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694 lines
23 KiB
Python
694 lines
23 KiB
Python
'''
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Bloch eigenmode solver/operators
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This module contains functions for generating and solving the
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3D Bloch eigenproblem. The approach is to transform the problem
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into the (spatial) fourier domain, transforming the equation
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1/mu * curl(1/eps * curl(H)) = (w/c)^2 H
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into
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conv(1/mu_k, ik x conv(1/eps_k, ik x H_k)) = (w/c)^2 H_k
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where:
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- the `_k` subscript denotes a 3D fourier transformed field
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- each component of `H_k` corresponds to a plane wave with wavevector `k`
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- `x` is the cross product
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- `conv()` denotes convolution
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Since `k` and `H` are orthogonal for each plane wave, we can use each
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`k` to create an orthogonal basis (k, m, n), with `k x m = n`, and
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`|m| = |n| = 1`. The cross products are then simplified with
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k @ h = kx hx + ky hy + kz hz = 0 = hk
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h = hk + hm + hn = hm + hn
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k = kk + km + kn = kk = |k|
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k x h = (ky hz - kz hy,
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kz hx - kx hz,
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kx hy - ky hx)
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= ((k x h) @ k, (k x h) @ m, (k x h) @ n)_kmn
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= (0, (m x k) @ h, (n x k) @ h)_kmn # triple product ordering
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= (0, kk (-n @ h), kk (m @ h))_kmn # (m x k) = -|k| n, etc.
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= |k| (0, -h @ n, h @ m)_kmn
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k x h = (km hn - kn hm,
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kn hk - kk hn,
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kk hm - km hk)_kmn
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= (0, -kk hn, kk hm)_kmn
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= (-kk hn)(mx, my, mz) + (kk hm)(nx, ny, nz)
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= |k| (hm * (nx, ny, nz) - hn * (mx, my, mz))
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where `h` is shorthand for `H_k`, `(...)_kmn` deontes the `(k, m, n)` basis,
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and e.g. `hm` is the component of `h` in the `m` direction.
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We can also simplify `conv(X_k, Y_k)` as `fftn(X * ifftn(Y_k))`.
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Using these results and storing `H_k` as `h = (hm, hn)`, we have
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e_xyz = fftn(1/eps * ifftn(|k| (hm * n - hn * m)))
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b_mn = |k| (-e_xyz @ n, e_xyz @ m)
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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 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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Typically you will want to do something like
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recip_lattice = numpy.diag(1/numpy.array(epsilon[0].shape * dx))
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n, v = bloch.eigsolve(5, k0, recip_lattice, epsilon)
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f = numpy.sqrt(-numpy.real(n[0]))
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n_eff = norm(recip_lattice @ k0) / f
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v2e = bloch.hmn_2_exyz(k0, recip_lattice, epsilon)
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e_field = v2e(v[0])
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k, f = find_k(frequency=1/1550,
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tolerance=(1/1550 - 1/1551),
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direction=[1, 0, 0],
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G_matrix=recip_lattice,
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epsilon=epsilon,
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band=0)
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'''
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from typing import Tuple, Callable
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import logging
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import numpy # type: ignore
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from numpy import pi, real, trace # type: ignore
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from numpy.fft import fftfreq # type: ignore
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import scipy # type: ignore
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import scipy.optimize # type: ignore
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from scipy.linalg import norm # type: ignore
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import scipy.sparse.linalg as spalg # type: ignore
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from ..fdmath import fdfield_t
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logger = logging.getLogger(__name__)
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try:
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import pyfftw.interfaces.numpy_fft # type: ignore
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import pyfftw.interfaces # type: ignore
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import multiprocessing
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logger.info('Using pyfftw')
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pyfftw.interfaces.cache.enable()
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pyfftw.interfaces.cache.set_keepalive_time(3600)
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fftw_args = {
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'threads': multiprocessing.cpu_count(),
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'overwrite_input': True,
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'planner_effort': 'FFTW_EXHAUSTIVE',
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}
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def fftn(*args, **kwargs):
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return pyfftw.interfaces.numpy_fft.fftn(*args, **kwargs, **fftw_args)
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def ifftn(*args, **kwargs):
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return pyfftw.interfaces.numpy_fft.ifftn(*args, **kwargs, **fftw_args)
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except ImportError:
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from numpy.fft import fftn, ifftn # type: ignore
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logger.info('Using numpy fft')
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def generate_kmn(k0: numpy.ndarray,
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G_matrix: numpy.ndarray,
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shape: numpy.ndarray
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) -> Tuple[numpy.ndarray, numpy.ndarray, numpy.ndarray]:
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"""
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Generate a (k, m, n) orthogonal basis for each k-vector in the simulation grid.
