Commit 35afcda0 authored by Martin Reinecke's avatar Martin Reinecke
Browse files

Merge branch '31-rewrite-unitloggauss' into 'NIFTy_5'

Resolve "Rewrite UnitLogGauss"

Closes #31

See merge request ift/nifty-dev!21
parents 0de313ef 2dfc4494
......@@ -172,7 +172,7 @@
" tol_abs_gradnorm=0.1)\n",
" # WienerFilterCurvature is (R.adjoint*N.inverse*R + Sh.inverse) plus some handy\n",
" # helper methods.\n",
" return ift.library.WienerFilterCurvature(R,N,Sh,iteration_controller=IC,iteration_controller_sampling=IC)"
" return ift.WienerFilterCurvature(R,N,Sh,iteration_controller=IC,iteration_controller_sampling=IC)"
]
},
{
......
%% Cell type:markdown id: tags:
# A NIFTy demonstration
%% Cell type:markdown id: tags:
## IFT: Big Picture
IFT starting point:
$$d = Rs+n$$
Typically, $s$ is a continuous field, $d$ a discrete data vector. Particularly, $R$ is not invertible.
IFT aims at **inverting** the above uninvertible problem in the **best possible way** using Bayesian statistics.
## NIFTy
NIFTy (Numerical Information Field Theory) is a Python framework in which IFT problems can be tackled easily.
Main Interfaces:
- **Spaces**: Cartesian, 2-Spheres (Healpix, Gauss-Legendre) and their respective harmonic spaces.
- **Fields**: Defined on spaces.
- **Operators**: Acting on fields.
%% Cell type:markdown id: tags:
## Wiener Filter: Formulae
### Assumptions
- $d=Rs+n$, $R$ linear operator.
- $\mathcal P (s) = \mathcal G (s,S)$, $\mathcal P (n) = \mathcal G (n,N)$ where $S, N$ are positive definite matrices.
### Posterior
The Posterior is given by:
$$\mathcal P (s|d) \propto P(s,d) = \mathcal G(d-Rs,N) \,\mathcal G(s,S) \propto \mathcal G (m,D) $$
where
$$\begin{align}
m &= Dj \\
D^{-1}&= (S^{-1} +R^\dagger N^{-1} R )\\
j &= R^\dagger N^{-1} d
\end{align}$$
Let us implement this in NIFTy!
%% Cell type:markdown id: tags:
## Wiener Filter: Example
- We assume statistical homogeneity and isotropy. Therefore the signal covariance $S$ is diagonal in harmonic space, and is described by a one-dimensional power spectrum, assumed here as $$P(k) = P_0\,\left(1+\left(\frac{k}{k_0}\right)^2\right)^{-\gamma /2},$$
with $P_0 = 0.2, k_0 = 5, \gamma = 4$.
- $N = 0.2 \cdot \mathbb{1}$.
- Number of data points $N_{pix} = 512$.
- reconstruction in harmonic space.
- Response operator:
$$R = FFT_{\text{harmonic} \rightarrow \text{position}}$$
%% Cell type:code id: tags:
``` python
N_pixels = 512 # Number of pixels
def pow_spec(k):
P0, k0, gamma = [.2, 5, 4]
return P0 / ((1. + (k/k0)**2)**(gamma / 2))
```
%% Cell type:markdown id: tags:
## Wiener Filter: Implementation
%% Cell type:markdown id: tags:
### Import Modules
%% Cell type:code id: tags:
``` python
import numpy as np
np.random.seed(40)
import nifty5 as ift
import matplotlib.pyplot as plt
%matplotlib inline
```
%% Cell type:markdown id: tags:
### Implement Propagator
%% Cell type:code id: tags:
``` python
def Curvature(R, N, Sh):
IC = ift.GradientNormController(iteration_limit=50000,
tol_abs_gradnorm=0.1)
# WienerFilterCurvature is (R.adjoint*N.inverse*R + Sh.inverse) plus some handy
# helper methods.
return ift.library.WienerFilterCurvature(R,N,Sh,iteration_controller=IC,iteration_controller_sampling=IC)
return ift.WienerFilterCurvature(R,N,Sh,iteration_controller=IC,iteration_controller_sampling=IC)
```
%% Cell type:markdown id: tags:
### Conjugate Gradient Preconditioning
- $D$ is defined via:
$$D^{-1} = \mathcal S_h^{-1} + R^\dagger N^{-1} R.$$
In the end, we want to apply $D$ to $j$, i.e. we need the inverse action of $D^{-1}$. This is done numerically (algorithm: *Conjugate Gradient*).
