Discrepency qcinv and actual solution weiner filtering

[Gabriel-Ducrocq]

Hi,

I have been trying to run qcinv and compare its output to the true solution when using a noise covariance matrix porportional to the identity matrix and no skymap - in this case the matrix A.T N^-1 A + C^-1 is diagonal.

Using a stopping criterion of an error lower than 1e-6, I am finding unexpected discrepencies between the solution found by qcinv and the one computed directly. Here you will find a minimal code:

import numpy as np
import healpy as hp
import qcinv
import utils
import matplotlib.pyplot as plt
from classy import Class
cosmo = Class()

## Setting params
LENSING = 'yes'
OUTPUT_CLASS = 'tCl pCl lCl'
observations = None
COSMO_PARAMS_NAMES = ["n_s", "omega_b", "omega_cdm", "100*theta_s", "ln10^{10}A_s", "tau_reio"]

nside=512
lmax=2*nside
npix = 12*nside**2

def generate_cls(theta, pol = False):
    params = {'output': OUTPUT_CLASS,
              "modes":"s,t",
              "r":0.001,
              'l_max_scalars': lmax,
              'lensing': LENSING}
    d = {name:val for name, val in zip(COSMO_PARAMS_NAMES, theta)}
    params.update(d)
    cosmo.set(params)
    cosmo.compute()
    cls = cosmo.lensed_cl(lmax)
    # 10^12 parce que les cls sont exprimés en kelvin carré, du coup ça donne une stdd en 10^6
    cls_tt = cls["tt"]*2.7255e6**2
    if not pol:
        cosmo.struct_cleanup()
        cosmo.empty()
        return cls_tt
    else:
        cls_ee = cls["ee"]*2.7255e6**2
        cls_bb = cls["bb"]*2.7255e6**2
        cls_te = cls["te"]*2.7255e6**2
        cosmo.struct_cleanup()
        cosmo.empty()
        return cls_tt, cls_ee, cls_bb, cls_te

COSMO_PARAMS_MEAN_PRIOR = np.array([0.9665, 0.02242, 0.11933, 1.04101, 3.047, 0.0561])
noise_covar_temp = 0.1**2*np.ones(npix)
inv_noise = (1/noise_covar_temp)

beam_fwhm = 0.35
fwhm_radians = (np.pi / 180) * 0.35
bl_gauss = hp.gauss_beam(fwhm=fwhm_radians, lmax=lmax)

### Generating skymap

theta_ = COSMO_PARAMS_MEAN_PRIOR
cls_ = generate_cls(theta_, False)
map_true = hp.synfast(cls_, nside=nside, lmax=lmax, fwhm=beam_fwhm, new=True)
d = map_true
d += np.random.normal(scale=np.sqrt(noise_covar_temp))

#### Setting solver's params
class cl(object):
    pass

s_cls = cl
s_cls.cltt = cls_

n_inv_filt = qcinv.opfilt_tt.alm_filter_ninv(inv_noise, bl_gauss)
chain_descr = [
    [0, ["diag_cl"], lmax, nside, np.inf, 1.0e-6, qcinv.cd_solve.tr_cg, qcinv.cd_solve.cache_mem()]]

chain = qcinv.multigrid.multigrid_chain(qcinv.opfilt_tt, chain_descr, s_cls, n_inv_filt,
                                        debug_log_prefix=('log_'))

soltn_complex = np.zeros(int(qcinv.util_alm.lmax2nlm(lmax)), dtype=np.complex)

#### Solving
chain.solve(soltn_complex,d)
hp.almxfl(soltn_complex, cls_, inplace=True)
weiner_map_pcg = soltn_complex

##### Computing the exact weiner map:

## comouting b part of the system
b_weiner = hp.almxfl(hp.map2alm(inv_noise * d, lmax=lmax)*(npix/(4*np.pi)), bl_gauss)

inv_cls = np.array([cl if cl != 0 else 0 for cl in cls_])

Sigma = 1 / (inv_cls + inv_noise[0] * (npix / (4 * np.pi)) * bl_gauss ** 2)

##### Solving:
weiner_map_diag = hp.almxfl(b_weiner, Sigma)

##Graphics

pix_map_pcg = hp.alm2map(weiner_map_pcg, lmax=lmax, nside=nside)
pix_map_diag = hp.alm2map(weiner_map_diag, lmax=lmax, nside=nside)

rel_error_pix = np.abs((pix_map_pcg-pix_map_diag)/pix_map_diag)
hp.mollview(rel_error_pix)
plt.show()

plt.boxplot(rel_error_pix, showfliers=True)
plt.show()
plt.boxplot(rel_error_pix, showfliers=False)
plt.show()

Note that with such a choice of beam, noise and resolution, the errors are acceptables on most of the pixels. Here are plots of the map and boxplot (with and without outliers) of relative differences:

Note that keeping all values equal but reducing the resolution increases the errors. For example, with nside = 32, we get the following graphics:

Now the fractional errors are about 1%, which is pretty big... Keeping a high resolution but increasing the noise level has the same effect.

Have you encountered such a behavior ? Am I missing something and using qcinv the wrong way ?

Thank you, Gabriel.

[carronj]

Hi Gabriel,

The matrix A.T N^-1 A matrix is never exactly diagonal (the pixels have finite size and the spherical harmonics are not exactly orthogonal on the pixelized sphere). One does expect some difference that goes away increasing the resolution.

Julien

[Gabriel-Ducrocq]

Hi Julien,

Thank you for replying quickly ! Okay that makes sense. The same behavior happens when keeping the same resolution with a greater noise. For example, keeping NSIDE = 512 but taking a noise level 10 times greater, we get:

Now most of the errors are of the order of few percents. Do you have an idea why increasing the noise level increases the errors ?

Intuitively, I would say that increasing the noise also increases the conditionning number of the matrix A^T N^-1 A + C^-1. However, even if this matrix is not perfectly diagonal, the diagonal preconditioner should be good enough to cancel problems due to bad conditioning, don't you think ?

Thank you, Gabriel.