import numpy as np
import lenstronomy.Util.util as util
import lenstronomy.Util.image_util as image_util
from scipy.optimize import minimize
from lenstronomy.LensModel.Solver.epl_shear_solver import solve_lenseq_pemd
__all__ = ["LensEquationSolver"]
[docs]
class LensEquationSolver(object):
"""Class to solve for image positions given lens model and source position."""
[docs]
def __init__(self, lensModel):
"""This class must contain the following definitions (with same syntax as the
standard LensModel() class: def ray_shooting() def hessian() def magnification()
:param lensModel: instance of a class according to
lenstronomy.LensModel.lens_model
"""
self.lensModel = lensModel
[docs]
def image_position_stochastic(
self,
source_x,
source_y,
kwargs_lens,
search_window=10,
precision_limit=10 ** (-10),
arrival_time_sort=True,
x_center=0,
y_center=0,
num_random=1000,
):
"""Solves the lens equation stochastic with the scipy minimization routine on
the quadratic distance between the backwards ray-shooted proposed image position
and the source position. Credits to Giulia Pagano.
:param source_x: source position
:param source_y: source position
:param kwargs_lens: lens model list of keyword arguments
:param search_window: angular size of search window
:param precision_limit: limit required on the precision in the source plane
:param arrival_time_sort: bool, if True sorts according to arrival time
:param x_center: center of search window
:param y_center: center of search window
:param num_random: number of random starting points of the non-linear solver in
the search window
:return: x_image, y_image
"""
kwargs_lens = self.lensModel.set_static(kwargs_lens)
x_solve, y_solve = [], []
for i in range(num_random):
x_init = (
np.random.uniform(-search_window / 2.0, search_window / 2) + x_center
)
y_init = (
np.random.uniform(-search_window / 2.0, search_window / 2) + y_center
)
xinitial = np.array([x_init, y_init])
result = minimize(
self._root,
xinitial,
args=(kwargs_lens, source_x, source_y),
tol=precision_limit**2,
method="Nelder-Mead",
)
if (
self._root(result.x, kwargs_lens, source_x, source_y)
< precision_limit**2
):
x_solve.append(result.x[0])
y_solve.append(result.x[1])
x_mins, y_mins = image_util.findOverlap(x_solve, y_solve, precision_limit)
if arrival_time_sort:
x_mins, y_mins = self.sort_arrival_times(x_mins, y_mins, kwargs_lens)
self.lensModel.set_dynamic()
return x_mins, y_mins
def _root(self, x, kwargs_lens, source_x, source_y):
"""
:param x: parameters [x-coord, y-coord]
:param kwargs_lens: list of keyword arguments of the lens model
:param source_x: source position
:param source_y: source position
:return: square distance between ray-traced image position and given source position
"""
x_, y_ = x
beta_x, beta_y = self.lensModel.ray_shooting(x_, y_, kwargs_lens)
return (beta_x - source_x) ** 2 + (beta_y - source_y) ** 2
[docs]
def candidate_solutions(
self,
sourcePos_x,
sourcePos_y,
kwargs_lens,
min_distance=0.1,
search_window=10,
verbose=False,
x_center=0,
y_center=0,
):
"""Finds pixels in the image plane possibly hosting a solution of the lens
equation, for the given source position and lens model.
:param sourcePos_x: source position in units of angle
:param sourcePos_y: source position in units of angle
:param kwargs_lens: lens model parameters as keyword arguments
:param min_distance: minimum separation to consider for two images in units of
angle
:param search_window: window size to be considered by the solver. Will not find
image position outside this window
:param verbose: bool, if True, prints some useful information for the user
:param x_center: float, center of the window to search for point sources
:param y_center: float, center of the window to search for point sources
:returns: (approximate) angular position of (multiple) images ra_pos, dec_pos in
units of angles, related ray-traced source displacements and pixel width
:raises: AttributeError, KeyError
"""
kwargs_lens = self.lensModel.set_static(kwargs_lens)
# compute number of pixels to cover the search window with the required min_distance
numPix = int(round(search_window / min_distance) + 0.5)
x_grid, y_grid = util.make_grid(numPix, min_distance)
x_grid += x_center
y_grid += y_center
# ray-shoot to find the relative distance to the required source position for each grid point
x_mapped, y_mapped = self.lensModel.ray_shooting(x_grid, y_grid, kwargs_lens)
absmapped = util.displaceAbs(x_mapped, y_mapped, sourcePos_x, sourcePos_y)
# select minima in the grid points and select grid points that do not deviate more than the
# width of the grid point to a solution of the lens equation
x_mins, y_mins, delta_map = util.local_minima_2d(absmapped, x_grid, y_grid)
# pixel width
pixel_width = x_grid[1] - x_grid[0]
return x_mins, y_mins, delta_map, pixel_width
[docs]
def image_position_analytical(
self,
x,
y,
kwargs_lens,
arrival_time_sort=True,
magnification_limit=None,
**kwargs_solver,
):
"""Solves the lens equation. Only supports EPL-like (plus shear) models. Uses a
specialized recipe that solves a one-dimensional lens equation that is easier
and more reliable to solve than the usual two-dimensional lens equation.