<!--
- One can define the *condition number* of a non-singular and normal matrix $A$:
$$\kappa (A) := \frac{|\lambda_{\text{max}}|}{|\lambda_{\text{min}}|},$$
where $\lambda_{\text{max}}$ and $\lambda_{\text{min}}$ are the largest and smallest eigenvalue of $A$, respectively.
- The larger $\kappa$ the slower Conjugate Gradient.
- By default, conjugate gradient solves: $D^{-1} m = j$ for $m$, where $D^{-1}$ can be badly conditioned. If one knows a non-singular matrix $T$ for which $TD^{-1}$ is better conditioned, one can solve the equivalent problem:
$$\tilde A m = \tilde j,$$
where $\tilde A = T D^{-1}$ and $\tilde j = Tj$.
- In our case $S^{-1}$ is responsible for the bad conditioning of $D$ depending on the chosen power spectrum. Thus, we choose
$$T = \mathcal F^\dagger S_h^{-1} \mathcal F.$$
-->
%% Cell type:markdown id: tags:
### Generate Mock data
- Generate a field $s$ and $n$ with given covariances.
- Calculate $d$.
%% Cell type:code id: tags:
``` python
s_space = ift.RGSpace(N_pixels)
h_space = s_space.get_default_codomain()
HT = ift.HarmonicTransformOperator(h_space, target=s_space)
# Operators
Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)
R = HT #*ift.create_harmonic_smoothing_operator((h_space,), 0, 0.02)
# Fields and data
sh = Sh.draw_sample()
noiseless_data=R(sh)
noise_amplitude = np.sqrt(0.2)
N = ift.ScalingOperator(noise_amplitude**2, s_space)
n = ift.Field.from_random(domain=s_space, random_type='normal',
std=noise_amplitude, mean=0)
d = noiseless_data + n
j = R.adjoint_times(N.inverse_times(d))
curv = Curvature(R=R, N=N, Sh=Sh)
D = curv.inverse
```
%% Cell type:markdown id: tags:
### Run Wiener Filter
%% Cell type:code id: tags:
``` python
m = D(j)
```
%% Cell type:markdown id: tags:
### Signal Reconstruction
%% Cell type:code id: tags:
``` python
# Get signal data and reconstruction data
s_data = HT(sh).to_global_data()
m_data = HT(m).to_global_data()
d_data = d.to_global_data()
plt.figure(figsize=(15,10))
plt.plot(s_data, 'r', label="Signal", linewidth=3)
plt.plot(d_data, 'k.', label="Data")
plt.plot(m_data, 'k', label="Reconstruction",linewidth=3)
plt.title("Reconstruction")
plt.legend()
plt.show()
```
%% Cell type:code id: tags:
``` python
plt.figure(figsize=(15,10))
plt.plot(s_data - s_data, 'r', label="Signal", linewidth=3)
plt.plot(d_data - s_data, 'k.', label="Data")
plt.plot(m_data - s_data, 'k', label="Reconstruction",linewidth=3)
plt.axhspan(-noise_amplitude,noise_amplitude, facecolor='0.9', alpha=.5)
plt.title("Residuals")
plt.legend()
plt.show()
```
%% Cell type:markdown id: tags:
### Power Spectrum
%% Cell type:code id: tags:
``` python
s_power_data = ift.power_analyze(sh).to_global_data()
m_power_data = ift.power_analyze(m).to_global_data()
plt.figure(figsize=(15,10))
plt.loglog()
plt.xlim(1, int(N_pixels/2))
ymin = min(m_power_data)
plt.ylim(ymin, 1)
xs = np.arange(1,int(N_pixels/2),.1)
plt.plot(xs, pow_spec(xs), label="True Power Spectrum", color='k',alpha=0.5)
plt.plot(s_power_data, 'r', label="Signal")
plt.plot(m_power_data, 'k', label="Reconstruction")
plt.axhline(noise_amplitude**2 / N_pixels, color="k", linestyle='--', label="Noise level", alpha=.5)
plt.axhspan(noise_amplitude**2 / N_pixels, ymin, facecolor='0.9', alpha=.5)
plt.title("Power Spectrum")
plt.legend()
plt.show()
```
%% Cell type:markdown id: tags:
## Wiener Filter on Incomplete Data
%% Cell type:code id: tags:
``` python
# Operators
Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)
N = ift.ScalingOperator(noise_amplitude**2,s_space)
# R is defined below
# Fields
sh = Sh.draw_sample()
s = HT(sh)
n = ift.Field.from_random(domain=s_space, random_type='normal',
std=noise_amplitude, mean=0)
```
%% Cell type:markdown id: tags:
### Partially Lose Data
%% Cell type:code id: tags:
``` python
l = int(N_pixels * 0.2)
h = int(N_pixels * 0.2 * 2)
mask = np.full(s_space.shape, 1.)
