def setUp(self):
# real-frequency grid and example spectrum
w = np.linspace(-10., 10., num=501, endpoint=True)
spec = 0.4 * np.exp(-0.5 * (w - 1.8)**2) + 0.6 * np.exp(-0.5 *
(w + 1.8)**2)
spec /= np.trapz(spec, w)
# Matsubara frequency grid and transformation of data
beta = 10.
niw = 20
iw = np.pi / beta * (2. * np.arange(niw) + 1.)
kernel = 1. / (1j * iw[:, None] - w[None, :])
giw = np.trapz(kernel * spec[None, :], w, axis=1)
# add noise to the data
rng = np.random.RandomState(4713)
noise_ampl = 0.0001
giw += noise_ampl * (rng.randn(niw) + 1j * rng.randn(niw))
# error bars and default model for analytic continuation
err = np.ones_like(iw) * noise_ampl
model = np.ones_like(w)
model /= np.trapz(model, w)
# specify the analytic continuation problem
probl = cont.AnalyticContinuationProblem(im_axis=iw,
re_axis=w,
im_data=giw,
kernel_mode='freq_fermionic')
# solve the problem
self.sol, _ = probl.solve(method='maxent_svd',
alpha_determination='chi2kink',
optimizer='newton',
model=model,
stdev=err,
interactive=False,
alpha_start=1e12,
alpha_end=1e-2,
preblur=True,
blur_width=0.5,
verbose=False)
self.A_opt_reference = np.load("tests/A_opt_maxent_fermionic.npy")
self.backtransform_reference = np.load(
"tests/backtransform_maxent_fermionic.npy")
self.chi2_reference = np.load("tests/chi2_maxent_fermionic.npy")
def setUp(self):
w_real = np.linspace(0., 5., num=2001, endpoint=True)
spec_real = np.exp(-(w_real)**2 / (2. * 0.2**2))
spec_real += 0.3 * np.exp(-(w_real - 1.5)**2 / (2. * 0.8**2))
spec_real += 0.3 * np.exp(-(w_real + 1.5)**2 /
(2. * 0.8**2)) # must be symmetric around 0!
spec_real /= np.trapz(spec_real, w_real) # normalization
beta = 10.
iw = 2. * np.pi / beta * np.arange(10)
noise_amplitude = 1e-4 # create gaussian noise
rng = np.random.RandomState(1234)
noise = rng.normal(0., noise_amplitude, iw.shape[0])
kernel = (w_real**2)[None, :] / ((iw**2)[:, None] +
(w_real**2)[None, :])
kernel[0, 0] = 1.
gf_bos = np.trapz(kernel * spec_real[None, :], w_real, axis=1) + noise
norm = gf_bos[0]
gf_bos /= norm
w = np.linspace(0., 5., num=501, endpoint=True)
probl = cont.AnalyticContinuationProblem(im_axis=iw,
re_axis=w,
im_data=gf_bos,
kernel_mode='freq_bosonic')
err = np.ones_like(iw) * noise_amplitude / norm
model = np.ones_like(w)
model /= np.trapz(model, w)
self.sol, _ = probl.solve(method='maxent_svd',
alpha_determination='chi2kink',
optimizer='newton',
stdev=err,
model=model,
interactive=False,
verbose=False)
self.A_opt_reference = np.load("tests/A_opt_maxent_bosonic.npy")
self.backtransform_reference = np.load(
"tests/backtransform_maxent_bosonic.npy")
self.chi2_reference = np.load("tests/chi2_maxent_bosonic.npy")
def setUp(self):
# real-frequency grid and example spectrum
w = np.linspace(-10., 10., num=501, endpoint=True)
spec = 0.4 * np.exp(-0.5 * (w - 1.8)**2) + 0.6 * np.exp(-0.5 *
(w + 1.8)**2)
spec /= np.trapz(spec, w)
# Matsubara frequency grid and transformation of data
beta = 10.
niw = 20
iw = np.pi / beta * (2. * np.arange(niw) + 1.)
kernel = 1. / (1j * iw[:, None] - w[None, :])
giw = np.trapz(kernel * spec[None, :], w, axis=1)
# add noise to the data
rng = np.random.RandomState(4713)
noise_ampl = 0.0001
giw += noise_ampl * (rng.randn(niw) + 1j * rng.randn(niw))
# error bars and default model for analytic continuation
err = np.ones_like(iw) * noise_ampl
# specify the analytic continuation problem
mats_ind = [0, 1, 2, 3, 4, 5, 6, 8,
9] # Matsubara indices of data points to use in Pade
probl = cont.AnalyticContinuationProblem(im_axis=iw[mats_ind],
re_axis=w,
im_data=giw[mats_ind],
kernel_mode='freq_fermionic')
# solve the problem
self.sol = probl.solve(method='pade')
check_axis = np.linspace(0., 1.25 * iw[mats_ind[-1]], num=500)
self.check = probl.solver.check(im_axis_fine=check_axis)
self.A_opt_reference = np.load("tests/A_opt_pade.npy")
self.check_reference = np.load("tests/check_pade.npy")
def main_function(self):
"""Main function for the analytic continuation procedure.
