def compute_eigenstate(parameters,
filename="eigenstates.hdf5",
computepq=True,
computePQ=True):
r"""
Special variables necessary in configuration:
* eigenstate_of_level (default: 0)
* eigenstates_indices (default: [0])
* starting_point (default: (2, ..., 2))
* hawp_template
* innerproduct
"""
D = parameters["dimension"]
if "eigenstate_of_level" in parameters:
N = parameters["eigenstate_of_level"]
else:
# Upper-most potential surface
N = 0
# Create output file now, in case this fails we did not waste computation time
IOM = IOManager()
IOM.create_file(filename)
# Save the simulation parameters
IOM.add_parameters()
IOM.save_parameters(parameters)
gid = IOM.create_group()
BF = BlockFactory()
# Create the potential
V = BF.create_potential(parameters)
V.calculate_local_quadratic()
# Compute position and momentum
if computepq:
# Minimize the potential to find q0
f = lambda x: real((squeeze(V.evaluate_at(x)[N])))
# Start with an offset because exact 0.0 values can give
# issues, especially with the Hessian evaluation. This way
# the minimizer will always stay away from zero a tiny bit.
# The current starting point can give issues if the potential
# is stationary at the point (2, ..., 2) but that is less likely.
if "starting_point" in parameters:
x0 = atleast_1d(parameters["starting_point"])
else:
x0 = 0.5 * ones(D)
q0 = fmin(f, x0, xtol=1e-12)
q0 = q0.reshape((D, 1))
# We are at the minimum with no momentum
p0 = zeros_like(q0)
else:
if "q0" in parameters:
q0 = atleast_2d(parameters["q0"])
else:
q0 = zeros((D, 1))
if "p0" in parameters:
p0 = atleast_2d(parameters["p0"])
else:
p0 = zeros((D, 1))
# Compute spreads
if computePQ:
# Q_0 = H^(-1/4)
H = V.evaluate_hessian_at(q0)
Q0 = inv(sqrtm(sqrtm(H)))
# P_0 = i Q_0^(-1)
P0 = 1.0j * inv(Q0)
else:
if "Q0" in parameters:
Q0 = atleast_2d(parameters["Q0"])
else:
Q0 = identity(D)
if "P0" in parameters:
P0 = atleast_2d(parameters["P0"])
else:
P0 = 1.0j * inv(Q0)
# The parameter set Pi
print(70 * "-")
print("Parameter values are:")
print("---------------------")
print(" q0:")
print(str(q0))
print(" p0:")
print(str(p0))
print(" Q0:")
print(str(Q0))
print(" P0:")
print(str(P0))
# Consistency check
print(" Consistency check:")
print(" P^T Q - Q^T P =?= 0")
print(dot(P0.T, Q0) - dot(Q0.T, P0))
print(" Q^H P - P^H Q =?= 2i")
print(
dot(transpose(conjugate(Q0)), P0) - dot(transpose(conjugate(P0)), Q0))
# Next find the new coefficients c'
HAWP = BF.create_wavepacket(parameters["hawp_template"])
# Set the parameter values
Pi = HAWP.get_parameters()
Pi[0] = q0
Pi[1] = p0
Pi[2] = Q0
Pi[3] = P0
HAWP.set_parameters(Pi)
# Next compute the matrix M_ij = <phi_i | T + V | phi_j>
# The potential part
HQ = BF.create_inner_product(parameters["innerproduct"])
opV = lambda x, q, entry: V.evaluate_at(x, entry=entry)
MV = HQ.build_matrix(HAWP, operator=opV)
# The kinetic part
MT = zeros_like(MV, dtype=complexfloating)
GR = GradientHAWP()
BS = HAWP.get_basis_shapes(component=N)
vects = {}
for i in BS:
z = zeros_like(HAWP.get_coefficient_vector(), dtype=complexfloating)
HAWP.set_coefficient_vector(z)
HAWP.set_coefficient(N, i, 1.0)
Kn, cnew = GR.apply_gradient(HAWP, component=N, as_packet=False)
vects[i] = cnew
for j in BS:
for k in BS:
cj = vects[j]
ck = vects[k]
entry = 0.5 * squeeze(sum(conjugate(cj) * ck))
MT[BS[j], BS[k]] = entry
# Find eigenvalues and eigenvectors of the whole matrix
M = MT + MV
ew, ev = eigh(M)