mask[l:h] = 0
mask = ift.Field.from_global_data(s_space, mask)
R = ift.DiagonalOperator(mask)*HT
n = n.to_global_data()
n[l:h] = 0
n = ift.Field.from_global_data(s_space, n)
d = R(sh) + n
```
%% Cell type:code id: tags:
``` python
curv = Curvature(R=R, N=N, Sh=Sh)
D = curv.inverse
j = R.adjoint_times(N.inverse_times(d))
m = D(j)
```
%% Cell type:markdown id: tags:
### Compute Uncertainty
%% Cell type:code id: tags:
``` python
m_mean, m_var = ift.probe_with_posterior_samples(curv, HT, 200)
```
%% Cell type:markdown id: tags:
### Get data
%% Cell type:code id: tags:
``` python
# Get signal data and reconstruction data
s_data = s.to_global_data()
m_data = HT(m).to_global_data()
m_var_data = m_var.to_global_data()
uncertainty = np.sqrt(m_var_data)
d_data = d.to_global_data()
# Set lost data to NaN for proper plotting
d_data[d_data == 0] = np.nan
```
%% Cell type:code id: tags:
``` python
fig = plt.figure(figsize=(15,10))
plt.axvspan(l, h, facecolor='0.8',alpha=0.5)
plt.fill_between(range(N_pixels), m_data - uncertainty, m_data + uncertainty, facecolor='0.5', alpha=0.5)
plt.plot(s_data, 'r', label="Signal", alpha=1, linewidth=3)
plt.plot(d_data, 'k.', label="Data")
plt.plot(m_data, 'k', label="Reconstruction", linewidth=3)
plt.title("Reconstruction of incomplete data")
plt.legend()
```
%% Cell type:markdown id: tags:
# 2d Example
%% Cell type:code id: tags:
``` python
N_pixels = 256 # Number of pixels
sigma2 = 2. # Noise variance
def pow_spec(k):
P0, k0, gamma = [.2, 2, 4]
return P0 * (1. + (k/k0)**2)**(-gamma/2)
s_space = ift.RGSpace([N_pixels, N_pixels])
```
%% Cell type:code id: tags:
``` python
h_space = s_space.get_default_codomain()
HT = ift.HarmonicTransformOperator(h_space,s_space)
# Operators
Sh = ift.create_power_operator(h_space, power_spectrum=pow_spec)
N = ift.ScalingOperator(sigma2,s_space)
# Fields and data
sh = Sh.draw_sample()
n = ift.Field.from_random(domain=s_space, random_type='normal',
std=np.sqrt(sigma2), mean=0)
# Lose some data
l = int(N_pixels * 0.33)
h = int(N_pixels * 0.33 * 2)
mask = np.full(s_space.shape, 1.)
mask[l:h,l:h] = 0.