This function is called when the "Do it" button is clicked.
It performs an analytical continuation for the present settings
and shows a plot.
"""
mats_ind = self.parse_mats_ind()
self.ana_cont_probl = cont.AnalyticContinuationProblem(
im_axis=self.input_data.mats[mats_ind],
im_data=self.input_data.value[mats_ind],
re_axis=self.realgrid.grid,
kernel_mode='freq_fermionic')
sol = self.ana_cont_probl.solve(method='pade')
check_axis = np.linspace(0.,
1.25 * self.input_data.mats[mats_ind[-1]],
num=500)
check = self.ana_cont_probl.solver.check(im_axis_fine=check_axis)
self.output_data.update(self.realgrid.grid, sol.A_opt, self.input_data)
fig, ax = plt.subplots(ncols=2, nrows=2,
figsize=(11.75, 8.25)) # A4 paper size
ax[0, 0].plot(self.realgrid.grid, sol.A_opt)
ax[0, 0].set_xlabel(r'$\omega$')
ax[0, 0].set_ylabel('spectrum')
ax[0, 1].plot(self.input_data.mats[mats_ind],
self.input_data.value.real[mats_ind],
color='red',
ls='None',
marker='.',
markersize=12,
alpha=0.33,
label='Re[selected data]')
ax[0, 1].plot(self.input_data.mats[mats_ind],
self.input_data.value.imag[mats_ind],
color='red',
ls='None',
marker='.',
markersize=12,
alpha=0.33,
label='Im[selected data]')
ax[0, 1].plot(self.input_data.mats,
self.input_data.value.real,
color='blue',
ls=':',
marker='x',
markersize=5,
label='Re[full data]')
ax[0, 1].plot(self.input_data.mats,
self.input_data.value.imag,
color='green',
ls=':',
marker='+',
markersize=5,
label='Im[full data]')
ax[1, 0].plot(self.input_data.mats[mats_ind],
self.input_data.value.real[mats_ind],
color='red',
ls='None',
marker='.',
markersize=12,
alpha=0.33,
label='Re[selected data]')
ax[1, 0].plot(self.input_data.mats[mats_ind],
self.input_data.value.imag[mats_ind],
color='red',
ls='None',
marker='.',
markersize=12,
alpha=0.33,
label='Im[selected data]')
# ax[1, 0].plot(self.input_data.mats, self.input_data.value.real,
# color='blue', ls=':', marker='x', markersize=5,
# label='Re[full data]')
# ax[1, 0].plot(self.input_data.mats, self.input_data.value.imag,
# color='green', ls=':', marker='+', markersize=5,
# label='Im[full data]')
ax[1, 0].plot(check_axis,
check.real,
ls='--',
color='gray',
label='Re[Pade interpolation]')
ax[1, 0].plot(check_axis,
check.imag,
color='gray',
label='Im[Pade interpolation]')
ax[1, 0].set_xlabel(r'$\nu_n$')
ax[1, 0].set_ylabel(self.input_data.data_type)
ax[1, 0].legend()
ax[1, 0].set_xlim(0., 1.05 * check_axis[-1])
# ax[1, 1].plot(self.input_data.mats, (self.input_data.value - sol[0].backtransform).real,
# ls='--', label='real part')
# ax[1, 1].plot(self.input_data.mats, (self.input_data.value - sol[0].backtransform).imag,
# label='imaginary part')
# ax[1, 1].set_xlabel(r'$\nu_n$')
# ax[1, 1].set_ylabel('data $-$ fit')
# ax[1, 1].legend()
plt.tight_layout()
plt.show()
def main_function(self):
"""Main function for the analytic continuation procedure.
This function is called when the "Do it" button is clicked.
It performs an analytical continuation for the present settings
and shows a plot.
