevals_act
interp_dict['fevals_adm'] = fevals_adm
interp_dict['enum_func'] = enum_func
interp_dict['mv_basis'] = mv_basis
return interp_dict
def alsace(func,
N,
jpdf,
tol=1e-22,
sample_type='R',
limit_cond=5,
max_fcalls=1000,
seed=123,
ed_file=None,
ed_fevals_file=None,
verbose=True,
pce_dict={}):
"""
ALSACE - Approximations via Lower-Set and Least-Squares-based Adaptive Chaos Expansions
func: function to be approximated.
N: number of parameters.
jpdf: joint probability density function.
limit_cond: maximum allowed condition number of tr(inv(D.T*D))
sample_type: 'R'-random, 'L'-LHS
seed: sampling seed
tol, max_fcalls: exit criteria, self-explanatory.
ed_file, ed_fevals_file: experimental design and corresponding evaluations
'act': activated, i.e. already part of the approximation.
'adm': admissible, i.e. candidates for the approximation's expansion.
"""
if not pce_dict: # if pce_dict is empty --> cold-start
idx_act = []
idx_act.append([0] * N) # start with 0 multi-index
idx_adm = []
# set seed
ot.RandomGenerator.SetSeed(seed)
ed_size = 2 * N # initial number of samples
# initial experimental design and coresponding evaluations
ed, ed_fevals = get_ed(func,
jpdf,
ed_size,
sample_type=sample_type,
knots=[],
values=[],
ed_file=ed_file,
ed_fevals_file=ed_fevals_file)
global_error_indicator = 1.0 # give arbitrary sufficiently large value
# get the distribution type of each random variable
dist_types = []
for i in range(N):
dist_type = jpdf.getMarginal(i).getName()
dist_types.append(dist_type)
# create orthogonal univariate bases
poly_collection = ot.PolynomialFamilyCollection(N)
for i in range(N):
pdf = jpdf.getDistributionCollection()[i]
algo = ot.AdaptiveStieltjesAlgorithm(pdf)
poly_collection[i] = ot.StandardDistributionPolynomialFactory(algo)
# create multivariate basis
mv_basis = ot.OrthogonalProductPolynomialFactory(
poly_collection, ot.EnumerateFunction(N))
# get enumerate function (multi-index handling)
enum_func = mv_basis.getEnumerateFunction()
else: # get data from dictionary
idx_act = pce_dict['idx_act']
idx_adm = pce_dict['idx_adm']
pce_coeff_act = pce_dict['pce_coeff_act']
pce_coeff_adm = pce_dict['pce_coeff_adm']
ed = pce_dict['ed']
ed_fevals = pce_dict['ed_fevals']
ed_size = len(ed_fevals)
# compute local and global error indicators
global_error_indicator = np.sum(np.array(pce_coeff_adm)**2)
enum_func = pce_dict['enum_func']
mv_basis = pce_dict['mv_basis']
#
while ed_size < max_fcalls and global_error_indicator > tol:
# the index added last to the activated set is the one to be refined
last_act_idx = idx_act[-1][:]
# get admissible neighbors of the lastly added index
adm_neighbors = admissible_neighbors(last_act_idx, idx_act)
# update admissible indices
idx_adm = idx_adm + adm_neighbors
# get polynomial basis for the LS problem
idx_ls = idx_act + idx_adm
idx_ls_single = transform_multi_index_set(idx_ls, enum_func)
ls_basis = mv_basis.getSubBasis(idx_ls_single)
ls_basis_size = len(ls_basis)
# construct the design matrix D and compute its QR decomposition
D = get_design_matrix(ls_basis, ed)
Q, R = sp.qr(D, mode='economic')
# construct information matrix A= D^T*D
A = np.matmul(D.T, D) / ed_size
trAinv_test = np.sum(1. / np.linalg.eig(A)[0])
trAinv = np.trace(np.linalg.inv(A))
print('new trace ', trAinv_test)
print('old trace ', trAinv)
# If tr(A) becomes too large, enrich the ED until tr(A) becomes
# acceptable or until ed_size reaches max_fcalls
while (trAinv > limit_cond
and ed_size < max_fcalls) or ed_size < ls_basis_size:
# inform user
if verbose:
print('WARNING: tr(inv(A)) = ', trAinv)
print('WARNING: cond(D) = ', np.linalg.cond(D))
print("")
# select new size for the ED
if ls_basis_size > ed_size:
ed_size = ls_basis_size + N
elif ed_size + N > max_fcalls:
ed_size = max_fcalls
else:
ed_size = ed_size + N
# expand ED
ed, ed_fevals = get_ed(func,
jpdf,
ed_size,
sample_type=sample_type,
knots=ed,
values=ed_fevals,
ed_file=ed_file,
ed_fevals_file=ed_fevals_file)
# construct the design matrix D and compute its QR decomposition
D = get_design_matrix(ls_basis, ed)
Q, R = sp.qr(D, mode='economic')
# construct information matrix A= D^T*D
A = np.matmul(D.T, D) / ed_size
trAinv = np.trace(np.linalg.inv(A))
# solve LS problem
c = Q.T.dot(ed_fevals)
pce_coeff_ls = sp.solve_triangular(R, c)
# find the multi-index with the largest contribution, add it to idx_act
# and delete it from idx_adm
pce_coeff_act = pce_coeff_ls[:len(idx_act)].tolist()
pce_coeff_adm = pce_coeff_ls[-len(idx_adm):].tolist()
help_idx = np.argmax(np.abs(pce_coeff_adm))
idx_add = idx_adm.pop(help_idx)
pce_coeff_add = pce_coeff_adm.pop(help_idx)
idx_act.append(idx_add)
pce_coeff_act.append(pce_coeff_add)
# store expansion data in dictionary
pce_dict = {}
pce_dict['idx_act'] = idx_act
pce_dict['idx_adm'] = idx_adm
pce_dict['pce_coeff_act'] = pce_coeff_act
pce_dict['pce_coeff_adm'] = pce_coeff_adm
pce_dict['ed'] = ed
pce_dict['ed_fevals'] = ed_fevals
pce_dict['enum_func'] = enum_func
pce_dict['mv_basis'] = mv_basis
return pce_dict
distribution = ot.ComposedDistribution(
[ot.Normal(), ot.Uniform(), ot.Gamma(2.75, 1.0), ot.Beta(2.5, 1.0, -1.0, 2.0)])
# %%
inputDimension = distribution.getDimension()
inputDimension
# %%
# STEP 1: Construction of the multivariate orthonormal basis
# ----------------------------------------------------------
# %%
# Create the univariate polynomial family collection which regroups the polynomial families for each direction.
# %%
polyColl = ot.PolynomialFamilyCollection(inputDimension)
# %%
# We could use the Krawtchouk and Charlier families (for discrete distributions).
# %%
polyColl[0] = ot.KrawtchoukFactory()
polyColl[1] = ot.CharlierFactory()
# %%
# We could also use the automatic selection of the polynomial which corresponds to the distribution: this is done with the `StandardDistributionPolynomialFactory` class.
# %%
for i in range(inputDimension):
marginal = distribution.getMarginal(i)
polyColl[i] = ot.StandardDistributionPolynomialFactory(marginal)
from mero_pdf import jpdf
from mero_pdf_ot import jpdf_ot
sys.path.append("../")
from pce_tools import transform_multi_index_set, get_design_matrix, PCE_Surrogate
# construct joint pdf
N = len(jpdf)
# get the distribution type of each random variable
dist_types = []
for i in range(N):
dist_type = jpdf_ot.getMarginal(i).getName()
dist_types.append(dist_type)
# create orthogonal univariate bases
poly_collection = ot.PolynomialFamilyCollection(N)
for i in range(N):
if dist_types[i] == 'Uniform':
poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily(
ot.LegendreFactory())
elif dist_types[i] == 'Normal':
poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily(
ot.HermiteFactory())
elif dist_types[i] == 'Beta':
poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily(
ot.JacobiFactory())
elif dist_types[i] == 'Gamma':
poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily(
ot.LaguerreFactory())
else:
pdf = jpdf_ot.getDistributionCollection()[i]
def _compute_coefficients_legendre(self,
sample_paths,
legendre_quadrature_order=None):
dimension = self._lower_bound.size
truncation_order = self._truncation_order
if legendre_quadrature_order is None:
legendre_quadrature_order = self._legendre_quadrature_order
elif type(legendre_quadrature_order) is not int \
or legendre_quadrature_order <= 0:
raise ValueError('legendre_quadrature_order must be a positive ' +
'integer.')