ind = argsort(ew)
# Build the requested energy levels and states
if "eigenstates_indices" in parameters:
states = parameters["eigenstates_indices"]
else:
# Groundstate only
states = [0]
BS = HAWP.get_basis_shapes(component=0)
KEY = ("q", "p", "Q", "P", "S", "adQ")
print(70 * "-")
for state in states:
if state > BS.get_basis_size():
print(
"Warning: can not compute energy level {} with basis size of {}"
.format((state, BS)))
continue
index = ind[state]
coeffs = ev[:, index]
energy = ew[index]
# Try to resolve ambiguities in sign
imax = argmax(abs(coeffs))
a = abs(angle(coeffs[imax]))
if a > pi / 2.0:
coeffs *= -1
print("State: {}".format(state))
print("Energy: {}".format(energy))
print("Coefficients: \n")
print(str(coeffs))
print(70 * "-")
HAWP.set_coefficient_vector(coeffs.reshape((-1, 1)))
# Save all the wavepacket data
bid = IOM.create_block(groupid=gid)
IOM.add_wavepacket(parameters, blockid=bid, key=KEY)
IOM.save_wavepacket(HAWP, 0, blockid=bid, key=KEY)
IOM.finalize()
# TODO: Find better criterion
if norm(q0) > 1000:
print("+----------------------------------+")
print("| Run-away minimum? |")
print("| Maybe try different: |")
print("| starting_point = [x0, y0, ...] |")
print("+----------------------------------+")
def compute_energy_inhawp(iom, blockid=0, eigentrafo=True, iseigen=True):
"""Compute the energies of a wavepacket timeseries.
This function is for inhomogeneous wavepackets.
:param iom: An :py:class:`IOManager` instance providing the simulation data.
:param blockid: The data block from which the values are read.
:type blockid: Integer, Default is ``0``
:param eigentrafo: Whether to make a transformation into the eigenbasis.
:type eigentrafo: Boolean, default is ``True``.
:param iseigen: Whether the data is assumed to be in the eigenbasis.
:type iseigen: Boolean, default is ``True``
"""
parameters = iom.load_parameters()
BF = BlockFactory()
# Number of time steps we saved
timesteps = iom.load_inhomogwavepacket_timegrid(blockid=blockid)
nrtimesteps = timesteps.shape[0]
# The potential used
Potential = BF.create_potential(parameters)
# Basis transformator
if eigentrafo is True:
BT = BasisTransformationHAWP(Potential)
# We want to save energies, thus add a data slot to the data file
iom.add_energy(parameters, timeslots=nrtimesteps, blockid=blockid)
# Initialize a Hagedorn wavepacket with the data
descr = iom.load_inhomogwavepacket_description(blockid=blockid)
HAWP = BF.create_wavepacket(descr)
# Inner product
if HAWP.get_innerproduct() is None:
IP = BF.create_inner_product(parameters["innerproduct"])
HAWP.set_innerproduct(IP)
if eigentrafo is True:
BT.set_matrix_builder(HAWP.get_innerproduct())
# Basis shapes
BS_descr = iom.load_inhomogwavepacket_basisshapes(blockid=blockid)
BS = {}
for ahash, descr in BS_descr.items():
BS[ahash] = BF.create_basis_shape(descr)
O = ObservablesHAWP()
KEY = ("q", "p", "Q", "P", "S", "adQ")
# Iterate over all timesteps
for i, step in enumerate(timesteps):
print(" Computing energies of timestep %d" % step)
# Retrieve simulation data
params = iom.load_inhomogwavepacket_parameters(timestep=step,
blockid=blockid,
key=KEY)
hashes, coeffs = iom.load_inhomogwavepacket_coefficients(
timestep=step, get_hashes=True, blockid=blockid)
# Configure the wavepacket
HAWP.set_parameters(params, key=KEY)
HAWP.set_basis_shapes([BS[int(ha)] for ha in hashes])
HAWP.set_coefficients(coeffs)
