def _build_mu(self, D, order):
r"""Build the list containing the lists :math:`\mu_i` where each
:math:`\mu_i` contains the :math:`D`-tuples in reverse lexicographical order.
:param D: The dimension :math:`D` of the wavepacket.
:param order: The maximal :math:`l_1` norm of any basis index :math:`\underline{k}`.
"""
return [list(lattice_points_norm(D, i)) for i in range(order + 1)]
def _build_mu(self, D, order):
r"""Build the list containing the lists :math:`\mu_i` where each
:math:`\mu_i` contains the :math:`D`-tuples in reverse lexicographical order.
:param D: The dimension :math:`D` of the wavepacket.
:param order: The maximal :math:`l_1` norm of any basis index :math:`\underline{k}`.
"""
return [list(lattice_points_norm(D, i)) for i in range(order + 1)]
def overlap(self, D, Pi, eps):
r"""Compute the overlap matrix:
.. math::
\mathbf{K}_{r,c} := \langle \psi_{\underline{e_r}}[\Pi] | \phi_{\underline{e_c}}[\Pi] \rangle
:param D: The dimension :math:`D`.
:param Pi: The parameter set :math:`\Pi`.
:param eps: The semiclassical scaling parameter :math:`\varepsilon`.
"""
ED = list(lattice_points_norm(D, 1))
BS = SimplexShape(D, 1)
WPo = HagedornWavepacket(D, 1, eps)
WPo.set_parameters(Pi)
WPo.set_basis_shapes([BS])
WPn = HagedornWavepacketPsi(D, 1, eps)
WPn.set_parameters(Pi)
WPn.set_basis_shapes([BS])
TPG = TensorProductQR(D * [GaussHermiteQR(2)])
DHQ = DirectHomogeneousQuadrature(TPG)
G = DHQ.transform_nodes(Pi, eps)
W = TPG.get_weights()
phi = WPo.evaluate_basis_at(G, component=0)
psi = WPn.evaluate_basis_at(G, component=0)
K = zeros((D, D), dtype=complexfloating)
for r, er in enumerate(ED):
for c, ec in enumerate(ED):
K[r, c] = eps**D * sum(
conjugate(psi[BS[er], :]) * phi[BS[ec], :] * W)
return K
def overlap(self, D, Pi, eps):
r"""Compute the overlap matrix:
.. math::
\mathbf{K}_{r,c} := \langle \psi_{\underline{e_r}}[\Pi] | \phi_{\underline{e_c}}[\Pi] \rangle
:param D: The dimension :math:`D`.
:param Pi: The parameter set :math:`\Pi`.
:param eps: The semiclassical scaling parameter :math:`\varepsilon`.
"""
ED = list(lattice_points_norm(D, 1))
BS = SimplexShape(D, 1)
WPo = HagedornWavepacket(D, 1, eps)
WPo.set_parameters(Pi)
WPo.set_basis_shapes([BS])
WPn = HagedornWavepacketPsi(D, 1, eps)
WPn.set_parameters(Pi)
WPn.set_basis_shapes([BS])
TPG = TensorProductQR(D * [GaussHermiteQR(2)])
DHQ = DirectHomogeneousQuadrature(TPG)
G = DHQ.transform_nodes(Pi, eps)
W = TPG.get_weights()
phi = WPo.evaluate_basis_at(G, component=0)
psi = WPn.evaluate_basis_at(G, component=0)
K = zeros((D, D), dtype=complexfloating)
for r, er in enumerate(ED):
for c, ec in enumerate(ED):
K[r, c] = eps**D * sum(conjugate(psi[BS[er], :]) * phi[BS[ec], :] * W)
return K
def construct_rule(self, K, tolerance=1e-15):
r"""Compute the quadrature nodes :math:`\{\gamma_i\}_i` and quadrature
weights :math:`\{\omega_i\}_i`.
:param K: The level :math:`K` of the Smolyak construction.
:param tolerance: Tolerance for dropping identical quadrature nodes.
.. note:: This is an internal method and there should be no reason
to explicitely call it manually.
.. warning:: This method is very expensive and may take a long time
to finish. Also, the quadrature nodes may use large amounts
of memory depending on the dimension and level parameters.
"""
# Check for valid level parameter
if not K >= 1:
raise ValueError("Smolyak level has to be 1 at least.")
if K > max(self._rules.keys()):
raise ValueError("Not enough quadrature rules to build Smolyak grid of level %d" % K)
self._level = K
D = self._dimension
allnodes = []
allweights = []
factors = []
# Index Set
for q in range(max(0, K - D), K):
S = lattice_points_norm(D, q)
for j, s in enumerate(S):
# Only use non-negative nodes for the construction.
# The quadrature nodes \gamma.
rules = [self._rules[si + 1] for si in s]
nodes = [rule.get_nodes() for rule in rules]
indices = [where(n >= 0) for n in nodes]
nodes = meshgrid_nd([n[i] for n, i in zip(nodes, indices)])
nodes = vstack([node.reshape(-1) for node in nodes])
# The quadrature weights \omega.
weights = [rule.get_weights() for rule in rules]
weights = meshgrid_nd([w[i] for w, i in zip(weights, indices)])
weights = reduce(multiply, weights)
allnodes.append(nodes)
allweights.append(weights.reshape(-1))
factors.append((-1)**(K - 1 - q) * binom(D - 1, K - 1 - q))
# Sort
allnodes = hstack(allnodes).reshape(D, -1)
allweights = hstack([f * w for f, w in zip(factors, allweights)]).reshape(1, -1)
I = lexsort(map(squeeze, vsplit(allnodes, D))[::-1])
allnodes = allnodes[:, I].reshape(D, -1)
allweights = allweights[:, I].reshape(1, -1)
# Remove duplicates
last = 0
I = [last]
no = norm(allnodes[:, :-1] - allnodes[:, 1:], axis=0)
for col in range(1, allnodes.shape[1]):
if no[col - 1] <= tolerance:
allweights[0, last] += allweights[0, col]
allweights[0, col] = 0
else:
last = col
I.append(last)
allnodes = allnodes[:, I]
allweights = allweights[:, I]
# Mirror points to all other hyperoctants
for d in range(D):
indices = abs(allnodes[d, :]) >= tolerance
mirrorn = allnodes[:, indices]
mirrorn[d, :] *= -1.0
mirrorw = allweights[:, indices]
allnodes = hstack([allnodes, mirrorn]).reshape(D, -1)
allweights = hstack([allweights, mirrorw]).reshape(1, -1)
self._nodes = allnodes
self._weights = allweights
self._number_nodes = allnodes.shape[1]