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Args:
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k0: [k0x, k0y, k0z], Bloch wavevector, in G basis.
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G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
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shape: [nx, ny, nz] shape of the simulation grid.
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Returns:
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`(|k|, m, n)` where `|k|` has shape `tuple(shape) + (1,)`
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and `m`, `n` have shape `tuple(shape) + (3,)`.
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All are given in the xyz basis (e.g. `|k|[0,0,0] = norm(G_matrix @ k0)`).
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"""
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k0 = numpy.array(k0)
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Gi_grids = numpy.meshgrid(*(fftfreq(n, 1 / n) for n in shape[:3]), indexing='ij')
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Gi = numpy.stack(Gi_grids, axis=3)
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k_G = k0[None, None, None, :] - Gi
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k_xyz = numpy.rollaxis(G_matrix @ numpy.rollaxis(k_G, 3, 2), 3, 2)
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m = numpy.broadcast_to([0, 1, 0], tuple(shape[:3]) + (3,)).astype(float)
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n = numpy.broadcast_to([0, 0, 1], tuple(shape[:3]) + (3,)).astype(float)
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xy_non0 = numpy.any(k_xyz[:, :, :, 0:1] != 0, axis=3)
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if numpy.any(xy_non0):
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u = numpy.cross(k_xyz[xy_non0], [0, 0, 1])
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m[xy_non0, :] = u / norm(u, axis=1)[:, None]
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z_non0 = numpy.any(k_xyz != 0, axis=3)
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if numpy.any(z_non0):
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v = numpy.cross(k_xyz[z_non0], m[z_non0])
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n[z_non0, :] = v / norm(v, axis=1)[:, None]
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k_mag = norm(k_xyz, axis=3)[:, :, :, None]
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return k_mag, m, n
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def maxwell_operator(k0: numpy.ndarray,
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G_matrix: numpy.ndarray,
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epsilon: fdfield_t,
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mu: fdfield_t = None
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) -> Callable[[numpy.ndarray], numpy.ndarray]:
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"""
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Generate the Maxwell operator
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conv(1/mu_k, ik x conv(1/eps_k, ik x ___))
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which is the spatial-frequency-space representation of
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1/mu * curl(1/eps * curl(___))
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The operator is a function that acts on a vector h_mn of size `2 * epsilon[0].size`
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See the `meanas.fdfd.bloch` docstring for more information.
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Args:
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k0: Bloch wavevector, `[k0x, k0y, k0z]`.
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G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
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epsilon: Dielectric constant distribution for the simulation.
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All fields are sampled at cell centers (i.e., NOT Yee-gridded)
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mu: Magnetic permability distribution for the simulation.
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Default None (1 everywhere).
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Returns:
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Function which applies the maxwell operator to h_mn.
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"""
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shape = epsilon[0].shape + (1,)
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k_mag, m, n = generate_kmn(k0, G_matrix, shape)
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epsilon = numpy.stack(epsilon, 3)
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if mu is not None:
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mu = numpy.stack(mu, 3)
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def operator(h: numpy.ndarray):
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"""
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Maxwell operator for Bloch eigenmode simulation.
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h is complex 2-field in (m, n) basis, vectorized
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Args:
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h: Raveled h_mn; size `2 * epsilon[0].size`.
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Returns:
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Raveled conv(1/mu_k, ik x conv(1/eps_k, ik x h_mn)).
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"""
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hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
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#{d,e,h}_xyz fields are complex 3-fields in (1/x, 1/y, 1/z) basis
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# cross product and transform into xyz basis
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d_xyz = (n * hin_m
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- m * hin_n) * k_mag
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# divide by epsilon
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e_xyz = fftn(ifftn(d_xyz, axes=range(3)) / epsilon, axes=range(3))
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# cross product and transform into mn basis
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b_m = numpy.sum(e_xyz * n, axis=3)[:, :, :, None] * -k_mag
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b_n = numpy.sum(e_xyz * m, axis=3)[:, :, :, None] * +k_mag
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if mu is None:
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h_m, h_n = b_m, b_n
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else:
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# transform from mn to xyz
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b_xyz = (m * b_m[:, :, :, None]
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+ n * b_n[:, :, :, None])
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# divide by mu
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h_xyz = fftn(ifftn(b_xyz, axes=range(3)) / mu, axes=range(3))
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# transform back to mn
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h_m = numpy.sum(h_xyz * m, axis=3)
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h_n = numpy.sum(h_xyz * n, axis=3)
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return numpy.hstack((h_m.ravel(), h_n.ravel()))
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return operator
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def hmn_2_exyz(k0: numpy.ndarray,
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G_matrix: numpy.ndarray,
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epsilon: fdfield_t,
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) -> Callable[[numpy.ndarray], fdfield_t]:
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"""
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Generate an operator which converts a vectorized spatial-frequency-space
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`h_mn` into an E-field distribution, i.e.
