# Load packages which form part of the Python 2.7 core
import pickle
import multiprocessing
import math
import logging
# Load packages which are a part of GeometricMD
from geometricmd.animation import write_xyz_animation
from geometricmd.geometry import convert_vector_to_atoms
# Load additional packages and check if they are installed
import numpy as np
from scipy.optimize import fmin_l_bfgs_b
[docs]def generate_points(x, start_point, end_point, rotation_matrix, total_number_of_points, co_dimension):
""" This function computes the local geodesic curve joining start_point to end_point using the L-BFGS method.
Args:
x (numpy.array) :
Array of vectors in co-dimension dimensional space, stacked flat. This vector characterises points of the
curve as translations from the line joining start_point to end_point.
start_point (numpy.array) :
The first end point of the curve.
end_point (numpy.array) :
The last end point of the curve.
rotation_matrix (numpy.array) :
A numpy.array describing the rotation from co_dimension + 1 dimensional space to the tangent space of the line
joining start_point to end_point.
total_number_of_points (int):
The total number of points used in the local geodesic computation, including endpoints.
co_dimension (int):
The dimension of the configuration space, less one.
Returns:
numpy.array :
The midpoint along the approximate local geodesic curve.
"""
# Compute tangent direction of line joining start and end points
tangent = np.subtract(end_point, start_point)/(total_number_of_points-1)
# Initialise list to store points
points = []
# Generate points that are uniformly distributed along the initial line
for i in xrange(total_number_of_points):
points.append(np.add(start_point, float(i)*tangent))
# Shift the points as encoded in x
for i in xrange(0, len(x)/co_dimension):
# Embed vector i into co_dimension + 1 dimensional space
unrotated_shift = np.hstack((np.zeros(1), x[i*co_dimension:(i+1)*co_dimension]))
# Convert vector, by rotation, from shift from e_1 basis direction to shift from tangent direction
shift = rotation_matrix.dot(unrotated_shift)
# Append point to list
points[i+1] = np.add(points[i+1], shift)
return points
[docs]def compute_metric(points, metric_function):
""" Takes a list of NumPy arrays describing points molecular configurations, evaluates the metric at each point and
returns a list of metric values at those points.
Args:
points (list) :
A list of NumPy arrays describing molecular configurations.
metric_function (func) :
A Python function which gives the value of sqrt(2(E - V)) at a given point.
Returns:
list :
A list of metric values at the corresponding points.
"""
# Initialise the list to store metric values
metric = []
# For each point, compute the metric at that point
for point in points:
metric.append(metric_function(point))
return metric
[docs]def norm(x, matrix):
""" Computes the value of sqrt(<x, matrix*x>).
Args:
x (numpy.array) :
A vector, stored as a NumPy array, to compute the norm for.
matrix (numpy.array) :
A matrix, stored as a NumPy array, used in the computation of <x, matrix*x>.
Returns:
float :
The value of sqrt(<x, matrix*x>).
"""
return math.sqrt(np.inner(x, matrix.dot(x)))
[docs]def norm_gradient(x, matrix):
""" Computes the gradient of sqrt(<x, matrix*x>).
Args:
x (numpy.array) :
A vector, stored as a NumPy array, to compute the norm for.
matrix (numpy.array) :
A matrix, stored as a NumPy array, used in the computation of <x, matrix*x>.
Returns:
numpy.array :
The gradient of sqrt(<x, matrix*x>).
"""
a = (matrix + matrix.transpose())
return a.dot(x) / (2 * norm(x, matrix))
[docs]def length(x, start_point, end_point, mass_matrix, rotation_matrix, total_number_of_points, co_dimension, metric):
""" This function computes the length of the local geodesic as a function of shifts from the line joining
start_point to end_point. It also returns the gradient of this function for the L-BFGS method.
Args:
x (numpy.array) :
Array of vectors in co-dimension dimensional space, stacked flat. This vector characterises points of the
curve as translations from the line joining start_point to end_point.
start_point (numpy.array) :
The first end point of the curve.
end_point (numpy.array) :
The last end point of the curve.
mass_matrix (numpy.array) :
A numpy.array describing the mass matrix for the molecule as a dynamical system.
rotation_matrix (numpy.array) :
A numpy.array describing the rotation from co_dimension + 1 dimensional space to the tangent space of the line
joining start_point to end_point.
total_number_of_points (int) :
The total number of points used in the local geodesic computation, including endpoints.
co_dimension (int) :
The dimension of the configuration space, less one.
metric (func) :
A Python function which when given a list of NumPy arrays, returns a list of metric values on those arrays.
Returns:
float :
The approximate length of the geodesic.
numpy.array :
The gradient of the approximate length of the geodesic.
