implement eigenvalue algorithm from Johnson paper. Could also use arpack + refinement, but that's also slow.
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@ -47,12 +47,10 @@ This module contains functions for generating and solving the
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h_mn = fftn(1/mu * ifftn(b_m * m + b_n * n))
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which forms the operator from the left side of the equation.
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We can then use ARPACK in shift-invert mode (via scipy.linalg.eigs)
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to find the eigenvectors for this operator.
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This approach is similar to the one used in MPB and derived at the start of
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SG Johnson and JD Joannopoulos, Block-iterative frequency-domain methods
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We can then use a preconditioned block Rayleigh iteration algorithm, as in
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SG Johnson and JD Joannopoulos, Block-iterative frequency-domain methods
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for Maxwell's equations in a planewave basis, Optics Express 8, 3, 173-190 (2001)
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(similar to that used in MPB) to find the eigenvectors for this operator.
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===
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@ -76,14 +74,19 @@ This module contains functions for generating and solving the
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'''
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from typing import List, Tuple, Callable, Dict
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import logging
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import numpy
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from numpy.fft import fftn, ifftn, fftfreq
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import scipy
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import scipy.optimize
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from scipy.linalg import norm
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import scipy.sparse.linalg as spalg
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from .eigensolvers import rayleigh_quotient_iteration
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from . import field_t
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logger = logging.getLogger(__name__)
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def generate_kmn(k0: numpy.ndarray,
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G_matrix: numpy.ndarray,
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@ -338,7 +341,8 @@ def eigsolve(num_modes: int,
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k0: numpy.ndarray,
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G_matrix: numpy.ndarray,
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epsilon: field_t,
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mu: field_t = None
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mu: field_t = None,
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tolerance = 1e-8,
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) -> Tuple[numpy.ndarray, numpy.ndarray]:
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"""
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Find the first (lowest-frequency) num_modes eigenmodes with Bloch wavevector
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@ -355,15 +359,102 @@ def eigsolve(num_modes: int,
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"""
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h_size = 2 * epsilon[0].size
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'''
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Generate the operators
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'''
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mop = maxwell_operator(k0=k0, G_matrix=G_matrix, epsilon=epsilon, mu=mu)
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imop = inverse_maxwell_operator_approx(k0=k0, G_matrix=G_matrix, epsilon=epsilon, mu=mu)
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scipy_op = spalg.LinearOperator(dtype=complex, shape=(h_size, h_size), matvec=mop)
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scipy_iop = spalg.LinearOperator(dtype=complex, shape=(h_size, h_size), matvec=imop)
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_eigvals, eigvecs = spalg.eigs(scipy_op, num_modes, sigma=0, OPinv=scipy_iop, which='LM')
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eigvals = numpy.sum(eigvecs * (scipy_op @ eigvecs), axis=0) / numpy.sum(eigvecs * eigvecs, axis=0)
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order = numpy.argsort(-eigvals)
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y_shape = (h_size, num_modes)
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def rayleigh_quotient(Z: numpy.ndarray, approx_grad: bool = True):
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"""
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Absolute value of the block Rayleigh quotient, and the associated gradient.
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See Johnson and Joannopoulos, Opt. Expr. 8, 3 (2001) for details (full
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citation in module docstring).
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===
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Notes on my understanding of the procedure:
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Minimize f(Y) = |trace((Y.H @ A @ Y)|, making use of Y = Z @ inv(Z.H @ Z)^(1/2)
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(a polar orthogonalization of Y). This gives f(Z) = |trace(Z.H @ A @ Z @ U)|,
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where U = inv(Z.H @ Z). We minimize the absolute value to find the eigenvalues
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with smallest magnitude.
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The gradient is P @ (A @ Z @ U), where P = (1 - Z @ U @ Z.H) is a projection
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onto the space orthonormal to Z. If approx_grad is True, the approximate
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inverse of the maxwell operator is used to precondition the gradient.
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"""
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z = Z.reshape(y_shape)
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U = numpy.linalg.inv(z.conj().T @ z)
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zU = z @ U
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AzU = scipy_op @ zU
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zTAzU = z.conj().T @ AzU
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f = numpy.real(numpy.trace(zTAzU))
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if approx_grad:
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df_dy = scipy_iop @ (AzU - zU @ zTAzU)
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else:
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df_dy = (AzU - zU @ zTAzU)
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return numpy.abs(f), numpy.sign(f) * df_dy.ravel()
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'''
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Use the conjugate gradient method and the approximate gradient calculation to
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quickly find approximate eigenvectors.
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'''
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result = scipy.optimize.minimize(rayleigh_quotient,
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numpy.random.rand(*y_shape),
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jac=True,
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method='CG',
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tol=1e-5,
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options={'maxiter': 30, 'disp':True})
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result = scipy.optimize.minimize(lambda y: rayleigh_quotient(y, False),
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result.x,
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jac=True,
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method='CG',
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tol=1e-13,
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options={'maxiter': 100, 'disp':True})
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z = result.x.reshape(y_shape)
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'''
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Recover eigenvectors from Z
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'''
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U = numpy.linalg.inv(z.conj().T @ z)
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y = z @ scipy.linalg.sqrtm(U)
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w = y.conj().T @ (scipy_op @ y)
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eigvals, w_eigvecs = numpy.linalg.eig(w)
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eigvecs = y @ w_eigvecs
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for i in range(len(eigvals)):
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v = eigvecs[:, i]
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n = eigvals[i]
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v /= norm(v)
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logger.info('eigness {}: {}'.format(i, norm(scipy_op @ v - (v.conj() @ (scipy_op @ v)) * v )))
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ev2 = eigvecs.copy()
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for i in range(len(eigvals)):
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logger.info('Refining eigenvector {}'.format(i))
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eigvals[i], ev2[:, i] = rayleigh_quotient_iteration(scipy_op,
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guess_vector=eigvecs[:, i],
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iterations=40,
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tolerance=tolerance * numpy.real(numpy.sqrt(eigvals[i])) * 2,
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solver = lambda A, b: spalg.bicgstab(A, b, maxiter=200)[0])
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eigvecs = ev2
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order = numpy.argsort(numpy.abs(eigvals))
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for i in range(len(eigvals)):
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v = eigvecs[:, i]
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n = eigvals[i]
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v /= norm(v)
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logger.info('eigness {}: {}'.format(i, norm(scipy_op @ v - (v.conj() @ (scipy_op @ v)) * v )))
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return eigvals[order], eigvecs.T[order]
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