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ifft(conv(1/eps_k, ik x h_mn))
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The operator is a function that acts on a vector `h_mn` of size `2 * epsilon[0].size`.
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See the `meanas.fdfd.bloch` docstring for more information.
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Args:
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k0: Bloch wavevector, `[k0x, k0y, k0z]`.
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G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
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epsilon: Dielectric constant distribution for the simulation.
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All fields are sampled at cell centers (i.e., NOT Yee-gridded)
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Returns:
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Function for converting `h_mn` into `E_xyz`
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"""
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shape = epsilon[0].shape + (1,)
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epsilon = numpy.stack(epsilon, 3)
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k_mag, m, n = generate_kmn(k0, G_matrix, shape)
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def operator(h: numpy.ndarray) -> fdfield_t:
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hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
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d_xyz = (n * hin_m
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- m * hin_n) * k_mag
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# divide by epsilon
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return numpy.array([ei for ei in numpy.rollaxis(ifftn(d_xyz, axes=range(3)) / epsilon, 3)]) # TODO avoid copy
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return operator
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def hmn_2_hxyz(k0: numpy.ndarray,
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G_matrix: numpy.ndarray,
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epsilon: fdfield_t
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) -> Callable[[numpy.ndarray], fdfield_t]:
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"""
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Generate an operator which converts a vectorized spatial-frequency-space
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`h_mn` into an H-field distribution, i.e.
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ifft(h_mn)
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The operator is a function that acts on a vector `h_mn` of size `2 * epsilon[0].size`.
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See the `meanas.fdfd.bloch` docstring for more information.
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Args:
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k0: Bloch wavevector, `[k0x, k0y, k0z]`.
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G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
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epsilon: Dielectric constant distribution for the simulation.
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Only `epsilon[0].shape` is used.
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Returns:
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Function for converting `h_mn` into `H_xyz`
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"""
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shape = epsilon[0].shape + (1,)
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_k_mag, m, n = generate_kmn(k0, G_matrix, shape)
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def operator(h: numpy.ndarray):
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hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
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h_xyz = (m * hin_m
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+ n * hin_n)
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return [ifftn(hi) for hi in numpy.rollaxis(h_xyz, 3)]
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return operator
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def inverse_maxwell_operator_approx(k0: numpy.ndarray,
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G_matrix: numpy.ndarray,
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epsilon: fdfield_t,
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mu: fdfield_t = None
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) -> Callable[[numpy.ndarray], numpy.ndarray]:
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"""
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Generate an approximate inverse of the Maxwell operator,
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ik x conv(eps_k, ik x conv(mu_k, ___))
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which can be used to improve the speed of ARPACK in shift-invert mode.
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See the `meanas.fdfd.bloch` docstring for more information.
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Args:
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k0: Bloch wavevector, `[k0x, k0y, k0z]`.
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G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
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epsilon: Dielectric constant distribution for the simulation.
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All fields are sampled at cell centers (i.e., NOT Yee-gridded)
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mu: Magnetic permability distribution for the simulation.
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Default None (1 everywhere).
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Returns:
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Function which applies the approximate inverse of the maxwell operator to `h_mn`.
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"""
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shape = epsilon[0].shape + (1,)
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epsilon = numpy.stack(epsilon, 3)
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k_mag, m, n = generate_kmn(k0, G_matrix, shape)
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if mu is not None:
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mu = numpy.stack(mu, 3)
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def operator(h: numpy.ndarray):
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"""
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Approximate inverse Maxwell operator for Bloch eigenmode simulation.
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h is complex 2-field in (m, n) basis, vectorized
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Args:
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h: Raveled h_mn; size `2 * epsilon[0].size`.