# Transform to the eigenbasis.
if eigentrafo is True:
BT.transform_to_eigen(HAWP)
# Compute the energies
O.set_innerproduct(HAWP.get_innerproduct())
O.set_gradient(HAWP.get_gradient_operator())
ekin = O.kinetic_energy(HAWP)
if iseigen is True:
epot = O.potential_energy(HAWP, Potential.evaluate_eigenvalues_at)
else:
epot = O.potential_energy(HAWP, Potential.evaluate_at)
iom.save_energy((ekin, epot), timestep=step, blockid=blockid)
def compute_eigenstate(parameters):
r"""
Special variables necessary in configuration:
* eigenstate_of_level (default: 0)
* states_indices (default: [0])
"""
D = parameters["dimension"]
if parameters.has_key("eigenstate_of_level"):
N = parameters["eigenstate_of_level"]
else:
# Upper-most potential surface
N = 0
# Create output file now, in case this fails we did not waste computations
IOM = IOManager()
IOM.create_file("eigenstates.hdf5")
# Save the simulation parameters
IOM.add_parameters()
IOM.save_parameters(parameters)
gid = IOM.create_group()
BF = BlockFactory()
# Create the potential
V = BF.create_potential(parameters)
V.calculate_local_quadratic()
# Minimize the potential to find q0
f = lambda x: real((squeeze(V.evaluate_at(x)[N])))
# Start with an offset because exact 0.0 values can give
# issues, especially with the Hessian evaluation. This way
# the minimizer will always stay away from zero a tiny bit.
# The current starting point can give issues if the potential
# is stationary at the point (2, ..., 2) but that is less likely.
x0 = 2.0*ones(D)
q0 = fmin(f, x0, xtol=1e-12)
q0 = q0.reshape((D,1))
# We are at the minimum with no momentum
p0 = zeros_like(q0)
# Compute spreads now
# Q_0 = H^(-1/4)
H = V.evaluate_hessian_at(q0)
Q0 = inv(sqrtm(sqrtm(H)))
# Take P_00 = i Q_0^(-1)
P0 = 1.0j * inv(Q0)
#
print(70*"-")
print("Parameter values are:")
print("---------------------")
print(" q0:")
print(str(q0))
print(" p0:")
print(str(p0))
print(" Q0:")
print(str(Q0))
print(" P0:")
print(str(P0))
# Consistency check
print(" consistency:")
print(str(conj(Q0)*P0 - conj(P0)*Q0))
print(70*"-")
# Next find the new coefficients c'
HAWP = BF.create_wavepacket(parameters["hawp_template"])
# Set the parameter values
Pi = HAWP.get_parameters()
Pi[0] = q0
Pi[1] = p0
Pi[2] = Q0
Pi[3] = P0
HAWP.set_parameters(Pi)
# Next compute the matrix M_ij = <phi_i | T + V | phi_j>
# The potential part
HQ = BF.create_inner_product(parameters["innerproduct"])
opV = lambda x, q, entry: V.evaluate_at(x, entry=entry)
MV = HQ.build_matrix(HAWP, operator=opV)
# The kinetic part
MT = zeros_like(MV, dtype=complexfloating)
GR = GradientHAWP()
BS = HAWP.get_basis_shapes(N)
vects = {}
for i in BS:
z = zeros_like(HAWP.get_coefficient_vector(), dtype=complexfloating)
HAWP.set_coefficient_vector(z)
HAWP.set_coefficient(N, i, 1.0)
Kn, cnew = GR.apply_gradient(HAWP, N)
vects[i] = cnew
for j in BS:
for k in BS:
cj = vects[j]
ck = vects[k]
entry = 0.5 * squeeze(sum(conj(cj) * ck))
MT[BS[j], BS[k]] = entry
# Find eigenvalues and eigenvectors of the whole matrix
M = MT + MV
ew, ev = eigh(M)
ind = argsort(ew)
# Build the requested energy levels and states
if parameters.has_key("eigenstates_indices"):
states = parameters["eigenstates_indices"]
else:
# Groundstate only
states = [0]
BS = HAWP.get_basis_shapes(component=0)
KEY = ("q","p","Q","P","S","adQ")
print(70*"-")
for state in states:
if state > BS.get_basis_size():
print("Warning: can not compute energy level "+state+" with basis size of "+str(BS))
continue
index = ind[state]
coeffs = ev[:,index]
energy = ew[index]
print("Level: "+str(state))
print("Energy: "+str(energy))
print("Coefficients: \n")
print(str(coeffs))
print(70*"-")
HAWP.set_coefficient_vector(coeffs.reshape((-1, 1)))
# Save all the wavepacket data
bid = IOM.create_block(groupid=gid)
IOM.add_wavepacket(parameters, blockid=bid, key=KEY)
IOM.save_wavepacket_description(HAWP.get_description(), blockid=bid)
for shape in HAWP.get_basis_shapes():
IOM.save_wavepacket_basisshapes(shape, blockid=bid)
IOM.save_wavepacket_parameters(HAWP.get_parameters(key=KEY), timestep=0, blockid=bid, key=KEY)
IOM.save_wavepacket_coefficients(HAWP.get_coefficients(), HAWP.get_basis_shapes(), timestep=0, blockid=bid)
IOM.finalize()
def compute_autocorrelation_hawp(iom, obsconfig, blockid=0, eigentrafo=True):
"""Compute the autocorrelation of a wavepacket timeseries.