mask = ift.Field.from_global_data(s_space, mask)
R = ift.DiagonalOperator(mask)*HT
n = n.to_global_data()
n[l:h, l:h] = 0
n = ift.Field.from_global_data(s_space, n)
curv = Curvature(R=R, N=N, Sh=Sh)
D = curv.inverse
d = R(sh) + n
j = R.adjoint_times(N.inverse_times(d))
# Run Wiener filter
m = D(j)
# Uncertainty
m_mean, m_var = ift.probe_with_posterior_samples(curv, HT, 20)
# Get data
s_data = HT(sh).to_global_data()
m_data = HT(m).to_global_data()
m_var_data = m_var.to_global_data()
d_data = d.to_global_data()
uncertainty = np.sqrt(np.abs(m_var_data))
```
%% Cell type:code id: tags:
``` python
cm = ['magma', 'inferno', 'plasma', 'viridis'][1]
mi = np.min(s_data)
ma = np.max(s_data)
fig, axes = plt.subplots(1, 2, figsize=(15, 7))
data = [s_data, d_data]
caption = ["Signal", "Data"]
for ax in axes.flat:
im = ax.imshow(data.pop(0), interpolation='nearest', cmap=cm, vmin=mi,
vmax=ma)
ax.set_title(caption.pop(0))
fig.subplots_adjust(right=0.8)
cbar_ax = fig.add_axes([0.85, 0.15, 0.05, 0.7])
fig.colorbar(im, cax=cbar_ax)
```
%% Cell type:code id: tags:
``` python
mi = np.min(s_data)
ma = np.max(s_data)
fig, axes = plt.subplots(3, 2, figsize=(15, 22.5))
sample = HT(curv.draw_sample(from_inverse=True)+m).to_global_data()
post_mean = (m_mean + HT(m)).to_global_data()
data = [s_data, m_data, post_mean, sample, s_data - m_data, uncertainty]
caption = ["Signal", "Reconstruction", "Posterior mean", "Sample", "Residuals", "Uncertainty Map"]
for ax in axes.flat:
im = ax.imshow(data.pop(0), interpolation='nearest', cmap=cm, vmin=mi, vmax=ma)
ax.set_title(caption.pop(0))
fig.subplots_adjust(right=0.8)
cbar_ax = fig.add_axes([.85, 0.15, 0.05, 0.7])
fig.colorbar(im, cax=cbar_ax)
```
%% Cell type:markdown id: tags:
### Is the uncertainty map reliable?
%% Cell type:code id: tags:
``` python
precise = (np.abs(s_data-m_data) < uncertainty)
print("Error within uncertainty map bounds: " + str(np.sum(precise) * 100 / N_pixels**2) + "%")
plt.figure(figsize=(15,10))
plt.imshow(precise.astype(float), cmap="brg")
plt.colorbar()
```
%% Cell type:markdown id: tags:
# Start Coding
## NIFTy Repository + Installation guide
https://gitlab.mpcdf.mpg.de/ift/NIFTy
NIFTy v5 **more or less stable!**
......
......@@ -2,7 +2,9 @@ import nifty5 as ift
import numpy as np
from global_newton.models_other.apply_data import ApplyData
from global_newton.models_energy.hamiltonian import Hamiltonian
from nifty5.library.unit_log_gauss import UnitLogGauss
from nifty5 import GaussianEnergy
if __name__ == '__main__':
# s_space = ift.RGSpace([1024])
s_space = ift.RGSpace([128,128])
......@@ -45,7 +47,7 @@ if __name__ == '__main__':
NWR = ApplyData(data, ift.Field(d_space,val=noise), Rs)
INITIAL_POSITION = ift.from_random('normal',total_domain)
likelihood = UnitLogGauss(INITIAL_POSITION, NWR)
likelihood = GaussianEnergy(INITIAL_POSITION, NWR)
IC = ift.GradientNormController(iteration_limit=500, tol_abs_gradnorm=1e-3)
inverter = ift.ConjugateGradient(controller=IC)
......
......@@ -16,7 +16,7 @@ from .minimization import *
from .sugar import *
from .plotting.plot import plot
from . import library
from .library import *
from . import extra
from .utilities import memo
......