n_sample_paths = len(sample_paths)
# Gauss-Legendre quadrature nodes and weights
polyColl = ot.PolynomialFamilyCollection([ot.LegendreFactory()] *
dimension)
polynoms = ot.OrthogonalProductPolynomialFactory(polyColl)
U, W = polynoms.getNodesAndWeights(
ot.Indices([legendre_quadrature_order] * dimension))
W = np.ravel(W)
U = np.array(U)
scale = (self._upper_bound - self._lower_bound) / 2.
shift = (self._upper_bound + self._lower_bound) / 2.
X = scale * U + shift
# Compute coefficients
try:
available_memory = int(.9 * get_available_memory())
except:
if self.verbose:
print('WRN: Available memory estimation failed! '
'Assuming 1Gb is available (first guess).')
available_memory = 1024**3
max_size = int(available_memory / 8 / truncation_order /
n_sample_paths)
batch_size = min(W.size, max_size)
if self.verbose and batch_size < W.size:
print('RAM: %d Mb available' % (available_memory / 1024**2))
print('RAM: %d allocable terms / %d total terms' %
(max_size, W.size))
print('RAM: %d loops required' % np.ceil(float(W.size) / max_size))
while True:
coefficients = np.zeros((n_sample_paths, truncation_order))
try:
n_done = 0
while n_done < W.size:
sample_paths_values = np.vstack([
np.ravel(sample_paths[i](X[n_done:(n_done +
batch_size)]))
for i in range(n_sample_paths)
])
mean_values = np.ravel(
self._mean(X[n_done:(n_done +
batch_size)]))[np.newaxis, :]
centered_sample_paths_values = \
sample_paths_values - mean_values
del sample_paths_values, mean_values
eigenelements_values = np.vstack([
self._eigenfunctions[k](
X[n_done:(n_done + batch_size)]) /
np.sqrt(self._eigenvalues[k])
for k in range(truncation_order)
])
coefficients += np.sum(
W[np.newaxis, np.newaxis, n_done:(n_done + batch_size)]
* centered_sample_paths_values[:, np.newaxis, :] *
eigenelements_values[np.newaxis, :, :],
axis=-1)
del centered_sample_paths_values, eigenelements_values
n_done += batch_size
break
except MemoryError:
batch_size /= 2
coefficients *= np.prod(self._upper_bound - self._lower_bound)
return coefficients
def _legendre_galerkin_scheme(self,
legendre_galerkin_order=10,
legendre_quadrature_order=None):
# Input checks
if legendre_galerkin_order <= 0:
raise ValueError('legendre_galerkin_order must be a positive ' +
'integer!')
if legendre_quadrature_order is not None:
if legendre_quadrature_order <= 0:
raise ValueError('legendre_quadrature_order must be a ' +
'positive integer!')
# Settings
dimension = self._lower_bound.size
truncation_order = self._truncation_order
galerkin_size = ot.EnumerateFunction(
dimension).getStrataCumulatedCardinal(legendre_galerkin_order)
if legendre_quadrature_order is None:
legendre_quadrature_order = 2 * legendre_galerkin_order + 1
# Check if the current settings are compatible
if truncation_order > galerkin_size:
raise ValueError(
'The truncation order must be less than or ' +
'equal to the size of the functional basis in the chosen ' +
'Legendre Galerkin scheme. Current size of the galerkin basis '
+ 'only allows to get %d terms in the KL expansion.' %
galerkin_size)
# Construction of the Galerkin basis: tensorized Legendre polynomials
tensorized_legendre_polynomial_factory = \
ot.PolynomialFamilyCollection([ot.LegendreFactory()] * dimension)
tensorized_legendre_polynomial_factory = \
ot.OrthogonalProductPolynomialFactory(
tensorized_legendre_polynomial_factory)
tensorized_legendre_polynomials = \
[tensorized_legendre_polynomial_factory.build(i)
for i in range(galerkin_size)]
# Compute matrix C coefficients using Gauss-Legendre quadrature
polyColl = ot.PolynomialFamilyCollection([ot.LegendreFactory()] *
dimension * 2)
polynoms = ot.OrthogonalProductPolynomialFactory(polyColl)
U, W = polynoms.getNodesAndWeights(
ot.Indices([legendre_quadrature_order] * dimension * 2))
W = np.ravel(W)
scale = (self._upper_bound - self._lower_bound) / 2.
shift = (self._upper_bound + self._lower_bound) / 2.