:param iom: An :py:class:`IOManager` instance providing the simulation data.
:param obsconfig: Configuration parameters describing f.e. the inner product to use.
:type obsconfig: A :py:class:`ParameterProvider` instance.
:param blockid: The data block from which the values are read.
:type blockid: Integer, Default is ``0``
:param eigentrafo: Whether to make a transformation into the eigenbasis.
:type eigentrafo: Boolean, default is ``True``.
"""
parameters = iom.load_parameters()
BF = BlockFactory()
# Number of time steps we saved
timesteps = iom.load_wavepacket_timegrid(blockid=blockid)
nrtimesteps = timesteps.shape[0]
# Basis transformator
if eigentrafo is True:
# The potential used
Potential = BF.create_potential(parameters)
BT = BasisTransformationHAWP(Potential)
# We want to save norms, thus add a data slot to the data file
iom.add_autocorrelation(parameters, timeslots=nrtimesteps, blockid=blockid)
# Initialize a Hagedorn wavepacket with the data
descr = iom.load_wavepacket_description(blockid=blockid)
HAWPo = BF.create_wavepacket(descr)
HAWPt = BF.create_wavepacket(descr)
if eigentrafo is True:
BT.set_matrix_builder(HAWPo.get_innerproduct())
# Basis shapes
BS_descr = iom.load_wavepacket_basisshapes(blockid=blockid)
BS = {}
for ahash, descr in BS_descr.items():
BS[ahash] = BF.create_basis_shape(descr)
# Comfigure the original wavepacket
KEY = ("q", "p", "Q", "P", "S", "adQ")
# Retrieve simulation data
params = iom.load_wavepacket_parameters(timestep=0,
blockid=blockid,
key=KEY)
hashes, coeffs = iom.load_wavepacket_coefficients(timestep=0,
get_hashes=True,
blockid=blockid)
# Configure the wavepacket
HAWPo.set_parameters(params, key=KEY)
HAWPo.set_basis_shapes([BS[int(ha)] for ha in hashes])
HAWPo.set_coefficients(coeffs)
# Set up the innerproduct for solving the integrals <phi_0 | phi_t>
IP = BF.create_inner_product(obsconfig["innerproduct"])
# Transform to the eigenbasis.
if eigentrafo is True:
BT.transform_to_eigen(HAWPo)
# Iterate over all timesteps
for i, step in enumerate(timesteps):
print(" Computing autocorrelation of timestep %d" % step)
# Retrieve simulation data
paramst = iom.load_wavepacket_parameters(timestep=step,
blockid=blockid,
key=KEY)
hashes, coeffs = iom.load_wavepacket_coefficients(timestep=step,
get_hashes=True,
blockid=blockid)
# Configure the wavepacket
HAWPt.set_parameters(paramst, key=KEY)
HAWPt.set_basis_shapes([BS[int(ha)] for ha in hashes])
HAWPt.set_coefficients(coeffs)
# Transform to the eigenbasis.
if eigentrafo is True:
BT.transform_to_eigen(HAWPt)
# Measure autocorrelations in the eigenbasis
acs = IP.quadrature(HAWPo, HAWPt, diagonal=True)
# Save the autocorrelations
iom.save_autocorrelation(acs, timestep=step, blockid=blockid)
def compute_autocorrelation_inhawp(iom, obsconfig, blockid=0, eigentrafo=True):
"""Compute the autocorrelation of a wavepacket timeseries.
This function is for inhomogeneous wavepackets.
:param iom: An :py:class:`IOManager` instance providing the simulation data.
:param obsconfig: Configuration parameters describing f.e. the inner product to use.
:type obsconfig: A :py:class:`ParameterProvider` instance.
:param blockid: The data block from which the values are read.
:type blockid: Integer, Default is ``0``
:param eigentrafo: Whether to make a transformation into the eigenbasis.
:type eigentrafo: Boolean, default is ``True``.
