def Esqr(cosmo, a):
r"""Square of the scale factor dependent factor E(a) in the Hubble
parameter.
Parameters
----------
a : array_like
Scale factor
Returns
-------
E^2 : ndarray, or float if input scalar
Square of the scaling of the Hubble constant as a function of
scale factor
Notes
-----
The Hubble parameter at scale factor `a` is given by
:math:`H^2(a) = E^2(a) H_o^2` where :math:`E^2` is obtained through
Friedman's Equation (see :cite:`2005:Percival`) :
.. math::
E^2(a) = \Omega_m a^{-3} + \Omega_k a^{-2} + \Omega_{de} a^{f(a)}
where :math:`f(a)` is the Dark Energy evolution parameter computed
by :py:meth:`.f_de`.
"""
return cosmo.Omega_m*np.power(a, -3) + cosmo.Omega_k*np.power(a, -2) + \
cosmo.Omega_de*np.power(a, f_de(cosmo, a))
def ft_phase_screen(r0, N, delta, L0, l0):
# Set PSD
del_f = 1/(N*delta)
fx = []
tmp = []
for i in range(512):
tmp.append(-256 + i)
for i in range(512):
fx.append(tmp)
fx = jnp.array(fx)
# frequency grid [1/m]
fy = jnp.transpose(fx)
f, th = cart2pol(fx, fy)
fm = 5.92/l0/(2*jnp.pi)
f0 = 1/L0
# modified von Karman atmospheric phase PSD
PSD_phi = 0.023*jnp.power(r0, -5/3) * jnp.exp(-jnp.power((f/fm), 2))/ jnp.power((jnp.power(f, 2) + jnp.power(f0, 2)), 11/6)
PSD_phi = np.array(PSD_phi)
PSD_phi[256][256] = 0
# random draws of Fourier coefficients
A = np.random.randn(N, N)
B = np.random.randn(N, N)
cn = (A + complex("j")*B) * jnp.sqrt(PSD_phi)*del_f
phz = jnp.real(ift2(cn,1))
return phz
def _case1(zagf):
z, alpha, _, flag = zagf
# dz = - dCDF(z; a) / pdf(z; a)
# pdf = z^(a-1) * e^(-z) / Gamma(a)
# CDF(z; a) = IncompleteGamma(a, z) / Gamma(a)
# dCDF(z; a) = (dIncompleteGamma - IncompleteGamma * Digamma(a)) / Gamma(a)
# =: unnormalized_dCDF / Gamma(a)
# IncompleteGamma ~ z^a [ 1/a - z/(a+1) + z^2/2!(a+2) - z^3/3!(a+3) + z^4/4!(a+4) - z^5/5!(a+5) ]
# =: z^a * term1
# dIncompleteGamma ~ z^a * log(z) * term1 - z^a [1/a^2 - z/(a+1)^2 + z^2/2!(a+2)^2
# - z^3/3!(a+3)^2 + z^4/4!(a+4)^2 - z^5/5!(a+5)^2 ]
# =: z^a * log(z) * term1 - z^a * term2
# unnormalized_dCDF = z^a { [log(z) - Digamma(a)] * term1 - term2 }
zi = 1.0
update = zi / alpha
term1 = update
term2 = update / alpha
for i in range(1, 6):
zi = -zi * z / i
update = zi / (alpha + i)
term1 = term1 + update
term2 = term2 + update / (alpha + i)
unnormalized_cdf_dot = np.power(z, alpha) * (
(np.log(z) - lax.digamma(alpha)) * term1 - term2)
unnormalized_pdf = np.power(z, alpha - 1) * np.exp(-z)
grad = -unnormalized_cdf_dot / unnormalized_pdf
return z, alpha, grad, ~flag
def objective(params: list, bparam: list, batch_input) -> float:
result = 25.0
for w1 in params:
result += np.mean(
np.divide(np.power(w1, 4), 4.0) +
bparam[0] * np.divide(np.power(w1, 2), 2.0))
return result