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
#
# Copyright(C) 2013-2018 Max-Planck-Society
#
# NIFTy is being developed at the Max-Planck-Institut fuer Astrophysik
# and financially supported by the Studienstiftung des deutschen Volkes.
from ..library.gaussian_energy import GaussianEnergy
from ..minimization.energy import Energy
from ..operators import InversionEnabler, SamplingEnabler
from ..models.variable import Variable
from ..operators import InversionEnabler, SamplingEnabler
from ..utilities import memo
from ..library.unit_log_gauss import UnitLogGauss
class Hamiltonian(Energy):
......@@ -15,11 +33,8 @@ class Hamiltonian(Energy):
super(Hamiltonian, self).__init__(lh.position)
self._lh = lh
self._ic = iteration_controller
if iteration_controller_sampling is None:
self._ic_samp = iteration_controller
else:
self._ic_samp = iteration_controller_sampling
self._prior = UnitLogGauss(Variable(self.position))
self._ic_samp = iteration_controller_sampling
self._prior = GaussianEnergy(Variable(self.position))
self._precond = self._prior.curvature
def at(self, position):
......@@ -39,8 +54,11 @@ class Hamiltonian(Energy):
@memo
def curvature(self):
prior_curv = self._prior.curvature
c = SamplingEnabler(self._lh.curvature, prior_curv.inverse,
self._ic_samp, prior_curv.inverse)
if self._ic_samp is None:
c = self._lh.curvature + prior_curv
else:
c = SamplingEnabler(self._lh.curvature, prior_curv.inverse,
self._ic_samp, prior_curv.inverse)
return InversionEnabler(c, self._ic, self._precond)
def __str__(self):
......
from .amplitude_model import make_amplitude_model
from .apply_data import ApplyData
from .gaussian_energy import GaussianEnergy
from .los_response import LOSResponse
from .nonlinear_wiener_filter_energy import NonlinearWienerFilterEnergy
from .unit_log_gauss import UnitLogGauss
from .point_sources import PointSources
from .poisson_log_likelihood import PoissonLogLikelihood
from .poissonian_energy import PoissonianEnergy
from .smooth_sky import make_smooth_mf_sky_model, make_smooth_sky_model
from .wiener_filter_curvature import WienerFilterCurvature
from .wiener_filter_energy import WienerFilterEnergy
def ApplyData(data, var, model_data):
from .. import DiagonalOperator, Constant, sqrt
# TODO This is rather confusing. Delete that eventually.
from ..operators.diagonal_operator import DiagonalOperator
from ..models.constant import Constant
from ..sugar import sqrt
sqrt_n = DiagonalOperator(sqrt(var))
data = Constant(model_data.position, data)
return sqrt_n.inverse(model_data - data)
......@@ -17,45 +17,50 @@
# and financially supported by the Studienstiftung des deutschen Volkes.
from ..minimization.energy import Energy
from ..operators.inversion_enabler import InversionEnabler
from ..operators.sandwich_operator import SandwichOperator
from ..utilities import memo
class UnitLogGauss(Energy):
def __init__(self, s, inverter=None):
class GaussianEnergy(Energy):
def __init__(self, inp, mean=None, covariance=None):
"""
s: Sky model object
inp: Model object
value = 0.5 * s.vdot(s), i.e. a log-Gauss distribution with unit
covariance
"""
super(UnitLogGauss, self).__init__(s.position)
self._s = s
self._inverter = inverter
super(GaussianEnergy, self).__init__(inp.position)
self._inp = inp
self._mean = mean
self._cov = covariance
def at(self, position):
return self.__class__(self._s.at(position), self._inverter)
return self.__class__(self._inp.at(position), self._mean, self._cov)
@property
@memo
def _gradient_helper(self):
return self._s.gradient
def residual(self):
if self._mean is not None:
return self._inp.value - self._mean
return self._inp.value
@property
@memo
def value(self):
return .5 * self._s.value.squared_norm()
if self._cov is None:
return .5 * self.residual.vdot(self.residual).real
return .5 * self.residual.vdot(self._cov.inverse(self.residual)).real
@property
@memo
def gradient(self):
return self._gradient_helper.adjoint(self._s.value)
if self._cov is None:
return self._inp.gradient.adjoint(self.residual)
return self._inp.gradient.adjoint(self._cov.inverse(self.residual))
@property
@memo
def curvature(self):
c = SandwichOperator.make(self._gradient_helper)
if self._inverter is None:
return c
return InversionEnabler(c, self._inverter)
if self._cov is None:
return SandwichOperator.make(self._inp.gradient, None)
return SandwichOperator.make(self._inp.gradient, self._cov.inverse)