"""
self.ana_cont_probl = cont.AnalyticContinuationProblem(
im_axis=self.input_data.mats,
im_data=self.input_data.value,
re_axis=self.realgrid.grid,
kernel_mode='freq_fermionic')
model = np.ones_like(self.realgrid.grid)
model /= np.trapz(model, self.realgrid.grid)
preblur, bw = self.get_preblur()
sol = self.ana_cont_probl.solve(method='maxent_svd',
optimizer='newton',
alpha_determination='chi2kink',
model=model,
stdev=self.input_data.error,
interactive=False,
alpha_start=1e14,
alpha_end=1e-3,
preblur=preblur,
blur_width=bw)
inp_str = 'atom {}, orb {}, spin {}, blur {}: '.format(
self.input_data.atom, self.input_data.orbital,
self.input_data.spin, bw)
all_chis = np.isfinite(np.array([s.chi2 for s in sol[1]]))
res_str = 'alpha_opt={:3.2f}, chi2(alpha_opt)={:3.2f}, min(chi2)={:3.2f}'.format(
sol[0].alpha, sol[0].chi2, np.amin(all_chis))
self.text_output.append(inp_str + res_str)
alphas = [s.alpha for s in sol[1]]
chis = [s.chi2 for s in sol[1]]
self.output_data.update(self.realgrid.grid, sol[0].A_opt,
self.input_data)
fig, ax = plt.subplots(ncols=2, nrows=2,
figsize=(11.75, 8.25)) # A4 paper size
ax[0, 0].loglog(alphas, chis, marker='s', color='black')
ax[0, 0].loglog(sol[0].alpha,
sol[0].chi2,
marker='*',
color='red',
markersize=15)
ax[0, 0].set_xlabel(r'$\alpha$')
ax[0, 0].set_ylabel(r'$\chi^2(\alpha)$')
ax[1, 0].plot(self.realgrid.grid, sol[0].A_opt)
ax[1, 0].set_xlabel(r'$\omega$')
ax[1, 0].set_ylabel('spectrum')
ax[0, 1].plot(self.input_data.mats,
self.input_data.value.real,
color='blue',
ls=':',
marker='x',
markersize=5,
label='Re[data]')
ax[0, 1].plot(self.input_data.mats,
self.input_data.value.imag,
color='green',
ls=':',
marker='+',
markersize=5,
label='Im[data]')
ax[0, 1].plot(self.input_data.mats,
sol[0].backtransform.real,
ls='--',
color='gray',
label='Re[fit]')
ax[0, 1].plot(self.input_data.mats,
sol[0].backtransform.imag,
color='gray',
label='Im[fit]')
ax[0, 1].set_xlabel(r'$\nu_n$')
ax[0, 1].set_ylabel(self.input_data.data_type)
ax[0, 1].legend()
ax[1, 1].plot(self.input_data.mats,
(self.input_data.value - sol[0].backtransform).real,
ls='--',
label='real part')
ax[1, 1].plot(self.input_data.mats,
(self.input_data.value - sol[0].backtransform).imag,
label='imaginary part')
ax[1, 1].set_xlabel(r'$\nu_n$')
ax[1, 1].set_ylabel('data $-$ fit')
ax[1, 1].legend()
plt.tight_layout()
plt.show()
gf_bos /= norm
fig,ax = plt.subplots(ncols=2, nrows=1, figsize=(12,4))
ax[0].plot(w_real, spec_real, label='spectrum')
ax[0].plot(w_real, w_real*spec_real, label='response function')
ax[0].legend()
ax[0].set_xlabel('real frequency')
ax[0].set_title('Spectrum')
ax[1].plot(iw, gf_bos.real, marker='+')
ax[1].set_xlabel('Matsubara frequency')
ax[1].set_title('Matsubara Greensfunction')
plt.show()
w = np.linspace(0., 5., num=501, endpoint=True)
probl = cont.AnalyticContinuationProblem(im_axis=iw, re_axis=w,
im_data=gf_bos, kernel_mode='freq_bosonic')
err = np.ones_like(iw) * noise_amplitude / norm
model = np.ones_like(w)
model /= np.trapz(model, w)
sol,_ = probl.solve(method='maxent_svd',
alpha_determination='chi2kink',
optimizer='newton',
stdev=err, model=model,
interactive=True)
np.save("A_opt_maxent_bosonic", sol.A_opt)
np.save("backtransform_maxent_bosonic", sol.backtransform)
np.save("chi2_maxent_bosonic", sol.chi2)
fig, ax = plt.subplots(ncols=3, nrows=1, figsize=(15,4))
kernel = 1. / (1j * iw[:, None] - w[None, :])
giw = np.trapz(kernel * spec[None, :], w, axis=1)
# add noise to the data
rng = np.random.RandomState(4713)
noise_ampl = 0.0001
giw += noise_ampl * (rng.randn(niw) + 1j * rng.randn(niw))
# error bars and default model for analytic continuation
err = np.ones_like(iw) * noise_ampl
# specify the analytic continuation problem
mats_ind = [0, 1, 2, 3, 4, 5, 6, 8,
9] # Matsubara indices of data points to use in Pade
probl = cont.AnalyticContinuationProblem(im_axis=iw[mats_ind],
re_axis=w,
im_data=giw[mats_ind],
kernel_mode='freq_fermionic')
# solve the problem
sol = probl.solve(method='pade')
check_axis = np.linspace(0., 1.25 * iw[mats_ind[-1]], num=500)
check = probl.solver.check(im_axis_fine=check_axis)
np.save("A_opt_pade", sol.A_opt)
np.save("check_pade", check)
# plot the results
fig, ax = plt.subplots(ncols=2, nrows=1, figsize=(12, 5))
ax[0].plot(w, spec, color='gray', label='real spectrum')
ax[0].plot(w, sol.A_opt, color='red', label='analytic continuation')
ax[0].set_xlabel('real frequency')