def mu_init(key, shape, rng):
low = -1 * 1 / jnp.power(x.shape[-1], 0.5)
high = 1 * 1 / jnp.power(x.shape[-1], 0.5)
return jax.random.uniform(rng,
shape=shape,
minval=low,
maxval=high)
def getlambda(self, hyperparams=None):
"""Computing the diagonal of the Lambda matrix (spectral values)
We're using jax.numpy routines since this function is used in the optimisation
Parameters
----------
hyperparams : ndarray
array with hyperparameters, if not passed the object attributes are used
Returns
-------
lambda : ndarray
(m,) array with spectral values
"""
# angular frequencies
omega = self.mperms * 0.5 * jnp.pi / jnp.array([[self.Lx, self.Ly]])
if hyperparams is None:
l = self.l
sf = self.sigma_f
else: # when optimising, we must be able to provide the parameters directly
l = hyperparams[1]
sf = hyperparams[0]
# compute the spectral density values
if self.covfunc.covtype == 'matern': # matern
return sf * jnp.power(
2 * self.covfunc.nu / l + jnp.sum(omega**2, 1),
-(self.covfunc.nu + 1)) / jnp.power(l, self.covfunc.nu)
if self.covfunc.covtype == 'se': # squared exponential
return sf * l * jnp.exp(-0.5 * l + jnp.sum(omega**2, 1))
def pacs_model(priors):
pointing_matrices = [([p.amat_row, p.amat_col], p.amat_data)
for p in priors]
flux_lower = np.asarray([p.prior_flux_lower for p in priors]).T
flux_upper = np.asarray([p.prior_flux_upper for p in priors]).T
bkg_mu = np.asarray([p.bkg[0] for p in priors]).T
bkg_sig = np.asarray([p.bkg[1] for p in priors]).T
with numpyro.plate('bands', len(priors)):
sigma_conf = numpyro.sample('sigma_conf', dist.HalfCauchy(1.0, 0.5))
bkg = numpyro.sample('bkg', dist.Normal(bkg_mu, bkg_sig))
with numpyro.plate('nsrc', priors[0].nsrc):
src_f = numpyro.sample('src_f',
dist.Uniform(flux_lower, flux_upper))
db_hat_psw = sp_matmul(pointing_matrices[0], src_f[:, 0][:, None],
priors[0].snpix).reshape(-1) + bkg[0]
db_hat_pmw = sp_matmul(pointing_matrices[1], src_f[:, 1][:, None],
priors[1].snpix).reshape(-1) + bkg[1]
sigma_tot_psw = jnp.sqrt(
jnp.power(priors[0].snim, 2) + jnp.power(sigma_conf[0], 2))
sigma_tot_pmw = jnp.sqrt(
jnp.power(priors[1].snim, 2) + jnp.power(sigma_conf[1], 2))
with numpyro.plate('psw_pixels', priors[0].sim.size): # as ind_psw:
numpyro.sample("obs_psw",
dist.Normal(db_hat_psw, sigma_tot_psw),
obs=priors[0].sim)
with numpyro.plate('pmw_pixels', priors[1].sim.size): # as ind_pmw:
numpyro.sample("obs_pmw",
dist.Normal(db_hat_pmw, sigma_tot_pmw),
obs=priors[1].sim)
def T_seperate(params_all, x, dx, coeff_mask, z0):
params_psi = params_all[:hyper_params['n_psi']]
sindy_coeff = params_all[-1][0]
z_opt = phi_vec(params_psi, x, z0)
Theta = sindy_library(z_opt)
dz = dz_func_vec(params_psi, x, dx, z0)