:param x: source position in units of angle, an array of positions is also supported.
:param y: source position in units of angle, an array of positions is also supported.
:param kwargs_lens: lens model parameters as keyword arguments
:param arrival_time_sort: bool, if True, sorts image position in arrival time (first arrival photon first listed)
:param magnification_limit: None or float, if set will only return image positions that have an
abs(magnification) larger than this number
:param kwargs_solver: additional kwargs to be supplied to the solver. Particularly relevant are Nmeas and Nmeas_extra
:returns: (exact) angular position of (multiple) images ra_pos, dec_pos in units of angle
Note: in contrast to the other solvers, generally the (heavily demagnified) central image will also be included, so
setting a a proper magnification_limit is more important. To get similar behaviour, a limit of 1e-1 is acceptable
"""
lens_model_list = list(self.lensModel.lens_model_list)
if lens_model_list not in (
["SIE", "SHEAR"],
["SIE"],
["EPL_NUMBA", "SHEAR"],
["EPL_NUMBA"],
["EPL", "SHEAR"],
["EPL"],
):
raise ValueError(
"Only SIE, EPL, EPL_NUMBA (+shear) supported in the analytical solver for now."
)
x_mins, y_mins = solve_lenseq_pemd((x, y), kwargs_lens, **kwargs_solver)
if arrival_time_sort:
x_mins, y_mins = self.sort_arrival_times(x_mins, y_mins, kwargs_lens)
if magnification_limit is not None:
mag = np.abs(self.lensModel.magnification(x_mins, y_mins, kwargs_lens))
x_mins = x_mins[mag >= magnification_limit]
y_mins = y_mins[mag >= magnification_limit]
return x_mins, y_mins
[docs]
def image_position_from_source(
self, sourcePos_x, sourcePos_y, kwargs_lens, solver="lenstronomy", **kwargs
):
"""Solves the lens equation, i.e. finds the image positions in the lens plane
that are mapped to a given source position.
:param sourcePos_x: source position in units of angle
:param sourcePos_y: source position in units of angle
:param kwargs_lens: lens model parameters as keyword arguments
:param solver: which solver to use, can be 'lenstronomy' (default), 'analytical'
or 'stochastic'.
:param kwargs: Any additional kwargs are passed to the chosen solver, see the
documentation of image_position_lenstronomy, image_position_analytical and
image_position_stochastic
:returns: (exact) angular position of (multiple) images ra_pos, dec_pos in units
of angle
"""
if solver == "lenstronomy":
return self.image_position_lenstronomy(
sourcePos_x, sourcePos_y, kwargs_lens, **kwargs
)
if solver == "analytical":
return self.image_position_analytical(
sourcePos_x, sourcePos_y, kwargs_lens, **kwargs
)
if solver == "stochastic":
return self.image_position_stochastic(
sourcePos_x, sourcePos_y, kwargs_lens, **kwargs
)
raise ValueError(f"{solver} is not a valid solver.")
[docs]
def image_position_lenstronomy(
self,
sourcePos_x,
sourcePos_y,
kwargs_lens,
min_distance=0.1,
search_window=10,
precision_limit=10 ** (-10),
num_iter_max=100,
arrival_time_sort=True,
initial_guess_cut=True,
verbose=False,
x_center=0,
y_center=0,
num_random=0,
non_linear=False,
magnification_limit=None,
):
"""Finds image position given source position and lens model. The solver first
samples does a grid search in the lens plane, and the grid points that are
closest to the supplied source position are fed to a specialized gradient-based
root finder that finds the exact solutions. Works with all lens models.
:param sourcePos_x: source position in units of angle
:param sourcePos_y: source position in units of angle
:param kwargs_lens: lens model parameters as keyword arguments
:param min_distance: minimum separation to consider for two images in units of
angle
:param search_window: window size to be considered by the solver. Will not find
image position outside this window
:param precision_limit: required precision in the lens equation solver (in units
of angle in the source plane).