"""
parameters = iom.load_parameters()
BF = BlockFactory()
# Number of time steps we saved
timesteps = iom.load_inhomogwavepacket_timegrid(blockid=blockid)
nrtimesteps = timesteps.shape[0]
# Basis transformator
if eigentrafo is True:
# The potential used
Potential = BF.create_potential(parameters)
BT = BasisTransformationHAWP(Potential)
# We want to save autocorrelations, thus add a data slot to the data file
iom.add_autocorrelation(parameters, timeslots=nrtimesteps, blockid=blockid)
# Initialize a Hagedorn wavepacket with the data
descr = iom.load_inhomogwavepacket_description(blockid=blockid)
HAWPo = BF.create_wavepacket(descr)
HAWPt = BF.create_wavepacket(descr)
if eigentrafo is True:
BT.set_matrix_builder(HAWPo.get_innerproduct())
# Basis shapes
BS_descr = iom.load_inhomogwavepacket_basisshapes(blockid=blockid)
BS = {}
for ahash, descr in BS_descr.items():
BS[ahash] = BF.create_basis_shape(descr)
# Comfigure the original wavepacket
# Retrieve simulation data
params = iom.load_inhomogwavepacket_parameters(timestep=0, blockid=blockid)
hashes, coeffs = iom.load_inhomogwavepacket_coefficients(timestep=0, get_hashes=True, blockid=blockid)
# Configure the wavepacket
HAWPo.set_parameters(params)
HAWPo.set_basis_shapes([BS[int(ha)] for ha in hashes])
HAWPo.set_coefficients(coeffs)
# Set up the innerproduct for solving the integrals <phi_0 | phi_t>
IP = BF.create_inner_product(obsconfig["innerproduct"])
# Iterate over all timesteps
for i, step in enumerate(timesteps):
print(" Computing autocorrelations of timestep %d" % step)
# Retrieve simulation data
params = iom.load_inhomogwavepacket_parameters(timestep=step, blockid=blockid)
hashes, coeffs = iom.load_inhomogwavepacket_coefficients(timestep=step, get_hashes=True, blockid=blockid)
# Configure the wavepacket
HAWPt.set_parameters(params)
HAWPt.set_basis_shapes([BS[int(ha)] for ha in hashes])
HAWPt.set_coefficients(coeffs)
# Transform to the eigenbasis.
if eigentrafo is True:
BT.transform_to_eigen(HAWPt)
# Measure autocorrelations in the eigenbasis
acs = IP.quadrature(HAWPo, HAWPt, diagonal=True)
# Save the autocorrelations
iom.save_autocorrelation(acs, timestep=step, blockid=blockid)
def load_from_file(filepath, blockid=0, timestep=0, sizeK=None):
r"""Utility script to load wavepacket parameters and coefficients
from another simulation result in a form suitable for the input
configuration of a new simulation. This is (mainly) used
to start simulations with previously computed eigenstates.
:param filepath: The path to the `.hdf5` file from which data will be read.
:param blockid: The `datablock` from which to read the data.
Default is the block with `blockid=0`.
:param timestep: Load the data corresponding to the given `timestep`.
The default timestep is `0`.
:param sizeK: Load at most 'sizeK' many coefficients. Note that the order
is defined by the linearization mapping :math:`\mu` of the
packet's current basis shape. We then pick the first `sizeK`
ones.
"""
IOM = IOManager()
IOM.open_file(filepath)
# Check if we have data
tg = IOM.load_wavepacket_timegrid(blockid=blockid)
if timestep not in tg:
raise ValueError("No data for timestep {}".format(timestep))
# Load data and assemble packet
BF = BlockFactory()
# Basis shapes
BS_descr = IOM.load_wavepacket_basisshapes(blockid=blockid)
BS = {}
for ahash, descr in BS_descr.items():
BS[ahash] = BF.create_basis_shape(descr)
# Create a packet
wpd = IOM.load_wavepacket_description(blockid=blockid)
HAWP = BF.create_wavepacket(wpd)
# Data
ha, ci = IOM.load_wavepacket_coefficients(blockid=blockid,
timestep=timestep,
get_hashes=True)
Pi = IOM.load_wavepacket_parameters(blockid=blockid, timestep=timestep)
HAWP.set_parameters(Pi)
HAWP.set_basis_shapes([BS[int(h)] for h in ha])
HAWP.set_coefficients(ci)
# Reformat data
C = []
for n in range(HAWP.get_number_components()):
B = HAWP.get_basis_shapes(component=n)
cn = HAWP.get_coefficients(component=n)
l = []
for i in range(B.get_basis_size()):
l.append((B[i], cn[i, 0]))
C.append(l)
if sizeK is not None:
# We load at most 'sizeK' coefficients.
# Note that this does NOT specify which
# ones in terms of multi-indices.
C = [c[:sizeK] for c in C]
return Pi, C
def load_from_file(filepath, blockid=0, timestep=0, sizeK=None):
r"""Utility script to load wavepacket parameters and coefficients
from another simulation result in a form suitable for the input
configuration of a new simulation. This is (mainly) used
to start simulations with previously computed eigenstates.