dz_pred = dz_pred_vec(Theta, coeff_mask, sindy_coeff)
x_rec = psi_vec(params_psi, z_opt)
dx_rec = dx_network_vec(params_psi, z_opt, dz_pred)
results = {}
results['x_loss'] = jnp.mean(jnp.power(x - x_rec, 2))
results['dx_loss'] = np.mean(jnp.power(dx - dx_rec, 2))
results['dz_loss'] = jnp.mean(jnp.power(dz - dz_pred, 2))
results['regul'] = jnp.mean(jnp.abs(sindy_coeff))
dx_loss = jnp.multiply(hyper_params['eta1'], results['dx_loss'])
dz_loss = jnp.multiply(hyper_params['eta2'], results['dz_loss'])
regul = jnp.multiply(hyper_params['eta3'], results['regul'])
results['loss'] = results['x_loss'] + dx_loss + dz_loss + regul
results['x_rec'] = x_rec
results['dx_rec'] = dx_rec
results['z'] = z_opt
results['dz'] = dz
results['dz_pred'] = dz_pred
return results
def quat2mat_jax(quat):
norm_quat = quat
norm_quat = norm_quat / jnp.linalg.norm(norm_quat, axis=1, keepdims=True)
w, x, y, z = norm_quat[:, 0], norm_quat[:, 1], norm_quat[:,
2], norm_quat[:,
3]
batch_size = quat.shape[0]
w2, x2, y2, z2 = jnp.power(w,
2), jnp.power(x,
2), jnp.power(y,
2), jnp.power(z, 2)
wx, wy, wz = w * x, w * y, w * z
xy, xz, yz = x * y, x * z, y * z
rotMat = jnp.stack(
[
w2 + x2 - y2 - z2,
2 * xy - 2 * wz,
2 * wy + 2 * xz,
2 * wz + 2 * xy,
w2 - x2 + y2 - z2,
2 * yz - 2 * wx,
2 * xz - 2 * wy,
2 * wx + 2 * yz,
w2 - x2 - y2 + z2,
],
axis=1,
).reshape(batch_size, 3, 3)
return rotMat
def loss_function(params, t, W, Xzero): # idea take M,D out by making X0
# X, Y, Y_tilde, Z, DYDT = vXYZpaths(params, t, W, Xzero)
X, Y, Y_tilde, Z = vXYZpaths(params, t, W, Xzero)
loss = jnp.sum(jnp.power(Y[:, 1:-1, :] - Y_tilde[:, 1:-1, :], 2))
loss += jnp.sum(jnp.power(Y[:, -1, :] - vg_tf(X[:, -1, :]), 2))
loss += jnp.sum(jnp.power(Z[:, -1, :] - vDg_tf(X[:, -1, :]), 2))
return loss
def apply_regularization_loss(reg_type,
regularization_values,
regularization_norm=None):
"""Calculates regularization of specified type on supplied values.
Args:
reg_type: which regularization type found in regualarization_fn_dict to use
regularization_values: values subject to regularization
regularization_norm: <p> value if lp-norm is used
Returns:
regularization loss value
Raises:
ValueError: if the specified reg_type is not in regualarization_fn_dict
"""
regualarization_fn_dict = {
'l1_norm':
lambda vals, _: sum(jnp.abs(vals)),
'l2_norm':
lambda vals, _: jnp.sqrt(sum(jnp.power(vals, 2))),
'lp_norm':
lambda vals, norm: jnp.power( # pylint: disable=g-long-lambda
sum(jnp.power(jnp.abs(vals), norm)), 1 / norm),
'1_minus_mean_sqr':
lambda vals, _: (1 - jnp.mean(vals))**2,
'1_minus_mean':
lambda vals, _: (1 - jnp.mean(vals))
}
if reg_type not in regualarization_fn_dict:
raise ValueError(f'Unrecognized regularization type: {reg_type}.')