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
#
# Copyright(C) 2013-2018 Max-Planck-Society
#
# NIFTy is being developed at the Max-Planck-Institut fuer Astrophysik
# and financially supported by the Studienstiftung des deutschen Volkes.
from ..models.constant import Constant
from .unit_log_gauss import UnitLogGauss
from ..energies.hamiltonian import Hamiltonian
def NonlinearWienerFilterEnergy(measured_data, data_model, sqrtN, iteration_controller):
d = measured_data.lock()
residual = Constant(data_model.position, d) - data_model
lh = UnitLogGauss(sqrtN.inverse(residual))
return Hamiltonian(lh, iteration_controller)
......@@ -23,7 +23,7 @@ from ..operators.sandwich_operator import SandwichOperator
from ..sugar import log, makeOp
class PoissonLogLikelihood(Energy):
class PoissonianEnergy(Energy):
def __init__(self, lamb, d):
"""
lamb: Sky model object
......@@ -31,7 +31,7 @@ class PoissonLogLikelihood(Energy):
value = 0.5 * s.vdot(s), i.e. a log-Gauss distribution with unit
covariance
"""
super(PoissonLogLikelihood, self).__init__(lamb.position)
super(PoissonianEnergy, self).__init__(lamb.position)
self._lamb = lamb
self._d = d
......
......@@ -164,10 +164,25 @@ class MultiField(object):
def __neg__(self):
return MultiField({key: -val for key, val in self.items()})
def __abs__(self):
return MultiField({key: abs(val) for key, val in self.items()})
def conjugate(self):
return MultiField({key: sub_field.conjugate()
for key, sub_field in self.items()})
def all(self):
for v in self.values():
if not v.all():
return False
return True
def any(self):
for v in self.values():
if v.any():
return True
return False
def isEquivalentTo(self, other):
"""Determines (as quickly as possible) whether `self`'s content is
identical to `other`'s content."""
......
......@@ -57,7 +57,7 @@ class Energy_Tests(unittest.TestCase):
tol_abs_gradnorm=1e-5)
S = ift.create_power_operator(hspace, power_spectrum=_flat_PS)
energy = ift.library.WienerFilterEnergy(
energy = ift.WienerFilterEnergy(
position=s0, d=d, R=R, N=N, S=S, iteration_controller=IC)
ift.extra.check_value_gradient_curvature_consistency(
energy, ntries=10)
......@@ -66,10 +66,10 @@ class Energy_Tests(unittest.TestCase):
ift.RGSpace(64, distances=.789),
ift.RGSpace([32, 32], distances=.789)],
[ift.Tanh, ift.Exponential, ift.Linear],
[1, 1e-2, 1e2],
[4, 78, 23]))
def testNonlinearMap(self, space, nonlinearity, seed):
def testGaussianEnergy(self, space, nonlinearity, noise, seed):
np.random.seed(seed)
f = nonlinearity()
dim = len(space.shape)
hspace = space.get_default_codomain()
ht = ift.HarmonicTransformOperator(hspace, target=space)
......@@ -77,23 +77,23 @@ class Energy_Tests(unittest.TestCase):
pspace = ift.PowerSpace(hspace, binbounds=binbounds)
Dist = ift.PowerDistributor(target=hspace, power_space=pspace)
xi0 = ift.Field.from_random(domain=hspace, random_type='normal')
xi0_var = ift.Variable(ift.MultiField({'xi':xi0}))['xi']
xi0_var = ift.Variable(ift.MultiField({'xi': xi0}))['xi']
def pspec(k): return 1 / (1 + k**2)**dim
pspec = ift.PS_field(pspace, pspec)
A = Dist(ift.sqrt(pspec))
n = ift.Field.from_random(domain=space, random_type='normal')
N = ift.ScalingOperator(noise, space)
n = N.draw_sample()
s = ht(ift.makeOp(A)(xi0_var))
R = ift.ScalingOperator(10., space)
sqrtN = ift.ScalingOperator(1., space)
d_model = R(ift.LocalModel(s, nonlinearity()))
d = d_model.value + n
IC = ift.GradientNormController(iteration_limit=100,
tol_abs_gradnorm=1e-5)
energy = ift.library.NonlinearWienerFilterEnergy(
d, d_model, sqrtN, IC)
if isinstance(nonlinearity, ift.Linear):
if noise == 1:
N = None
energy = ift.GaussianEnergy(d_model, d, N)
if isinstance(nonlinearity(), ift.Linear):
ift.extra.check_value_gradient_curvature_consistency(
energy, ntries=10)
else:
......
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