"""
# Convert the shifts x into points in the full dimensional space
points = generate_points(x, start_point, end_point, rotation_matrix, total_number_of_points, co_dimension)
# Pre-compute the metric values to minimise repeated metric evaluations
a = compute_metric(points, metric)
# Compute quantities used to determine the length and gradient
n = np.subtract(points[1], points[0])
b = norm(n, mass_matrix)
c = norm_gradient(n, mass_matrix)
u = (a[1][0]+a[0][0])
# Initialise the length with the trapezoidal approximation of the first line segments length
l = u * b
# Initialise a list to store the gradient
g = []
for i in xrange(1, len(points)-1):
# Compute the quantities needed for the next trapezoidal rule approximation.
n = np.subtract(points[i+1], points[i])
d = norm(n, mass_matrix)
e = norm_gradient(n, mass_matrix)
v = (a[i+1][0]+a[i][0])
# Add length of line segment to total length
l += v * norm(n, mass_matrix)
# Compute next gradient component and update gradient
g.append(rotation_matrix.transpose().dot(a[i][1] * (b + d) + u * c - v * e)[1:])
# Pass back calculated values for efficiency
b = d
c = e
u = v
return 0.5 * l, 0.5 * np.asarray(g).flatten()
[docs]def get_rotation(start_point, end_point, dimension):
""" Computes the transformation from dimension dimensional space to the tangent space of the line
joining start_point to end_point.
Args:
start_point (numpy.array) :
The first end point of the curve.
end_point (numpy.array) :
The last end point of the curve.
dimension (int) :
The dimension of the configuration space.
Returns:
numpy.array :
The matrix representing the linear transformation from dimension dimensional space to the tangent space of
the line joining start_point to end_point.
"""
# Compute tangent direction of line joining start and end points
tangent = np.subtract(end_point, start_point)
# Set the first column of our output matrix as tangent
mx = tangent
# Find the first non-zero entry of the tangent vector (exists as start and endpoints are different)
j = np.nonzero(mx)[0][0]
# For the remaining dim - 1 columns choose unit basis vectors of the form (0,...,0,1,0,...,0) with the nonzero entry
# not in position j.
for i in xrange(1, dimension):
if j != i:
e = np.zeros(dimension)
e[i] = 1
mx = np.vstack((mx, e))
mx = mx.transpose()
# With the resulting matrix, perform the Gram-Schmidt orthonormalisation procedure on the transpose of the matrix
# and return it.
m, n = np.shape(mx)
Q = np.zeros([m, n])
R = np.zeros([n, n])
v = np.zeros(m)
for j in range(n):
v[:] = mx[:,j]
for i in range(j):
r = np.dot(Q[:,i], mx[:,j]); R[i,j] = r
v[:] = v[:] - r*Q[:,i]
r = np.linalg.norm(v); R[j,j]= r
Q[:,j] = v[:]/r
return Q
[docs]def find_geodesic_midpoint(start_point, end_point, number_of_inner_points, dimension, mass_matrix, molecule, energy,
node_number, length_function):
""" This function computes the local geodesic curve joining start_point to end_point using the L-BFGS method.
Args:
start_point (numpy.array) :
The first end point of the curve.
end_point (numpy.array) :
The last end point of the curve.
number_of_inner_points (int) :
The number of nodes along the curve, less the end points.
dimension (int) :
The dimension of the problem. Computed from the atomistic simulation environment.
mass_matrix (numpy.array) :
A diagonal NumPy array containing the masses of the molecular system as computed in the SimulationClient
object.
molecule (ase.atoms) :
The ASE atoms object corresponding to the molecule being simulated.
energy (float) :
The total energy of the system.
node_number (int) :
The node number for which we are calculating a new position for.
length_function (func) :
A Python function that estimates the length of a curve and also returns it's gradient.
Returns:
int :
The node number for which the returned midpoint corresponds to.
numpy.array :
The midpoint along the approximate local geodesic curve.
"""
# Define a function that returns sqrt(2(E-V)) and it's gradient based on a given configuration
def metric(point):
# Update molecular configuration based on given configuration
molecule.set_positions(convert_vector_to_atoms(point))
# Evaluate the value of sqrt(2(E-V)), replacing E-V with 1E-9 if V > E.
cf = math.sqrt(max([2*(energy - molecule.get_potential_energy()), 1E-9]))
# Return sqrt(2(E-V)) and it's gradient
return [cf, molecule.get_forces().flatten()/cf]
# Obtain the transformation from dimension dimensional space to the tangent space of the line
# joining start_point to end_point.
Q = get_rotation(start_point, end_point, dimension)
# Perform L-BFGS optimisation on length_function, returning a new geodesic midpoint
geodesic, f_min, detail = fmin_l_bfgs_b(func=length_function,
x0=np.zeros(number_of_inner_points*(dimension-1)),
args=(start_point,
end_point,
mass_matrix,
Q,
number_of_inner_points+2,
dimension-1,
metric))
# If something went wrong with the L-BFGS algorithm print an error message for the end user
if detail['warnflag'] != 0:
print 'BFGS Warning:' + detail['task']
# Convert the obtained geodesic from it's shift description to the full point description
points = np.reshape(generate_points(geodesic, start_point, end_point, Q, number_of_inner_points+2, dimension-1),
(number_of_inner_points+2, dimension))
# Compute the midpoint
if number_of_inner_points % 2 == 1:
# If there is an odd number of inner points then return the middle element of the array
midpoint = points[(number_of_inner_points + 1) / 2]
else:
# If there is an even number of inner points return the midpoint of the two middle points - this prevents
# artificial movement of the curve due to the algorithm.
midpoint = 0.5 * (points[number_of_inner_points / 2] + points[(number_of_inner_points / 2) + 1])
# Return the node number and new midpoint
return [node_number, midpoint]
[docs]def compute_trajectory(trajectory, local_num_nodes, energy, tol, filename, configuration, length_function=length):
""" This function updates the trajectory object positions to represent the shortest curve.