reg_fn = regualarization_fn_dict[reg_type]
return reg_fn(regularization_values, regularization_norm)
def d3(charges, coordinates, rs6, rs8, s6, s8, rab, rcov, r2r4, coefficients):
bohr_to_angstrom = constants.conversion_factor("bohr", "angstrom")
coordinates_angstrom = list(
map(lambda x: x * bohr_to_angstrom, coordinates))
coordination_numbers = compute_coordination_numbers(
coordinates_angstrom, charges, rcov)
result = 0.0
for j, charge1 in enumerate(charges):
for k, charge2 in enumerate(charges):
if k > j:
dx = coordinates[3 * j + 0] - coordinates[3 * k + 0]
dy = coordinates[3 * j + 1] - coordinates[3 * k + 1]
dz = coordinates[3 * j + 2] - coordinates[3 * k + 2]
dist = jnp.sqrt(dx * dx + dy * dy + dz * dz)
c6jk = get_c6jk(
coefficients[(charge1, charge2)],
coordination_numbers[j],
coordination_numbers[k],
)
rr = rab[(charge1, charge2)] / (dist * bohr_to_angstrom)
r1 = (-s6 * c6jk / (jnp.power(dist, 6) *
(1.0 + 6.0 * jnp.power(rs6 * rr, 14))))
c8jk = 3.0 * c6jk * r2r4[charge1] * r2r4[charge2]
r2 = (-s8 * c8jk / (jnp.power(dist, 8) *
(1.0 + 6.0 * jnp.power(rs8 * rr, 16))))
result += r1 + r2
return result
def mu_init(key, shape):
# Initialization of mean noise parameters (Section 3.2)
low = -1 / jnp.power(x.shape[0], 0.5)
high = 1 / jnp.power(x.shape[0], 0.5)
return jax.random.uniform(key,
minval=low,
maxval=high,
shape=shape)
def _diag_grad_xy(
self, x: jnp.ndarray, y: jnp.ndarray, c: Union[float, jnp.ndarray],
beta: Union[float, jnp.ndarray],
bandwidth: Union[float, jnp.ndarray]) -> Union[float, jnp.ndarray]:
diff = (x - y) / bandwidth
base = c + 0.5 * jnp.sum(jnp.square(diff), axis=-1)
return (- beta * jnp.power(base, beta - 1) + beta * (beta - 1) * diff ** 2 * jnp.power(base, beta - 2)) \
/ bandwidth ** 2
def loss_function(params, t, W, Xzero):
alpha = 1
beta = 1
X, Y, Y_tilde, Z = vXYZpaths(params, t, W, Xzero)
loss = jnp.sum(jnp.power(Y[:, 1:-1, :] - Y_tilde[:, 1:-1, :], 2))
loss += alpha * jnp.sum(jnp.power(Y[:, -1, :] - vg_tf(X[:, -1, :]), 2))
loss += beta * jnp.sum(jnp.power(Z[:, -1, :] - vDg_tf(X[:, -1, :]), 2))
return loss
def loss_function(params, t, W, Xzero): #idea take M,D out by making X0
X,Y,Y_tilde,Z = vXYZpaths(params, t, W, Xzero)
# loss += jnp.sum(jnp.power(Y[:, :, 1:] - Y_tilde, 2)) # Y is 51, Y_tilde is 50 but only sum up to penultimate
loss = jnp.sum(jnp.power(Y[:, 1:-1, :] - Y_tilde[:, 1:-1, :], 2)) + jnp.sum(jnp.power(Y[:,-1,:] - vg_tf(X[:,-1,:]), 2))
loss += jnp.sum(jnp.power(Z[:,-1,:] - vDg_tf(X[:,-1,:]), 2)) #terminal 1st order condition remove for now
# loss += jnp.sum(jnp.power(jnp.dot(Z[:,-1,:].T, sigma_tf(T, X[:,-1,:], Y[:,-1,:])) - vDg_tf(X[:,-1,:]), 2)) #terminal 1st order condition remove for now
return loss
def _tempred_cross_entropy_loss(unused_activations):
loss_values = jnp.multiply(
labels,
log_t(labels + 1e-10, t1) -
log_t(probabilities, t1)) - 1.0 / (2.0 - t1) * (jnp.power(
labels, 2.0 - t1) - jnp.power(probabilities, 2.0 - t1))
loss_values = jnp.sum(loss_values, -1)
return loss_values