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Returns:
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Raveled ik x conv(eps_k, ik x conv(mu_k, h_mn))
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"""
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hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
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#{d,e,h}_xyz fields are complex 3-fields in (1/x, 1/y, 1/z) basis
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if mu is None:
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b_m, b_n = hin_m, hin_n
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else:
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# transform from mn to xyz
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h_xyz = (m * hin_m[:, :, :, None]
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+ n * hin_n[:, :, :, None])
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# multiply by mu
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b_xyz = fftn(ifftn(h_xyz, axes=range(3)) * mu, axes=range(3))
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# transform back to mn
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b_m = numpy.sum(b_xyz * m, axis=3)
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b_n = numpy.sum(b_xyz * n, axis=3)
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# cross product and transform into xyz basis
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e_xyz = (n * b_m
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- m * b_n) / k_mag
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# multiply by epsilon
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d_xyz = fftn(ifftn(e_xyz, axes=range(3)) * epsilon, axes=range(3))
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# cross product and transform into mn basis crossinv_t2c
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h_m = numpy.sum(d_xyz * n, axis=3)[:, :, :, None] / +k_mag
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h_n = numpy.sum(d_xyz * m, axis=3)[:, :, :, None] / -k_mag
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return numpy.hstack((h_m.ravel(), h_n.ravel()))
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return operator
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def find_k(frequency: float,
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tolerance: float,
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direction: numpy.ndarray,
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G_matrix: numpy.ndarray,
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epsilon: fdfield_t,
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mu: fdfield_t = None,
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band: int = 0,
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k_min: float = 0,
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k_max: float = 0.5,
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solve_callback: Callable = None
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) -> Tuple[numpy.ndarray, float]:
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"""
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Search for a bloch vector that has a given frequency.
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Args:
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frequency: Target frequency.
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tolerance: Target frequency tolerance.
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direction: k-vector direction to search along.
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G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
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epsilon: Dielectric constant distribution for the simulation.
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All fields are sampled at cell centers (i.e., NOT Yee-gridded)
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mu: Magnetic permability distribution for the simulation.
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Default None (1 everywhere).
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band: Which band to search in. Default 0 (lowest frequency).
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Returns:
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`(k, actual_frequency)`
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The found k-vector and its frequency.
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"""
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direction = numpy.array(direction) / norm(direction)
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def get_f(k0_mag: float, band: int = 0):
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k0 = direction * k0_mag
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n, v = eigsolve(band + 1, k0, G_matrix=G_matrix, epsilon=epsilon, mu=mu)
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f = numpy.sqrt(numpy.abs(numpy.real(n[band])))
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if solve_callback:
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solve_callback(k0_mag, n, v, f)
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return f
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res = scipy.optimize.minimize_scalar(lambda x: abs(get_f(x, band) - frequency),
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(k_min + k_max) / 2,
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method='Bounded',
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bounds=(k_min, k_max),
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options={'xatol': abs(tolerance)})
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return res.x * direction, res.fun + frequency
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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: fdfield_t,
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mu: fdfield_t = None,
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tolerance: float = 1e-20,
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max_iters: int = 10000,
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reset_iters: int = 100,
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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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k0 of the specified structure.
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Args:
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k0: Bloch wavevector, `[k0x, k0y, k0z]`.
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G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
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epsilon: Dielectric constant distribution for the simulation.
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All fields are sampled at cell centers (i.e., NOT Yee-gridded)
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|
mu: Magnetic permability distribution for the simulation.
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Default `None` (1 everywhere).