:param num_iter_max: maximum iteration of lens-source mapping conducted by
solver to match the required precision
:param arrival_time_sort: bool, if True, sorts image position in arrival time
(first arrival photon first listed)
:param initial_guess_cut: bool, if True, cuts initial local minima selected by
the grid search based on distance criteria from the source position
:param verbose: bool, if True, prints some useful information for the user
:param x_center: float, center of the window to search for point sources
:param y_center: float, center of the window to search for point sources
:param num_random: int, number of random positions within the search window to
be added to be starting positions for the gradient decent solver
:param non_linear: bool, if True applies a non-linear solver not dependent on
Hessian computation
:param magnification_limit: None or float, if set will only return image
positions that have an abs(magnification) larger than this number
:returns: (exact) angular position of (multiple) images ra_pos, dec_pos in units
of angle
:raises: AttributeError, KeyError
"""
# find pixels in the image plane possibly hosting a solution of the lens equation, related source distances and
# pixel width
x_mins, y_mins, delta_map, pixel_width = self.candidate_solutions(
sourcePos_x,
sourcePos_y,
kwargs_lens,
min_distance,
search_window,
verbose,
x_center,
y_center,
)
if verbose:
print(
"There are %s regions identified that could contain a solution of the lens equation with"
"coordinates %s and %s " % (len(x_mins), x_mins, y_mins)
)
if len(x_mins) < 1:
return x_mins, y_mins
if initial_guess_cut:
mag = np.abs(self.lensModel.magnification(x_mins, y_mins, kwargs_lens))
mag[mag < 1] = 1
x_mins = x_mins[delta_map <= min_distance * mag * 5]
y_mins = y_mins[delta_map <= min_distance * mag * 5]
if verbose:
print(
"The number of regions that meet the plausibility criteria are %s"
% len(x_mins)
)
x_mins = np.append(
x_mins,
np.random.uniform(
low=-search_window / 2 + x_center,
high=search_window / 2 + x_center,
size=num_random,
),
)
y_mins = np.append(
y_mins,
np.random.uniform(
low=-search_window / 2 + y_center,
high=search_window / 2 + y_center,
size=num_random,
),
)
# iterative solving of the lens equation for the selected grid points
# print("Candidates:", x_mins.shape, y_mins.shape)
x_mins, y_mins, solver_precision = self._find_gradient_decent(
x_mins,
y_mins,
sourcePos_x,
sourcePos_y,
kwargs_lens,
precision_limit,
num_iter_max,
verbose=verbose,
min_distance=min_distance,
non_linear=non_linear,
)
# only select iterative results that match the precision limit
x_mins = x_mins[solver_precision <= precision_limit]
y_mins = y_mins[solver_precision <= precision_limit]
# find redundant solutions within the min_distance criterion
x_mins, y_mins = image_util.findOverlap(x_mins, y_mins, min_distance)
if arrival_time_sort:
x_mins, y_mins = self.sort_arrival_times(x_mins, y_mins, kwargs_lens)
if magnification_limit is not None:
mag = np.abs(self.lensModel.magnification(x_mins, y_mins, kwargs_lens))
x_mins = x_mins[mag >= magnification_limit]
y_mins = y_mins[mag >= magnification_limit]
self.lensModel.set_dynamic()
return x_mins, y_mins
def _find_gradient_decent(
self,
x_min,
y_min,
sourcePos_x,
sourcePos_y,
kwargs_lens,
precision_limit=10 ** (-10),
num_iter_max=200,
verbose=False,
min_distance=0.01,
non_linear=False,
):
"""Given a 'good guess' of a solution of the lens equation (expected image
position given a fixed source position) this routine iteratively performs a ray-
tracing with second order correction (effectively gradient decent) to find a
precise solution to the lens equation.
:param x_min: np.array, list of 'good guess' solutions of the lens equation
:param y_min: np.array, list of 'good guess' solutions of the lens equation
:param sourcePos_x: source position for which to solve the lens equation
:param sourcePos_y: source position for which to solve the lens equation
:param kwargs_lens: keyword argument list of the lens model
:param precision_limit: float, required match in the solution in the source
plane
:param num_iter_max: int, maximum number of iterations before the algorithm
stops
:param verbose: bool, if True inserts print statements about the behavior of the
solver
:param min_distance: maximum correction applied per step (to avoid over-shooting
in unstable regions)
:param non_linear: bool, if True, uses scipy.miminize instead of the directly
implemented gradient decent approach.
