def gradient_S3_pullback(gamma, psi, sigma):
"""
Compute the gradient on S3 by first calculating the gradient on S2
and pulling back the result to S3.
"""
x = hopf(psi)
grad_S2 = gradient_S2_slow(gamma, x, sigma)
return np.einsum('ij, abj, ib -> ia', grad_S2, pauli, psi)
def setUp(self):
# Random initial conditions
self.gamma = np.random.rand(self.N)
phi = np.random.rand(self.N, 2) + 1j*np.random.rand(self.N, 2)
# Normalize phi
norms = np.sum(phi.conj()*phi, axis=1)**.5
phi = row_product(1./norms, phi)
# Perturb away from the unit sphere somewhat
phi *= 1.01
self.phi = phi
# Compute projected point
self.x = hopf(phi)
import numpy as np
from hopf.lie_algebras.su2_geometry import hopf
from hopf.util.vectors import row_product
from hopf.util.matlab_io import save_variables
def create_random_initial_conditions(N):
"""
Create an ensemble of `N` randomly chosen vortices and vortex strengths.
"""
phi = np.random.rand(N, 2) + 1j*np.random.rand(N, 2)
gamma = np.random.rand(N)
# Normalize
norms = np.sum(phi.conj()*phi, axis=1)**.5
phi = row_product(1./norms, phi)
return phi, gamma
if __name__ == '__main__':
print "Creating 40 random vortices of strength 1/8..."
phi, _ = create_random_initial_conditions(40)
gamma = 1./8 * np.ones(40)
sigma = 0.1
x = hopf(phi)
save_variables('random40.mat', {'gamma': gamma, 'X0': x, 'sigma': sigma})