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|
tolerance: Solver stops when fractional change in the objective
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`trace(Z.H @ A @ Z @ inv(Z Z.H))` is smaller than the tolerance
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|
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Returns:
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`(eigenvalues, eigenvectors)` where `eigenvalues[i]` corresponds to the
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vector `eigenvectors[i, :]`
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"""
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h_size = 2 * epsilon[0].size
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|
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kmag = norm(G_matrix @ k0)
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|
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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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y_shape = (h_size, num_modes)
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|
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prev_E = 0
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d_scale = 1
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prev_traceGtKG = 0
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#prev_theta = 0.5
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D = numpy.zeros(shape=y_shape, dtype=complex)
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|
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y0 = None
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if y0 is None:
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Z = numpy.random.rand(*y_shape) + 1j * numpy.random.rand(*y_shape)
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else:
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Z = y0
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while True:
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Z *= num_modes / norm(Z)
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ZtZ = Z.conj().T @ Z
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try:
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U = numpy.linalg.inv(ZtZ)
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except numpy.linalg.LinAlgError:
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Z = numpy.random.rand(*y_shape) + 1j * numpy.random.rand(*y_shape)
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|
continue
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|
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trace_U = real(trace(U))
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|
if trace_U > 1e8 * num_modes:
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Z = Z @ scipy.linalg.sqrtm(U).conj().T
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|
prev_traceGtKG = 0
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continue
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|
break
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|
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|
for i in range(max_iters):
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ZtZ = Z.conj().T @ Z
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U = numpy.linalg.inv(ZtZ)
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AZ = scipy_op @ Z
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AZU = AZ @ U
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ZtAZU = Z.conj().T @ AZU
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E_signed = real(trace(ZtAZU))
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|
sgn = numpy.sign(E_signed)
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|
E = numpy.abs(E_signed)
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G = (AZU - Z @ U @ ZtAZU) * sgn
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|
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|
if i > 0 and abs(E - prev_E) < tolerance * 0.5 * (E + prev_E + 1e-7):
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|
logger.info('Optimization succeded: {} - 5e-8 < {} * {} / 2'.format(abs(E - prev_E), tolerance, E + prev_E))
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|
break
|
|
|
|
KG = scipy_iop @ G
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|
traceGtKG = _rtrace_AtB(G, KG)
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|
|
|
if prev_traceGtKG == 0 or i % reset_iters == 0:
|
|
logger.info('CG reset')
|
|
gamma = 0
|
|
else:
|
|
gamma = traceGtKG / prev_traceGtKG
|
|
|
|
D = gamma / d_scale * D + KG
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|
d_scale = num_modes / norm(D)