:return: x_position array, y_position array, error in the source plane array
"""
num_candidates = len(x_min)
x_mins = np.zeros(num_candidates)
y_mins = np.zeros(num_candidates)
solver_precision = np.zeros(num_candidates)
for i in range(len(x_min)):
x_guess, y_guess, delta, l = self._solve_single_proposal(
x_min[i],
y_min[i],
sourcePos_x,
sourcePos_y,
kwargs_lens,
precision_limit,
num_iter_max,
max_step=min_distance,
non_linear=non_linear,
)
if verbose:
print(
"Solution found for region %s with required precision at iteration %s"
% (i, l)
)
x_mins[i] = x_guess
y_mins[i] = y_guess
solver_precision[i] = delta
return x_mins, y_mins, solver_precision
def _solve_single_proposal(
self,
x_guess,
y_guess,
source_x,
source_y,
kwargs_lens,
precision_limit,
num_iter_max,
max_step,
non_linear=False,
):
"""Gradient decent solution of a single proposed starting point (close to a true
solution)
:param x_guess: starting guess position in the image plane
:param y_guess: starting guess position in the image plane
:param source_x: source position to solve for in the image plane
:param source_y: source position to solve for in the image plane
:param kwargs_lens: keyword argument list of the lens model
:param precision_limit: float, required match in the solution in the source
plane
:param num_iter_max: int, maximum number of iterations before the algorithm
stops
:param max_step: maximum correction applied per step (to avoid over-shooting in
instable regions)
:param non_linear: bool, if True, uses scipy.miminize instead of the directly
implemented gradient decent approach.
:return: x_position, y_position, error in the source plane, steps required (for
gradient decent)
"""
l = 0
if non_linear:
xinitial = np.array([x_guess, y_guess])
result = minimize(
self._root,
xinitial,
args=(kwargs_lens, source_x, source_y),
tol=precision_limit**2,
method="Nelder-Mead",
)
delta = self._root(result.x, kwargs_lens, source_x, source_y)
x_guess, y_guess = result.x[0], result.x[1]
else:
x_mapped, y_mapped = self.lensModel.ray_shooting(
x_guess, y_guess, kwargs_lens
)
delta = np.sqrt((x_mapped - source_x) ** 2 + (y_mapped - source_y) ** 2)
while delta > precision_limit and l < num_iter_max:
x_mapped, y_mapped = self.lensModel.ray_shooting(
x_guess, y_guess, kwargs_lens
)
delta = np.sqrt((x_mapped - source_x) ** 2 + (y_mapped - source_y) ** 2)
f_xx, f_xy, f_yx, f_yy = self.lensModel.hessian(
x_guess, y_guess, kwargs_lens
)
DistMatrix = np.array([[1 - f_yy, f_yx], [f_xy, 1 - f_xx]])
det = (1 - f_xx) * (1 - f_yy) - f_xy * f_yx
deltaVec = np.array([x_mapped - source_x, y_mapped - source_y])
image_plane_vector = DistMatrix.dot(deltaVec) / det
dist = np.sqrt(image_plane_vector[0] ** 2 + image_plane_vector[1] ** 2)
if dist > max_step:
image_plane_vector *= max_step / dist
x_guess, y_guess, delta, l = self._gradient_step(
x_guess,
y_guess,
source_x,
source_y,
delta,
image_plane_vector,
kwargs_lens,
l,
num_iter_max,
)
return x_guess, y_guess, delta, l
def _gradient_step(
self,
x_guess,
y_guess,
source_x,
source_y,
delta_init,
image_plane_vector,
kwargs_lens,
iter_num,
num_iter_max,
):
"""
:param x_guess: float, current best fit solution in the image plane
:param y_guess: float, current best fit solution in the image plane
:param source_x: float, source position to be matched
:param source_y: float, source position ot be matched
:param delta_init: current precision in the source plane of the mapped solution
:param image_plane_vector: correction vector in the image plane based on the Hessian operator and the deviation
in the source plane
:param kwargs_lens: lens model keyword argument list
:param iter_num: int, current iteration number
:param num_iter_max: int, maximum iteration number before aborting the process
:return: updated image position in x, updated image position in y, updated precision in the source plane,
total iterations done after this call
"""
x_new = x_guess - image_plane_vector[0]
y_new = y_guess - image_plane_vector[1]
x_mapped, y_mapped = self.lensModel.ray_shooting(x_new, y_new, kwargs_lens)
delta_new = np.sqrt((x_mapped - source_x) ** 2 + (y_mapped - source_y) ** 2)
iter_num += 1
if delta_new > delta_init:
if num_iter_max < iter_num:
return x_guess, y_guess, delta_init, iter_num
else:
# if the new proposal is worse than the previous one, it randomly draws a new proposal in a different
# direction and tries again
image_plane_vector[0] *= np.random.normal(loc=0, scale=0.5)
image_plane_vector[1] *= np.random.normal(loc=0, scale=0.5)
return self._gradient_step(
x_guess,
y_guess,
source_x,
source_y,
delta_init,
image_plane_vector,
kwargs_lens,
iter_num,
num_iter_max,
)
else:
return x_new, y_new, delta_new, iter_num
[docs]
def findBrightImage(
self,
sourcePos_x,
sourcePos_y,
kwargs_lens,
numImages=4,
min_distance=0.01,
search_window=5,
precision_limit=10 ** (-10),
num_iter_max=10,
arrival_time_sort=True,
x_center=0,
y_center=0,
num_random=0,
non_linear=False,
magnification_limit=None,
initial_guess_cut=True,
verbose=False,
):
"""
:param sourcePos_x: source position in units of angle
:param sourcePos_y: source position in units of angle
:param kwargs_lens: lens model parameters as keyword arguments
:param min_distance: minimum separation to consider for two images in units of angle
:param search_window: window size to be considered by the solver. Will not find image position outside this window
:param precision_limit: required precision in the lens equation solver (in units of angle in the source plane).
:param num_iter_max: maximum iteration of lens-source mapping conducted by solver to match the required precision
:param arrival_time_sort: bool, if True, sorts image position in arrival time (first arrival photon first listed)
:param initial_guess_cut: bool, if True, cuts initial local minima selected by the grid search based on distance criteria from the source position
:param verbose: bool, if True, prints some useful information for the user
:param x_center: float, center of the window to search for point sources
:param y_center: float, center of the window to search for point sources
:param num_random: int, number of random positions within the search window to be added to be starting
positions for the gradient decent solver
:param non_linear: bool, if True applies a non-linear solver not dependent on Hessian computation
:param magnification_limit: None or float, if set will only return image positions that have an
abs(magnification) larger than this number
:returns: (exact) angular position of (multiple) images ra_pos, dec_pos in units of angle
"""
x_mins, y_mins = self.image_position_from_source(
sourcePos_x,
sourcePos_y,
kwargs_lens,
min_distance=min_distance,
search_window=search_window,
precision_limit=precision_limit,
num_iter_max=num_iter_max,
arrival_time_sort=arrival_time_sort,
initial_guess_cut=initial_guess_cut,
verbose=verbose,
x_center=x_center,
y_center=y_center,
num_random=num_random,
non_linear=non_linear,
magnification_limit=magnification_limit,
)
mag_list = []
for i in range(len(x_mins)):
mag = self.lensModel.magnification(x_mins[i], y_mins[i], kwargs_lens)
mag_list.append(abs(mag))
mag_list = np.array(mag_list)
x_mins_sorted = util.selectBest(x_mins, mag_list, numImages)
y_mins_sorted = util.selectBest(y_mins, mag_list, numImages)
if arrival_time_sort:
x_mins_sorted, y_mins_sorted = self.sort_arrival_times(
x_mins_sorted, y_mins_sorted, kwargs_lens
)
return x_mins_sorted, y_mins_sorted
[docs]
def sort_arrival_times(self, x_mins, y_mins, kwargs_lens):
"""Sort arrival times (fermat potential) of image positions in increasing order
of light travel time.
:param x_mins: ra position of images
:param y_mins: dec position of images
:param kwargs_lens: keyword arguments of lens model
:return: sorted lists of x_mins and y_mins
"""
if hasattr(self.lensModel, "_no_potential"):
raise Exception(
"Instance of `LensModel` passed to this class does not compute the lensing potential, "
"and therefore cannot compute time delays."
)
if len(x_mins) <= 1:
return x_mins, y_mins
if self.lensModel.multi_plane:
arrival_time = self.lensModel.arrival_time(x_mins, y_mins, kwargs_lens)
else:
fermat_pot = self.lensModel.fermat_potential(x_mins, y_mins, kwargs_lens)
arrival_time = fermat_pot
idx = np.argsort(arrival_time)
x_mins = np.array(x_mins)[idx]
y_mins = np.array(y_mins)[idx]
return x_mins, y_mins