Args:
trajectory (curve) :
A GeometricMD curve object describing the initial trajectory between start and end configurations.
local_num_nodes (int) :
The number of points to use when computing the local geodesics.
energy (float) :
The total energy of the system.
tol (float) :
The tolerance by which if the total curve movement falls below this number then the Birkhoff method stops.
filename (str) :
The filename for the output files from the simulation.
configuration (dict) :
A dictionary containing additional parameters for the simulation. Accepts: 'processes' - the number of
processors to use (defaults to 1), 'write_to_log' - a boolean value, if true writes to a logfile, otherwise
prints to console (defaults to False) and 'save_every' - an integer indicating the program will save after
every 'save_every'th iteration of the Birkhoff algorithm (defaults to 1).
length_function (func) :
A Python function that approximates the length of a curve.
"""
# Extract a copy of the ASE atoms object to determine forces
molecule = trajectory.configuration['molecule']
# Determine the dimension of the Hamiltonian system
dimension = len(trajectory.get_points()[0])
# Compute the mass matrix for the Hamiltonian system
mass_matrix = np.diag(np.dstack((molecule.get_masses(),) * (dimension /
len(molecule.get_masses()))).flatten())
# Set counter for saving
i = 0
# Attempt to extract additional parameters, otherwise set default behaviours
try:
processes = configuration['processes']
except KeyError:
processes = 1
try:
write_to_log = configuration['write_to_log']
except KeyError:
write_to_log = False
try:
save_frequency = configuration['save_every']
except KeyError:
save_frequency = 1
# Initialise logging based on whether the user indicated whether they would like it printing to stout or not.
if write_to_log:
logging.basicConfig(format='%(asctime)s %(message)s', filename='out/'+filename+'.log', level=logging.INFO)
else:
logging.basicConfig(format='%(asctime)s %(message)s', level=logging.INFO)
# If the user intends to use the algorithm on one core then...
if processes == 1:
# Main loop of the Birkhoff algorithm, continues until curve.movement < tol then breaks out
while True:
# Iterating over each node in the trajectory find a new position based on the geodesic midpoint
# joining it's neighbours
for node_number in trajectory:
trajectory.set_node_position(node_number, find_geodesic_midpoint(trajectory.points[node_number - 1],
trajectory.points[node_number + 1],
local_num_nodes,
dimension,
mass_matrix,
molecule,
energy,
node_number,
length_function)[1])
# Once all the nodes in the curve have been tested, print the node movement
logging.info('Curve Movement: ' + str(trajectory.movement))
# If the required save frequency has passed then have both a pickle of the current curve object and
# write out the animation.
if i % save_frequency == 0:
pickle.dump(trajectory, open(filename+'.pkl', "wb"))
write_xyz_animation(filename+'.pkl',filename+'.xyz')
# If the movement of the curve is below the tol threshold then exit the main loop
if trajectory.movement < tol:
break
# Indicate that the next iteration is to be completed
trajectory.set_node_movable()
# Otherwise the user has indicated they would like to perform a parallel computation...
else:
# Create a callback function that updates the node position once it is calculated
def update_curve(result):
trajectory.set_node_position(result[0], result[1])
# Create a pool of worker processes to work in parallel
pool = multiprocessing.Pool(processes=processes)
# Main loop of the Birkhoff algorithm, continues until curve.movement < tol then breaks out
while True:
# Iterating over each node in the trajectory create a task to find a new position based on the
# geodesic midpoint joining it's neighbours. Add this task to the pool queue.
for node_number in trajectory:
pool.apply_async(func=find_geodesic_midpoint,
args=(trajectory.points[node_number - 1], trajectory.points[node_number + 1], local_num_nodes,
dimension, mass_matrix, molecule, energy, node_number, length_function,),
callback=update_curve)
# If all the nodes in the trajectory have been moved...
if trajectory.all_nodes_moved():
# Once all the nodes in the curve have been tested, print the node movement
logging.info('Curve Movement: ' + str(trajectory.movement))
# If the required save frequency has passed then have both a pickle of the current curve object and
# write out the animation.
if i % save_frequency == 0:
pickle.dump(trajectory, open(filename+'.pkl', "wb"))
write_xyz_animation(filename+'.pkl',filename+'.xyz')
# If the movement of the curve is below the tol threshold then exit the main loop
if trajectory.movement < tol:
break
# Indicate that the next iteration is to be completed
trajectory.set_node_movable()
# Once the algorithm has executed close the pool
pool.close()
pool.join()