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|
D *= d_scale
|
|
|
|
ZtAZ = Z.conj().T @ AZ
|
|
|
|
AD = scipy_op @ D
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DtD = D.conj().T @ D
|
|
DtAD = D.conj().T @ AD
|
|
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symZtD = _symmetrize(Z.conj().T @ D)
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|
symZtAD = _symmetrize(Z.conj().T @ AD)
|
|
|
|
Qi_memo = [None, None]
|
|
|
|
def Qi_func(theta):
|
|
nonlocal Qi_memo
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|
if Qi_memo[0] == theta:
|
|
return Qi_memo[1]
|
|
|
|
c = numpy.cos(theta)
|
|
s = numpy.sin(theta)
|
|
Q = c * c * ZtZ + s * s * DtD + 2 * s * c * symZtD
|
|
try:
|
|
Qi = numpy.linalg.inv(Q)
|
|
except numpy.linalg.LinAlgError:
|
|
logger.info('taylor Qi')
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|
# if c or s small, taylor expand
|
|
if c < 1e-4 * s and c != 0:
|
|
DtDi = numpy.linalg.inv(DtD)
|
|
Qi = DtDi / (s * s) - 2 * c / (s * s * s) * (DtDi @ (DtDi @ symZtD).conj().T)
|
|
elif s < 1e-4 * c and s != 0:
|
|
ZtZi = numpy.linalg.inv(ZtZ)
|
|
Qi = ZtZi / (c * c) - 2 * s / (c * c * c) * (ZtZi @ (ZtZi @ symZtD).conj().T)
|
|
else:
|
|
raise Exception('Inexplicable singularity in trace_func')
|
|
Qi_memo[0] = theta
|
|
Qi_memo[1] = Qi
|
|
return Qi
|
|
|
|
def trace_func(theta):
|
|
c = numpy.cos(theta)
|
|
s = numpy.sin(theta)
|
|
Qi = Qi_func(theta)
|
|
R = c * c * ZtAZ + s * s * DtAD + 2 * s * c * symZtAD
|
|
trace = _rtrace_AtB(R, Qi)
|
|
return numpy.abs(trace)
|
|
|
|
'''
|
|
def trace_deriv(theta):
|
|
Qi = Qi_func(theta)
|
|
c2 = numpy.cos(2 * theta)
|
|
s2 = numpy.sin(2 * theta)
|
|
F = -0.5*s2 * (ZtAZ - DtAD) + c2 * symZtAD
|
|
trace_deriv = _rtrace_AtB(Qi, F)
|
|
|
|
G = Qi @ F.conj().T @ Qi.conj().T
|
|
H = -0.5*s2 * (ZtZ - DtD) + c2 * symZtD
|
|
trace_deriv -= _rtrace_AtB(G, H)
|
|
|
|
trace_deriv *= 2
|
|
return trace_deriv * sgn
|
|
|
|
U_sZtD = U @ symZtD
|
|
|
|
dE = 2.0 * (_rtrace_AtB(U, symZtAD) -
|
|
_rtrace_AtB(ZtAZU, U_sZtD))
|
|
|
|
d2E = 2 * (_rtrace_AtB(U, DtAD) -
|
|
_rtrace_AtB(ZtAZU, U @ (DtD - 4 * symZtD @ U_sZtD)) -
|
|
4 * _rtrace_AtB(U, symZtAD @ U_sZtD))
|
|
|
|
# Newton-Raphson to find a root of the first derivative:
|
|
theta = -dE/d2E
|
|
|
|
if d2E < 0 or abs(theta) >= pi:
|
|
theta = -abs(prev_theta) * numpy.sign(dE)
|
|
|
|
# theta, new_E, new_dE = linmin(theta, E, dE, 0.1, min(tolerance, 1e-6), 1e-14, 0, -numpy.sign(dE) * K_PI, trace_func)
|
|
theta, n, _, new_E, _, _new_dE = scipy.optimize.line_search(trace_func, trace_deriv, xk=theta, pk=numpy.ones((1,1)), gfk=dE, old_fval=E, c1=min(tolerance, 1e-6), c2=0.1, amax=pi)
|
|
'''
|
|
result = scipy.optimize.minimize_scalar(trace_func, bounds=(0, pi), tol=tolerance)
|
|
new_E = result.fun
|
|
theta = result.x
|
|
|
|
improvement = numpy.abs(E - new_E) * 2 / numpy.abs(E + new_E)
|
|
logger.info('linmin improvement {}'.format(improvement))
|
|
Z *= numpy.cos(theta)
|
|
Z += D * numpy.sin(theta)
|
|
|
|
prev_traceGtKG = traceGtKG
|
|
#prev_theta = theta
|
|
prev_E = E
|
|
|
|
'''
|
|
Recover eigenvectors from Z
|
|
'''
|
|
U = numpy.linalg.inv(Z.conj().T @ Z)
|
|
Y = Z @ scipy.linalg.sqrtm(U)
|
|
W = Y.conj().T @ (scipy_op @ Y)
|
|
|
|
eigvals, W_eigvecs = numpy.linalg.eig(W)
|
|
eigvecs = Y @ W_eigvecs
|
|
|
|
for i in range(len(eigvals)):
|
|
v = eigvecs[:, i]
|
|
n = eigvals[i]
|
|
v /= norm(v)
|
|
eigness = norm(scipy_op @ v - (v.conj() @ (scipy_op @ v)) * v)
|
|
f = numpy.sqrt(-numpy.real(n))
|
|
df = numpy.sqrt(-numpy.real(n + eigness))
|
|
neff_err = kmag * (1 / df - 1 / f)
|
|
logger.info('eigness {}: {}\n neff_err: {}'.format(i, eigness, neff_err))
|
|
|
|
order = numpy.argsort(numpy.abs(eigvals))
|
|
return eigvals[order], eigvecs.T[order]
|
|
|
|
|
|
'''
|
|
def linmin(x_guess, f0, df0, x_max, f_tol=0.1, df_tol=min(tolerance, 1e-6), x_tol=1e-14, x_min=0, linmin_func):
|
|
if df0 > 0:
|
|
x0, f0, df0 = linmin(-x_guess, f0, -df0, -x_max, f_tol, df_tol, x_tol, -x_min, lambda q, dq: -linmin_func(q, dq))
|
|
return -x0, f0, -df0
|
|
elif df0 == 0:
|
|
return 0, f0, df0
|
|
else:
|
|
x = x_guess
|
|
fx = f0
|
|
dfx = df0
|
|
|
|
isave = numpy.zeros((2,), numpy.intc)
|
|
dsave = numpy.zeros((13,), float)
|
|
|
|
x, fx, dfx, task = minpack2.dsrch(x, fx, dfx, f_tol, df_tol, x_tol, task,
|
|
x_min, x_max, isave, dsave)
|
|
for i in range(int(1e6)):
|
|
if task != 'F':
|
|
logging.info('search converged in {} iterations'.format(i))
|
|
break
|
|
fx = f(x, dfx)
|
|
x, fx, dfx, task = minpack2.dsrch(x, fx, dfx, f_tol, df_tol, x_tol, task,
|
|
x_min, x_max, isave, dsave)
|
|
|
|
return x, fx, dfx
|
|
'''
|
|
|
|
def _rtrace_AtB(A, B):
|
|
return real(numpy.sum(A.conj() * B))
|
|
|
|
def _symmetrize(A):
|
|
return (A + A.conj().T) * 0.5
|
|
|