def loss_function(params, t, W, Xzero): #idea take M,D out by making X0
X, Y, Y_tilde = vXYpaths(params, t, W, Xzero)
# loss += jnp.sum(jnp.power(Y[:, :, 1:] - Y_tilde, 2)) # Y is 51, Y_tilde is 50 but only sum up to penultimate
# + terminal condition
loss = jnp.sum(jnp.power(Y[:, :, 1:-2] - Y_tilde[:, :, :-2], 2)) + jnp.sum(
jnp.power(Y[:, :, N] - g_tf(X[:, :, N]), 2))
# loss += torch.sum(torch.pow(Z1 - Dg_tf(X1), 2)) #terminal 1st order condition remove for now
return loss
def func(var=np.array([0.5, 1.])):
temp = np.subtract(arg, var[1])
temp = np.power(temp, 2)
temp_sum = np.sum(temp)
divider = np.power(var[0], 2)
res = np.divide(temp_sum, divider)
common = np.log(np.divide(1, var[0]))
return np.dot(res, common)
def for_body(i, state): x, first_moment, second_moment = state g = grad_f(x) first_moment = beta_1*first_moment + (1 - beta_1)*g second_moment = beta_2*second_moment + (1-beta_2)*(g**2) true_first_moment = first_moment/(1. - np.power(beta_1, i+1)) true_second_moment = second_moment/(1. - np.power(beta_2, i+1)) x = x - lr*true_first_moment/(np.sqrt(true_second_moment) + epsilon) return (x, first_moment, second_moment)
def odeInt(f, x0, y0, xT):
# 1993 Solving Ordinary Differential Equations I, page 169
# initial step size
f0 = f(x0, y0)
d0 = rmsNorm(y0)
d1 = rmsNorm(f0)
if d0 < 1e-5 or d1 < 1e-5:
h0 = 1e-6
else:
h0 = 1e-2 * d0 / d1
y1 = y0 + f0 * h0
f1 = f(x0 + h0, y1)
d2 = rmsNorm(f1 - f0) / h0
maxD = np.maximum(d1, d2)
if maxD <= 1e-15:
h1 = np.maximum(1e-6, h0 * 1e-3)
else:
h1 = np.power(1e-2 / maxD, 1 / (P + 1))
step = np.minimum(1e2 * h0, h1)
# integrate
x = x0
y = y0
k1 = f0
while x < xT:
rejected = False
accepted = False
while not accepted:
ks, y1, y1h = odeIntStep(step, f, x, y, k1)
xNew = x + step
scale = ABS_TOL + np.maximum(np.abs(y1), np.abs(y1h)) * REL_TOL
errNorm = rmsNorm((y1 - y1h) / scale)
if errNorm < 1:
accepted = True
if errNorm == 0:
updateFactor = MAX_UPDATE_FACTOR
else:
updateFactor = np.minimum(
MAX_UPDATE_FACTOR,
SAFETY_FACTOR * np.power(errNorm, ERROR_EXP))
if rejected:
updateFactor = np.minimum(1, updateFactor)
step *= updateFactor
else:
rejected = True
updateFactor = np.maximum(
MIN_UPDATE_FACTOR,
SAFETY_FACTOR * np.power(errNorm, ERROR_EXP))
step *= updateFactor
# interpolate
# update
x = xNew
y = y1
k1 = ks[6]
step = xT - x
ks, y1, y1h = odeIntStep(step, f, x, y, k1)
return y1
def sum_f(var):
ret = 0
for i in range(10**4):
# x = 10 / 10**4 * i - 5
x = np.subtract(np.divide(10, np.dot(np.power(10, 4), i)), 5)
# ret -= (x - var[1])**2 / var[0]**2
ret -= np.subtract(
ret,
np.divide(np.power(np.subtract(x, var[1]), 2), np.power(var[0],
2)))
return np.dot(np.log(np.divide(1, var[0])), ret)
def sersic(x, y, mux=0, muy=0, roll=0, q=1, c=2, I=1, Re=1, n=1):
rm = __rotation_matrix(roll)
qm = jnp.array(((1 / q, 0), (0, 1)))
mu = jnp.array((mux, muy))
P = jnp.stack((x.ravel(), y.ravel()))
dp = (jnp.expand_dims(mu, 1) - P)
R = jnp.power(
jnp.sum(jnp.power(jnp.abs(jnp.dot(qm, jnp.dot(rm, dp))), c), axis=0),
1 / c).reshape(x.shape)