U = np.array(U)
X = np.repeat(scale, 2) * U + np.repeat(shift, 2)
if self.verbose:
print('Computing matrix C...')
try:
available_memory = int(.9 * get_available_memory())
except:
if self.verbose:
print('WRN: Available memory estimation failed! '
'Assuming 1Gb is available (first guess).')
available_memory = 1024**3
max_size = int(available_memory / 8 / galerkin_size**2)
batch_size = min(W.size, max_size)
if self.verbose and batch_size < W.size:
print('RAM: %d Mb available' % (available_memory / 1024**2))
print('RAM: %d allocable terms / %d total terms' %
(max_size, W.size))
print('RAM: %d loops required' % np.ceil(float(W.size) / max_size))
while True:
C = np.zeros((galerkin_size, galerkin_size))
try:
n_done = 0
while n_done < W.size:
covariance_at_X = self._covariance(X[n_done:(n_done +
batch_size)])
H1 = np.vstack([
np.ravel(tensorized_legendre_polynomials[i](
U[n_done:(n_done + batch_size), :dimension]))
for i in range(galerkin_size)
])
H2 = np.vstack([
np.ravel(tensorized_legendre_polynomials[i](
U[n_done:(n_done + batch_size), dimension:]))
for i in range(galerkin_size)
])
C += np.sum(W[np.newaxis, np.newaxis,
n_done:(n_done + batch_size)] *
covariance_at_X[np.newaxis, np.newaxis, :] *
H1[np.newaxis, :, :] * H2[:, np.newaxis, :],
axis=-1)
del covariance_at_X, H1, H2
n_done += batch_size
break
except MemoryError:
batch_size /= 2
C *= np.prod(self._upper_bound - self._lower_bound)**2.
# Matrix B is orthonormal up to some constant
B = np.diag(
np.repeat(np.prod(self._upper_bound - self._lower_bound),
galerkin_size))
# Solve the generalized eigenvalue problem C D = L B D in L, D
if self.verbose:
print('Solving generalized eigenvalue problem...')
eigenvalues, eigenvectors = linalg.eigh(C, b=B, lower=True)
eigenvalues, eigenvectors = eigenvalues.real, eigenvectors.real
# Sort the eigensolutions in the descending order of eigenvalues
order = eigenvalues.argsort()[::-1]
eigenvalues = eigenvalues[order]
eigenvectors = eigenvectors[:, order]
# Truncate the expansion
eigenvalues = eigenvalues[:truncation_order]
eigenvectors = eigenvectors[:, :truncation_order]
# Eliminate unsignificant negative eigenvalues
if eigenvalues.min() <= 0.:
if eigenvalues.min() > .01 * eigenvalues.max():
raise Exception(
'The smallest significant eigenvalue seems ' +
'to be negative... Check the positive definiteness of the '
+ 'covariance function.')
else:
truncation_order = np.nonzero(eigenvalues <= 0)[0][0]
eigenvalues = eigenvalues[:truncation_order]
eigenvectors = eigenvectors[:, :truncation_order]
self._truncation_order = truncation_order
print('WRN: truncation_order was too large.')
print('It has been reset to: %d' % truncation_order)
# Define eigenfunctions
class LegendrePolynomialsBasedEigenFunction():
def __init__(self, vector):
self._vector = vector
def __call__(self, x):
x = np.asanyarray(x)
if x.ndim <= 1:
x = np.atleast_2d(x).T
u = (x - shift) / scale
return np.sum([
np.ravel(tensorized_legendre_polynomials[i](u)) *
self._vector[i] for i in range(truncation_order)
],
axis=0)
# Set attributes
self._eigenvalues = eigenvalues
self._eigenfunctions = [
LegendrePolynomialsBasedEigenFunction(vector)
for vector in eigenvectors.T
]
self._legendre_galerkin_order = legendre_galerkin_order
self._legendre_quadrature_order = legendre_quadrature_order