:param filepath: The path to the `.hdf5` file from which data will be read.
:param blockid: The `datablock` from which to read the data.
Default is the block with `blockid=0`.
:param timestep: Load the data corresponding to the given `timestep`.
The default timestep is `0`.
:param sizeK: Load at most 'sizeK' many coefficients. Note that the order
is defined by the linearization mapping :math:`\mu` of the
packet's current basis shape. We then pick the first `sizeK`
ones.
"""
IOM = IOManager()
IOM.open_file(filepath)
# Check if we have data
tg = IOM.load_wavepacket_timegrid(blockid=blockid)
if timestep not in tg:
raise ValueError("No data for timestep {}".format(timestep))
# Load data and assemble packet
BF = BlockFactory()
# Basis shapes
BS_descr = IOM.load_wavepacket_basisshapes(blockid=blockid)
BS = {}
for ahash, descr in BS_descr.items():
BS[ahash] = BF.create_basis_shape(descr)
# Create a packet
wpd = IOM.load_wavepacket_description(blockid=blockid)
HAWP = BF.create_wavepacket(wpd)
# Data
ha, ci = IOM.load_wavepacket_coefficients(blockid=blockid, timestep=timestep, get_hashes=True)
Pi = IOM.load_wavepacket_parameters(blockid=blockid, timestep=timestep)
HAWP.set_parameters(Pi)
HAWP.set_basis_shapes([BS[int(h)] for h in ha])
HAWP.set_coefficients(ci)
# Reformat data
C = []
for n in range(HAWP.get_number_components()):
B = HAWP.get_basis_shapes(component=n)
cn = HAWP.get_coefficients(component=n)
l = []
for i in range(B.get_basis_size()):
l.append((B[i], cn[i, 0]))
C.append(l)
if sizeK is not None:
# We load at most 'sizeK' coefficients.
# Note that this does NOT specify which
# ones in terms of multi-indices.
C = [c[:sizeK] for c in C]
return Pi, C
def plot_frames(PP, iom, blockid=0, load=False, limits=None):
r"""
"""
parameters = iom.load_parameters()
BF = BlockFactory()
if not parameters["dimension"] == 2:
print("No wavepacket of two space dimensions, silent return!")
return
if PP is None:
PP = parameters
if load is True:
# TODO: Implement reshaping
raise NotImplementedError("Loading of 2D grids is not implemented")
#G = iom.load_grid(blockid=blockid)
#G = grid.reshape((1, -1))
else:
G = BF.create_grid(PP)
u, v = map(squeeze, G.get_axes())
V = BF.create_potential(parameters)
BT = BasisTransformationHAWP(V)
wpd = iom.load_wavepacket_description(blockid=blockid)
HAWP = BF.create_wavepacket(wpd)
# Basis shapes
BS_descr = iom.load_wavepacket_basisshapes(blockid=blockid)
BS = {}
for ahash, descr in BS_descr.iteritems():
BS[ahash] = BF.create_basis_shape(descr)
timegrid = iom.load_wavepacket_timegrid(blockid=blockid)
N = HAWP.get_number_components()
for step in timegrid:
print(" Plotting frame of timestep # " + str(step))
hi, ci = iom.load_wavepacket_coefficients(timestep=step, get_hashes=True, blockid=blockid)
Pi = iom.load_wavepacket_parameters(timestep=step, blockid=blockid)
HAWP.set_parameters(Pi)
HAWP.set_basis_shapes([ BS[int(ha)] for ha in hi ])
HAWP.set_coefficients(ci)
psi = HAWP.evaluate_at(G, prefactor=True, component=0)
fig = figure()
for level in xrange(N):
z = psi[level]
z = z.reshape(G.get_number_nodes())
subplot(N,1,level+1)
#plotcm(z.reshape(G.get_number_nodes()), darken=0.3)
plotcf2d(u, v, z, darken=0.3, limits=limits)
savefig("wavepacket_block_"+str(blockid)+"_level_"+str(level)+"_timestep_"+(5-len(str(step)))*"0"+str(step)+".png")
close(fig)
print(" Plotting frames finished")
def compute_eigenstate(parameters, filename="eigenstates.hdf5", computepq=True, computePQ=True):
r"""
Special variables necessary in configuration:
* eigenstate_of_level (default: 0)
* eigenstates_indices (default: [0])
* starting_point (default: (2, ..., 2))
* hawp_template
* innerproduct
"""
D = parameters["dimension"]
if "eigenstate_of_level" in parameters:
N = parameters["eigenstate_of_level"]
else:
# Upper-most potential surface
N = 0
# Create output file now, in case this fails we did not waste computation time
IOM = IOManager()
IOM.create_file(filename)
# Save the simulation parameters
IOM.add_parameters()
IOM.save_parameters(parameters)
gid = IOM.create_group()
BF = BlockFactory()
# Create the potential
V = BF.create_potential(parameters)
V.calculate_local_quadratic()
# Compute position and momentum
if computepq:
# Minimize the potential to find q0
f = lambda x: real((squeeze(V.evaluate_at(x)[N])))