intensity = I * jnp.exp(-_b(n) * ((R / Re)**(1 / n) - 1))
return intensity
def compute_sigma(num_examples, iterations, lipshitz, strong, diameter,
epsilon, delta):
"""Theorem 3.1 https://arxiv.org/pdf/2007.02923.pdf"""
gamma = (smooth - strong) / (smooth + strong)
numerator = 4 * np.sqrt(2) * (lipshitz + smooth * diameter) * np.power(
gamma, iterations)
denominator = (strong * num_examples *
(1 - np.power(gamma, iterations))) * (
(np.sqrt(np.log(1 / delta) + epsilon)) -
np.sqrt(np.log(1 / delta)))
return numerator / denominator
def model():
with numpyro.plate("plate", 2):
loc_latent = numpyro.sample(
"loc_latent", FakeNormal(loc0, jnp.power(lam0, -0.5)))
for i, x in enumerate(data):
numpyro.sample(
"obs_{}".format(i),
dist.Normal(loc_latent, jnp.power(lam, -0.5)),
obs=x,
)
return loc_latent
def test_tracegraph_normal_normal():
# normal-normal; known covariance
lam0 = jnp.array([0.1, 0.1]) # precision of prior
loc0 = jnp.array([0.0, 0.5]) # prior mean
# known precision of observation noise
lam = jnp.array([6.0, 4.0])
data = []
data.append(jnp.array([-0.1, 0.3]))
data.append(jnp.array([0.0, 0.4]))
data.append(jnp.array([0.2, 0.5]))
data.append(jnp.array([0.1, 0.7]))
n_data = len(data)
sum_data = data[0] + data[1] + data[2] + data[3]
analytic_lam_n = lam0 + n_data * lam
analytic_log_sig_n = -0.5 * jnp.log(analytic_lam_n)
analytic_loc_n = sum_data * (lam / analytic_lam_n) + loc0 * (
lam0 / analytic_lam_n)
class FakeNormal(dist.Normal):
reparametrized_params = []
def model():
with numpyro.plate("plate", 2):
loc_latent = numpyro.sample(
"loc_latent", FakeNormal(loc0, jnp.power(lam0, -0.5)))
for i, x in enumerate(data):
numpyro.sample(
"obs_{}".format(i),
dist.Normal(loc_latent, jnp.power(lam, -0.5)),
obs=x,
)
return loc_latent
def guide():
loc_q = numpyro.param("loc_q",
analytic_loc_n + jnp.array([0.334, 0.334]))
log_sig_q = numpyro.param(
"log_sig_q", analytic_log_sig_n + jnp.array([-0.29, -0.29]))
sig_q = jnp.exp(log_sig_q)
with numpyro.plate("plate", 2):
loc_latent = numpyro.sample("loc_latent", FakeNormal(loc_q, sig_q))
return loc_latent
adam = optim.Adam(step_size=0.0015, b1=0.97, b2=0.999)
svi = SVI(model, guide, adam, loss=TraceGraph_ELBO())
svi_result = svi.run(jax.random.PRNGKey(0), 5000)
loc_error = jnp.sum(
jnp.power(analytic_loc_n - svi_result.params["loc_q"], 2.0))
log_sig_error = jnp.sum(
jnp.power(analytic_log_sig_n - svi_result.params["log_sig_q"], 2.0))
assert_allclose(loc_error, 0, atol=0.05)
assert_allclose(log_sig_error, 0, atol=0.05)
def loss_function(params, t, W, Xzero): #idea take M,D out by making X0
loss = 0
X, Y, Y_tilde = vXYpaths(params, t, W, Xzero)
loss += jnp.sum(jnp.power(Y[:, :, 1:] - Y_tilde,
2)) # Y is 51, Y_tilde is 50
loss += jnp.sum(jnp.power(Y[:, :, N] - g_tf(X[:, :, N]),
2)) #terminal condition
# loss += torch.sum(torch.pow(Z1 - Dg_tf(X1), 2)) #terminal 1st order condition remove for now
return loss
def harmonic_angle(conf, params, box, angle_idxs, param_idxs, cos_angles=True):
"""
Compute the harmonic bond energy given a collection of molecules.
