def set_quadrature(self, quadrature): r""" Set the ``InhomogeneousQuadrature`` instance used for evaluating brakets. :param quadrature: The new ``InhomogeneousQuadrature`` instance. May be ``None`` to use a dafault one with a quadrature rule of order :math:`K+4`. """ # TODO: Put an "extra accuracy" parameter into global defaults with value of 4. # TODO: Improve on the max(basis_size) later # TODO: Rethink if wavepackets should contain a QR if quadrature is None: self.quadrature = InhomogeneousQuadrature(order=max(self.basis_size) + 4) else: self.quadrature = quadrature
def create_quadrature(self, description): try: qe_type = description["type"] except: # Default setting qe_type = "InhomogeneousQuadrature" # TODO: Maybe denest QR initialization? if qe_type == "HomogeneousQuadrature": from HomogeneousQuadrature import HomogeneousQuadrature QR = self.create_quadrature_rule(description["qr"]) QE = HomogeneousQuadrature(QR) elif qe_type == "InhomogeneousQuadrature": from InhomogeneousQuadrature import InhomogeneousQuadrature QR = self.create_quadrature_rule(description["qr"]) QE = InhomogeneousQuadrature(QR) else: raise ValueError("Unknown basis shape type " + str(qe_type)) return QE
def spawn_basis_projection(self, mother, child, component): r""" Update the superposition coefficients of mother and spawned wavepacket. We do a full basis projection to the basis of the spawned wavepacket here. """ c_old = mother.get_coefficients(component=component) # Mother packet c_new_m = np.zeros(c_old.shape, dtype=np.complexfloating) c_new_m[:self.K,:] = c_old[:self.K,:] # Spawned packet c_new_s = np.zeros((child.get_basis_size(component=component),1), dtype=np.complexfloating) # The quadrature quadrature = InhomogeneousQuadrature() # Quadrature rule. Assure the "right" quadrature is choosen if # mother and child have different basis sizes if mother.get_basis_size(component=component) > child.get_basis_size(component=component): quadrature.set_qr(mother.get_quadrature().get_qr()) else: quadrature.set_qr(child.get_quadrature().get_qr()) # The quadrature nodes and weights q0, QS = quadrature.mix_parameters(mother.get_parameters(), child.get_parameters()) nodes = quadrature.transform_nodes(mother.get_parameters(), child.get_parameters(), mother.eps) weights = quadrature.get_qr().get_weights() # Basis sets for both packets basis_m = mother.evaluate_basis_at(nodes, prefactor=True) basis_s = child.evaluate_basis_at(nodes, prefactor=True) max_order = min(child.get_basis_size(component=component), self.max_order) # Project to the basis of the spawned wavepacket # Original, inefficient code for projection # R = QR.get_order() # for i in xrange(max_order): # # Loop over all quadrature points # tmp = 0.0j # for r in xrange(R): # tmp += np.conj(np.dot(c_old[self.K:,0], basis_m[self.K:,r])) * basis_s[i,r] * weights[r] # # c_new_s[i,0] = self.eps * QS * tmp # Optimised and vectorised code (in ugly formatting) c_new_s[:max_order,:] = self.eps * QS * ( np.reshape( np.sum( np.transpose( np.reshape( np.conj( np.sum(c_old[self.K:,:] * basis_m[self.K:,:],axis=0) ) ,(-1,1) ) ) * (basis_s[:max_order,:] * weights[:,:]), axis=1 ) ,(-1,1) )) # Reassign the new coefficients mother.set_coefficients(c_new_m, component=component) child.set_coefficients(c_new_s, component=component) return (mother, child)
class HagedornWavepacketInhomogeneous(Wavepacket): r""" This class represents inhomogeneous vector valued wavepackets :math:`|\Psi\rangle`. """ def __init__(self, parameters): r""" Initialize the ``HagedornWavepacketInhomogeneous`` object that represents :math:`|\Psi\rangle`. :param parameters: A :py:class:`ParameterProvider` instance or a dict containing simulation parameters. :raise ValueError: For :math:`N < 1` or :math:`K < 2`. """ #: Number of components :math:`\Phi_i` the wavepacket :math:`|\Psi\rangle` has got. self.number_components = parameters["ncomponents"] if self.number_components < 1: raise ValueError("Number of components of the Hagedorn wavepacket has to be >= 1.") # Size of the basis from which we construct the wavepacket. # If there is a key "basis_size" in the input parameters, the corresponding # value can be either a single int or a list of ints. If there is no such key # we use the values from the global defaults. if parameters.has_key("basis_size"): bs = parameters["basis_size"] if type(bs) is list or type(bs) is tuple: if not len(bs) == self.number_components: raise ValueError("Number of value(s) for basis size(s) does not match.") self.basis_size = bs[:] else: self.basis_size = self.number_components * [ bs ] else: self.basis_size = self.number_components * [ GD.default_basis_size ] if any([bs < 2 for bs in self.basis_size]): raise ValueError("Number of basis fucntions for Hagedorn wavepacket has to be >= 2.") # Cache the parameter values epsilon we will use over and over again. self.eps = parameters["eps"] #: Data structure that contains the Hagedorn parameter sets :math:`\Pi_i` of each component :math:`\Phi_i`. #: The parameter values are initialized to the Harmonic Oscillator Eigenfunctions self.parameters = [ GD.default_Pi for i in xrange(self.number_components) ] #: The coefficients :math:`c^i` of the linear combination for each component :math:`\Phi_i`. self.coefficients = [ zeros((self.basis_size[index],1), dtype=complexfloating) for index in xrange(self.number_components) ] #: An object that can compute brakets via quadrature. self.quadrature = None self._cont_sqrt_cache = [ 0.0 for i in xrange(self.number_components) ] def __str__(self): r""" :return: A string describing the Hagedorn wavepacket. """ s = "Inhomogeneous Hagedorn wavepacket with "+str(self.number_components)+" components\n" return s def clone(self, keepid=False): # Parameters of this packet params = {"ncomponents": self.number_components, "eps": self.eps} # Create a new Packet other = HagedornWavepacketInhomogeneous(params) # If we wish to keep the packet ID if keepid is True: other.set_id(self.get_id()) # And copy over all (private) data other.set_basis_size(self.get_basis_size()) other.set_quadrature(self.get_quadrature()) other.set_parameters(self.get_parameters()) other.set_coefficients(self.get_coefficients()) other._cont_sqrt_cache = [ cache for cache in self._cont_sqrt_cache ] return other def get_parameters(self, component=None, aslist=False): r""" Get the Hagedorn parameters :math:`\Pi_i` of each component :math:`\Phi_i` of the wavepacket :math:`\Psi`. :param component: The index :math:`i` of the component whose parameters :math:`\Pi_i` we want to get. :param aslist: Dummy parameter for API compatibility with the homogeneous packets. :return: A list with all the sets :math:`\Pi_i` or a single set. """ if component is None: result = [ tuple(item) for item in self.parameters ] else: result = self.parameters[component] return tuple(result) def set_parameters(self, parameters, component=None): r""" Set the Hagedorn parameters :math:`\Pi_i` of each component :math:`\Phi_i` of the wavepacket :math:`\Psi`. :param parameters: A list or a single set of Hagedorn parameters. :param component: The index :math:`i` of the component whose parameters :math:`\Pi_i` we want to update. """ if component is None: for index, item in enumerate(parameters): self.parameters[index] = item[:] else: self.parameters[component] = parameters[:] def set_quadrature(self, quadrature): r""" Set the ``InhomogeneousQuadrature`` instance used for evaluating brakets. :param quadrature: The new ``InhomogeneousQuadrature`` instance. May be ``None`` to use a dafault one with a quadrature rule of order :math:`K+4`. """ # TODO: Put an "extra accuracy" parameter into global defaults with value of 4. # TODO: Improve on the max(basis_size) later # TODO: Rethink if wavepackets should contain a QR if quadrature is None: self.quadrature = InhomogeneousQuadrature(order=max(self.basis_size) + 4) else: self.quadrature = quadrature def get_quadrature(self): r""" Return the ``InhomogeneousQuadrature`` instance used for evaluating brakets. :return: The current instance ``InhomogeneousQuadrature``. """ return self.quadrature def evaluate_basis_at(self, nodes, component, prefactor=False): r""" Evaluate the Hagedorn functions :math:`\phi_k` recursively at the given nodes :math:`\gamma`. :param nodes: The nodes :math:`\gamma` at which the Hagedorn functions are evaluated. :param component: The index :math:`i` of the component whose basis functions :math:`\phi^i_k` we want to evaluate. :param prefactor: Whether to include a factor of :math:`\left(\det\left(Q_i\right)\right)^{-\frac{1}{2}}`. :return: Returns a twodimensional array :math:`H` where the entry :math:`H[k,i]` is the value of the :math:`k`-th Hagedorn function evaluated at the node :math:`i`. """ H = zeros((self.basis_size[component], nodes.size), dtype=complexfloating) (P, Q, S, p, q) = self.parameters[component] Qinv = Q**(-1.0) Qbar = conj(Q) nodes = nodes.reshape((1,nodes.size)) H[0] = pi**(-0.25)*self.eps**(-0.5) * exp(1.0j/self.eps**2 * (0.5*P*Qinv*(nodes-q)**2 + p*(nodes-q))) H[1] = Qinv*sqrt(2.0/self.eps**2) * (nodes-q) * H[0] for k in xrange(2, self.basis_size[component]): H[k] = Qinv*sqrt(2.0/self.eps**2)*1.0/sqrt(k) * (nodes-q) * H[k-1] - Qinv*Qbar*sqrt((k-1.0)/k) * H[k-2] if prefactor is True: sqrtQ, self._cont_sqrt_cache[component] = cont_sqrt(Q, reference=self._cont_sqrt_cache[component]) H = 1.0/sqrtQ*H return H def evaluate_at(self, nodes, component=None, prefactor=False): r""" Evaluete the Hagedorn wavepacket :math:`\Psi` at the given nodes :math:`\gamma`. :param nodes: The nodes :math:`\gamma` at which the Hagedorn wavepacket gets evaluated. :param component: The index :math:`i` of a single component :math:`\Phi_i` to evaluate. (Defaults to 'None' for evaluating all components.) :param prefactor: Whether to include a factor of :math:`\left(\det\left(Q_i\right)\right)^{-\frac{1}{2}}`. :return: A list of arrays or a single array containing the values of the :math:`\Phi_i` at the nodes :math:`\gamma`. """ nodes = nodes.reshape((1,nodes.size)) if component is not None: # Avoid the expensive evaluation of unused other bases (P, Q, S, p, q) = self.parameters[component] basis = self.evaluate_basis_at(nodes, component, prefactor=prefactor) phase = exp(1.0j*S/self.eps**2) values = phase * sum(self.coefficients[component] * basis, axis=0) else: values = [ zeros(nodes.shape) for index in xrange(self.number_components) ] for index in xrange(self.number_components): (P, Q, S, p, q) = self.parameters[index] basis = self.evaluate_basis_at(nodes, index, prefactor=prefactor) phase = exp(1.0j*S/self.eps**2) values[index] = phase * sum(self.coefficients[index] * basis, axis=0) return values def get_norm(self, component=None, summed=False): r""" Calculate the :math:`L^2` norm of the wavepacket :math:`|\Psi\rangle`. :param component: The component :math:`\Phi_i` of which the norm is calculated. :param summed: Whether to sum up the norms of the individual components :math:`\Phi_i`. :return: A list containing the norms of all components :math:`\Phi_i` or the overall norm of :math:`\Psi`. """ if component is None: result = [ norm(item) for item in self.coefficients ] if summed is True: result = reduce(lambda x,y: x+conj(y)*y, result, 0) result = sqrt(result) else: result = norm(self.coefficients[component]) return result def potential_energy(self, potential, summed=False): r""" Calculate the potential energy :math:`\langle\Psi|V|\Psi\rangle` of the wavepacket componentwise. :param potential: The potential energy operator :math:`V` as function. :param summed: Wheter to sum up the individual integrals :math:`\langle\Phi_i|V_{i,j}|\Phi_j\rangle`. :return: The potential energy of the wavepacket's components :math:`\Phi_i` or the overall potential energy of :math:`\Psi`. """ f = partial(potential, as_matrix=True) Q = self.quadrature.quadrature(self, self, f) N = self.number_components epot = [ sum(Q[i*N:(i+1)*N]) for i in xrange(N) ] if summed is True: epot = sum(epot) return epot def kinetic_energy(self, summed=False): r""" Calculate the kinetic energy :math:`\langle\Psi|T|\Psi\rangle` of the wavepacket componentwise. :param summed: Wheter to sum up the individual integrals :math:`\langle\Phi_i|T_{i,j}|\Phi_j\rangle`. :return: The kinetic energy of the wavepacket's components :math:`\Phi_i` or the overall kinetic energy of :math:`\Psi`. """ tmp = [ self.grady(component) for component in xrange(self.number_components) ] # TODO: Check 0.25 vs orig 0.5! ekin = [ 0.25*norm(item)**2 for item in tmp ] if summed is True: ekin = sum(ekin) return ekin def grady(self, component): r""" Calculate the effect of :math:`-i \epsilon^2 \frac{\partial}{\partial x}` on a component :math:`\Phi_i` of the Hagedorn wavepacket :math:`\Psi`. :param component: The index :math:`i` of the component :math:`\Phi_i` on which we apply the above operator. :return: The modified coefficients. """ (P,Q,S,p,q) = self.parameters[component] sh = array(self.coefficients[component].shape) sh = sh+1 c = zeros(sh, dtype=complexfloating) k = 0 c[k] = c[k] + p*self.coefficients[component][k] c[k+1] = c[k+1] + sqrt(k+1)*P*sqrt(self.eps**2*0.5)*self.coefficients[component][k] for k in xrange(1,self.basis_size[component]): c[k] = c[k] + p*self.coefficients[component][k] c[k+1] = c[k+1] + sqrt(k+1)*P*sqrt(self.eps**2*0.5)*self.coefficients[component][k] c[k-1] = c[k-1] + sqrt(k)*conj(P)*sqrt(self.eps**2*0.5)*self.coefficients[component][k] return c def project_to_canonical(self, potential): r""" Project the Hagedorn wavepacket into the canonical basis. :param potential: The potential :math:`V` whose eigenvectors :math:`nu_l` are used for the transformation. .. note:: This function is expensive and destructive! It modifies the coefficients of the ``self`` instance. """ # No projection for potentials with a single energy level. # The canonical and eigenbasis are identical here. if potential.get_number_components() == 1: return potential.calculate_eigenvectors() # Basically an ugly hack to overcome some shortcomings of the matrix function # and of the data layout. def f(q, x, component): x = x.reshape((self.quadrature.get_qr().get_number_nodes(),)) z = potential.evaluate_eigenvectors_at(x) (row, col) = component return z[col][row,:] F = transpose(conj(self.quadrature.build_matrix(self,self,f))) c = self.get_coefficient_vector() d = dot(F, c) self.set_coefficient_vector(d) def project_to_eigen(self, potential): r""" Project the Hagedorn wavepacket into the eigenbasis of a given potential :math:`V`. :param potential: The potential :math:`V` whose eigenvectors :math:`nu_l` are used for the transformation. .. note:: This function is expensive and destructive! It modifies the coefficients of the ``self`` instance. """ # No projection for potentials with a single energy level. # The canonical and eigenbasis are identical here. if potential.get_number_components() == 1: return potential.calculate_eigenvectors() # Basically an ugly hack to overcome some shortcomings of the matrix function # and of the data layout. def f(q, x, component): x = x.reshape((self.quadrature.get_qr().get_number_nodes(),)) z = potential.evaluate_eigenvectors_at(x) (row, col) = component return z[col][row,:] F = self.quadrature.build_matrix(self,self,f) c = self.get_coefficient_vector() d = dot(F, c) self.set_coefficient_vector(d) def to_fourier_space(self, assign=True): r""" Transform the wavepacket to Fourier space. :param assign: Whether to assign the transformation to this packet or return a cloned packet. .. note:: This is the inverse of the method ``to_real_space()``. """ # The Fourier transformed parameters Pihats = [ (1.0j*Q, -1.0j*P, S, -q, p) for P,Q,S,p,q in self.parameters ] # The Fourier transformed coefficients coeffshat = [] for index in xrange(self.number_components): k = arange(0, self.basis_size[index]).reshape((self.basis_size[index], 1)) # Compute phase arising from the transformation phase = (-1.0j)**k * exp(-1.0j*self.parameters[index][3]*self.parameters[index][4] / self.eps**2) # Absorb phase into the coefficients coeffshat.append(phase * self.get_coefficients(component=index)) if assign is True: self.set_parameters(Pihats) self.set_coefficients(coeffshat) else: FWP = self.clone() FWP.set_parameters(Pihats) FWP.set_coefficients(coeffshat) return FWP def to_real_space(self, assign=True): r""" Transform the wavepacket to real space. :param assign: Whether to assign the transformation to this packet or return a cloned packet. .. note:: This is the inverse of the method ``to_fourier_space()``. """ # The inverse Fourier transformed parameters Pi = [ (1.0j*Q, -1.0j*P, S, q, -p) for P,Q,S,p,q in self.parameters ] # The inverse Fourier transformed coefficients coeffs = [] for index in xrange(self.number_components): k = arange(0, self.basis_size[index]).reshape((self.basis_size[index], 1)) # Compute phase arising from the transformation phase = (-1.0j)**k * exp(-1.0j*self.parameters[index][3]*self.parameters[index][4] / self.eps**2) # Absorb phase into the coefficients coeffs.append(phase * self.get_coefficients(component=index)) if assign is True: self.set_parameters(Pi) self.set_coefficients(coeffs) else: RWP = self.clone() RWP.set_parameters(Pi) RWP.set_coefficients(coeffs) return RWP