# Start with an offset because exact 0.0 values can give
# issues, especially with the Hessian evaluation. This way
# the minimizer will always stay away from zero a tiny bit.
# The current starting point can give issues if the potential
# is stationary at the point (2, ..., 2) but that is less likely.
if "starting_point" in parameters:
x0 = atleast_1d(parameters["starting_point"])
else:
x0 = 0.5 * ones(D)
q0 = fmin(f, x0, xtol=1e-12)
q0 = q0.reshape((D, 1))
# We are at the minimum with no momentum
p0 = zeros_like(q0)
else:
if "q0" in parameters:
q0 = atleast_2d(parameters["q0"])
else:
q0 = zeros((D, 1))
if "p0" in parameters:
p0 = atleast_2d(parameters["p0"])
else:
p0 = zeros((D, 1))
# Compute spreads
if computePQ:
# Q_0 = H^(-1/4)
H = V.evaluate_hessian_at(q0)
Q0 = inv(sqrtm(sqrtm(H)))
# P_0 = i Q_0^(-1)
P0 = 1.0j * inv(Q0)
else:
if "Q0" in parameters:
Q0 = atleast_2d(parameters["Q0"])
else:
Q0 = identity(D)
if "P0" in parameters:
P0 = atleast_2d(parameters["P0"])
else:
P0 = 1.0j * inv(Q0)
# The parameter set Pi
print(70 * "-")
print("Parameter values are:")
print("---------------------")
print(" q0:")
print(str(q0))
print(" p0:")
print(str(p0))
print(" Q0:")
print(str(Q0))
print(" P0:")
print(str(P0))
# Consistency check
print(" Consistency check:")
print(" P^T Q - Q^T P =?= 0")
print(dot(P0.T, Q0) - dot(Q0.T, P0))
print(" Q^H P - P^H Q =?= 2i")
print(dot(transpose(conjugate(Q0)), P0) - dot(transpose(conjugate(P0)), Q0))
# Next find the new coefficients c'
HAWP = BF.create_wavepacket(parameters["hawp_template"])
# Set the parameter values
Pi = HAWP.get_parameters()
Pi[0] = q0
Pi[1] = p0
Pi[2] = Q0
Pi[3] = P0
HAWP.set_parameters(Pi)
# Next compute the matrix M_ij = <phi_i | T + V | phi_j>
# The potential part
HQ = BF.create_inner_product(parameters["innerproduct"])
opV = lambda x, q, entry: V.evaluate_at(x, entry=entry)
MV = HQ.build_matrix(HAWP, operator=opV)
# The kinetic part
MT = zeros_like(MV, dtype=complexfloating)
GR = GradientHAWP()
BS = HAWP.get_basis_shapes(component=N)
vects = {}
for i in BS:
z = zeros_like(HAWP.get_coefficient_vector(), dtype=complexfloating)
HAWP.set_coefficient_vector(z)
HAWP.set_coefficient(N, i, 1.0)
Kn, cnew = GR.apply_gradient(HAWP, component=N, as_packet=False)
vects[i] = cnew
for j in BS:
for k in BS:
cj = vects[j]
ck = vects[k]
entry = 0.5 * squeeze(sum(conjugate(cj) * ck))
MT[BS[j], BS[k]] = entry
# Find eigenvalues and eigenvectors of the whole matrix
M = MT + MV
ew, ev = eigh(M)
ind = argsort(ew)
# Build the requested energy levels and states
if "eigenstates_indices" in parameters:
states = parameters["eigenstates_indices"]
else:
# Groundstate only
states = [0]
BS = HAWP.get_basis_shapes(component=0)
KEY = ("q", "p", "Q", "P", "S", "adQ")
print(70 * "-")
for state in states:
if state > BS.get_basis_size():
print("Warning: can not compute energy level {} with basis size of {}".format((state, BS)))
continue
index = ind[state]
coeffs = ev[:, index]
energy = ew[index]
# Try to resolve ambiguities in sign
imax = argmax(abs(coeffs))
a = abs(angle(coeffs[imax]))
if a > pi / 2.0:
coeffs *= -1
print("State: {}".format(state))