This implements a harmonic angle potential: V(t) = k*(t - t0)^2 or V(t) = k*(cos(t)-cos(t0))^2
Parameters:
-----------
conf: shape [num_atoms, 3] np.array
atomic coordinates
params: shape [num_params,] np.array
unique parameters
box: shape [3, 3] np.array
periodic boundary vectors, if not None
angle_idxs: shape [num_angles, 3] np.array
each element (a, b, c) is a unique angle in the conformation. atom b is defined
to be the middle atom.
param_idxs: shape [num_angles, 2] np.array
each element (k_idx, t_idx) maps into params for angle constants and ideal angles
cos_angles: True (default)
if True, then this instead implements V(t) = k*(cos(t)-cos(t0))^2. This is far more
numerically stable when the angle is pi.
"""
ci = conf[angle_idxs[:, 0]]
cj = conf[angle_idxs[:, 1]]
ck = conf[angle_idxs[:, 2]]
kas = params[param_idxs[:, 0]]
a0s = params[param_idxs[:, 1]]
vij = delta_r(ci, cj, box)
vjk = delta_r(ck, cj, box)
top = np.sum(np.multiply(vij, vjk), -1)
bot = np.linalg.norm(vij, axis=-1) * np.linalg.norm(vjk, axis=-1)
tb = top / bot
# (ytz): we used the squared version so that we make this energy being strictly positive
if cos_angles:
energies = kas / 2 * np.power(tb - np.cos(a0s), 2)
else:
angle = np.arccos(tb)
energies = kas / 2 * np.power(angle - a0s, 2)
return np.sum(energies, -1) # reduce over all angles
def lens_against_ft(Uin, wvl, d1, f):
N = len(Uin[:,0])
k = 2 * jnp.pi / wvl
fx = []
x2 = []
y2 = []
for i in range(512):
fx.append((-256 + i) * wvl * f)
for i in range(512):
x2.append(fx)
y2 = jnp.transpose(x2)
Uout = jnp.exp(complex("j") * k / (2*f) * (jnp.power(x2, 2) + jnp.power(y2, 2))) / (complex("j")*wvl*f) * ft2(Uin, d1)
return Uout, np.array(x2), np.array(y2)
def func(var):
# (var[0]-args[:,0])**3
temp = np.sin(np.power(np.subtract(var[0], args[:, 0]), 3))
ele1 = np.sum(temp)
# sin(var[1]**3 - var[2]**2) * args[:,1]
temp = np.sin(np.subtract(np.power(var[1], 3), np.power(var[2], 2)))
ele2 = np.sum(np.dot(temp, args[:, 1]))
# log(args[:,1]*args[:,2]*var[0]*var[1]*var[2])
temp = np.multiply(args[:, 1], args[:, 2])
ele3 = np.sum(np.log(1 + np.abs(np.dot(var[0] * var[1] * var[2], temp))))
# print("\n\n", ele1, "\n", ele2, "\n", ele3)
# print(res, type(res))
return ele1 + ele2 + ele3