print("Energy: {}".format(energy))
print("Coefficients: \n")
print(str(coeffs))
print(70 * "-")
HAWP.set_coefficient_vector(coeffs.reshape((-1, 1)))
# Save all the wavepacket data
bid = IOM.create_block(groupid=gid)
IOM.add_wavepacket(parameters, blockid=bid, key=KEY)
IOM.save_wavepacket(HAWP, 0, blockid=bid, key=KEY)
IOM.finalize()
# TODO: Find better criterion
if norm(q0) > 1000:
print("+----------------------------------+")
print("| Run-away minimum? |")
print("| Maybe try different: |")
print("| starting_point = [x0, y0, ...] |")
print("+----------------------------------+")
def compute_energy_inhawp(iom, blockid=0, eigentrafo=True, iseigen=True):
"""Compute the energies of a wavepacket timeseries.
This function is for inhomogeneous wavepackets.
:param iom: An :py:class:`IOManager` instance providing the simulation data.
:param blockid: The data block from which the values are read.
:type blockid: Integer, Default is ``0``
:param eigentrafo: Whether to make a transformation into the eigenbasis.
:type eigentrafo: Boolean, default is ``True``.
:param iseigen: Whether the data is assumed to be in the eigenbasis.
:type iseigen: Boolean, default is ``True``
"""
parameters = iom.load_parameters()
BF = BlockFactory()
# Number of time steps we saved
timesteps = iom.load_inhomogwavepacket_timegrid(blockid=blockid)
nrtimesteps = timesteps.shape[0]
# The potential used
Potential = BF.create_potential(parameters)
# Basis transformator
if eigentrafo is True:
BT = BasisTransformationHAWP(Potential)
# We want to save energies, thus add a data slot to the data file
iom.add_energy(parameters, timeslots=nrtimesteps, blockid=blockid)
# Initialize a Hagedorn wavepacket with the data
descr = iom.load_inhomogwavepacket_description(blockid=blockid)
HAWP = BF.create_wavepacket(descr)
# Inner product
if HAWP.get_innerproduct() is None:
IP = BF.create_inner_product(parameters["innerproduct"])
HAWP.set_innerproduct(IP)
if eigentrafo is True:
BT.set_matrix_builder(HAWP.get_innerproduct())
# Basis shapes
BS_descr = iom.load_inhomogwavepacket_basisshapes(blockid=blockid)
BS = {}
for ahash, descr in BS_descr.items():
BS[ahash] = BF.create_basis_shape(descr)
O = ObservablesHAWP()
KEY = ("q", "p", "Q", "P", "S", "adQ")
# Iterate over all timesteps
for i, step in enumerate(timesteps):
print(" Computing energies of timestep %d" % step)
# Retrieve simulation data
params = iom.load_inhomogwavepacket_parameters(timestep=step, blockid=blockid, key=KEY)
hashes, coeffs = iom.load_inhomogwavepacket_coefficients(timestep=step, get_hashes=True, blockid=blockid)
# Configure the wavepacket
HAWP.set_parameters(params, key=KEY)
HAWP.set_basis_shapes([BS[int(ha)] for ha in hashes])
HAWP.set_coefficients(coeffs)
# Transform to the eigenbasis.
if eigentrafo is True:
BT.transform_to_eigen(HAWP)
# Compute the energies
O.set_innerproduct(HAWP.get_innerproduct())
ekin = O.kinetic_energy(HAWP)
if iseigen is True:
epot = O.potential_energy(HAWP, Potential.evaluate_eigenvalues_at)
else:
epot = O.potential_energy(HAWP, Potential.evaluate_at)
iom.save_energy((ekin, epot), timestep=step, blockid=blockid)