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\begin{document}

\centerline{\bf{\large{Scientific/Technical/Management Section }}}

\section{Motivation, Objectives, and Intellectual Merit}



Cosmic microwave background (CMB) polarization measurements offer
a unique possibility to reconstruct the physical conditions in the
very early universe, and to test the application of fundamental physics
laws toward Planck energy scales ($10^{19}$ GeV) that are far beyond
the reach of any laboratory tests or particle colliders like the Large
Hadron Collider (LHC) at CERN with targeted energy scales around 14 TeV.
In particular, the detection of a primary $B$-mode (curl-type) polarization
and the determination of its source
are major challenging tasks for the
CMB community; see Ref.~\cite{Poletti:2016xhi} for a recent study.

Primary gravitational wave (GW) detection through
$B$-mode polarization \cite{Kamionkowski:1996zd,
Kamionkowski:1996ks,Seljak:1996gy,Zaldarriaga:1996xe} would be a
direct proof of the amplification of quantum-mechanical fluctuations
\cite{Grishchuk:1974ny} during a very early inflationary
\cite{Guth:1980zm,Linde:1981mu} epoch
 and therefore open a window on physics
at ultra-high energy scales and the very earliest moments of the universe.
In 2014, the BICEP2 collaboration announced a
possible detection of relic GWs through the
$B$-polarization signal~\cite{Ade:2014xna}, but further
investigation showed that this signal can be fully explained
by galactic dust rather than being of cosmological origin
\cite{Mortonson:2014bja,Flauger:2014qra,Tassis:2014bea,Ade:2015tva}.
The presence of foreground dust emission does not exclude the presence
of a cosmological $B$-polarization signal, but, given the presence of $B$-modes from dust
and galactic synchrotron emission,
an unambiguous detection of a primordial signal will require  multi-frequency
measurements and sophisticated modeling of foreground polarization
signals;\footnote{
\small The $B$-polarization signal from thermal dust and synchrotron emission dominate over the primordial signal at high and
low frequencies, respectively \cite{Kandel:2017xjx}}
see e.g.\ \cite{Lizarraga:2014waa,Aghanim:2016cps,Ghosh:2016jsx,
Vansyngel:2016fbn,
Stompor:2016hhw,Akrami:2018wkt,Akrami:2018mcd,
Ade:2018gkx,vonHausegger:2018tjq}.


Even in the absence of foregrounds, extracting the $B$-mode signal from
a cut sky presents formidable challenges, especially in the current case when both the intensity (Stokes $I$) and the $E$-modes are significantly
larger than the $B$-modes.
As such, this proposal is focused on two main objectives:
\begin{itemize}\itemsep-2pt
\item Understanding the nature of the dust polarization signal
and properly account for synchrotron emission
by
applying both state-of-the-art simulations and careful comparisons with
data products.
\item Creating new techniques for extracting the faint $B$-mode signal
in the realistic case where it competes with larger signals and noise.
\end{itemize}

One of the most intriguing features of {\it Planck}'s dust-polarization
measurements is that the $E$-mode polarization exceeds the
$B$-mode by almost a factor of two \cite{Adam:2014bub}.
This was in contrast to the theoretical expectations that dust
contributions to the $E$- and $B$- modes should be similar. The {\it Planck}
collaboration suggested that the spectral $E$- and $B$- mode
asymmetry, as well as the positive value of the temperature ($T$)
cross correlation with $E$-polarization ($TE$), might be due to the alignment
of the cold medium filamentary structure with the magnetic field
 \cite{Ade:2015mbc}.
This possibility has been quantified theoretically in
Ref.~\cite{Caldwell:2016xkd}.
However, the relevant mechanism that might produce this alignment is unclear. Further,
analyzes of the latest version of the {\it Planck} data
suggest that the hypothesis of the alignment between density and magnetic
field structures cannot explain the $TE$ cross correlation
at low multipoles $\ell$ ( $< 40$ ) which characterize the dust polarization
on angular scales $\sim10^{\circ}$, larger than those typical
of the interstellar filaments (although it remains plausible for
$\ell > 40$; see Ref.~\cite{clark2014}). Moreover, recent analysis of the {\it Planck} data also shows
a non-vanishing cross correlation between temperature and $B$-polarization ($TB$)  (\cite{Akrami:2018wkt}).
This is an important, yet unexplored, result, which
suggests the presence of parity-violating processes in the interstellar medium (ISM).

There are therefore a number of mysteries surrounding the {\em Planck} dust maps, and we propose to address these
with a combination of enhanced analysis techniques and simulations.
Efforts to explain these puzzles have relied on studying the interaction of magnetohydrodynamic (MHD)
turbulence with dust grains.
The standard model for calculating the dust polarization foreground is based on the effect of interstellar
 turbulence. In this case, dust grains align perpendicular to the magnetic field lines
(see \cite{Andersson:2015} and references therein).
However, a recent study has pointed out that the observed ratio between $E$- and
$B$-mode polarization spectra together with a positive $TE$ cross correlation
cannot normally be achieved within simple MHD turbulence models
\cite{Caldwell:2016xkd}.
Therefore, more complex MHD turbulence models are called for. Interstellar turbulence can be magnetically
dominated and strongly anisotropic toward small scales
\cite{Goldreich:1995gs,Cho:2000cv,Maron:2001mg}.
In that case, compressible MHD turbulence can in principle lead to the
observed ratio between $E$- and $B$-mode spectra ($EE/BB$) \cite{Kandel:2017klp}.
In addition,  it has been shown that interstellar turbulence can be
sub-Alfv\'enic at high galactic latitudes, and density and magnetic
field perturbations are weakly correlated \cite{Kandel:2017xjx}.
On the other hand, the $EE/BB$ ratio exhibits an almost scale-invariant nature.
Hence, a successful turbulent model of galactic dust polarization maps
should account for weak scale-invariance of the turbulence anisotropy.
Another major obstacle to detect primary B-polarization mode is the
galactic synchrotron emission \cite{Krachmalnicoff:2018imw}.


Previously we have performed numerical simulations of compressible hydrodynamic
and MHD turbulence in the early universe (in post-inflationary
epochs, during cosmological phases transitions, until recombination)
using the {\sc Pencil Code}, adopting various initial conditions
\cite{Kahniashvili:2010gp,Kahniashvili:2012vt,
Tevzadze:2012kk,Kahniashvili:2012uj,Brandenburg:2014mwa,
Brandenburg:2018ptt,Brandenburg:2017neh,
Brandenburg:2017rnt,Brandenburg:2017rcb,Kahniashvili:2016bkp,
Brandenburg:2016odr,Kahniashvili:2015msa}.
We will utilize and develop the "interstellar" module of the
{\sc Pencil Code}, building upon our earlier work \cite{BKM07,KBSTN99}
and that of others in the {\sc Pencil Code} community \cite{Gent13a,Gent13b}.
Recently we studied
nonhelical turbulence and suggested that a strong non-vanishing  skewness of the $E$-polarization is
responsible for an enhanced ratio $EE/BB$  in both helical and nonhelical cases
\cite{Bracco:2018vyi,Brandenburg:2018khp}.

Understanding the properties of dust polarization is interesting in its own right, but also can serve as a tool for better
component separation,
see \cite{Liu:2018oqp} for a recent study and references therein.
Even for all-sky data coverage, the extraction of $E$-
and $B$-mode dust polarization maps is a complex task:
handling the contribution to the
spectra of point sources or very bright objects,
such as the galactic
disk, is not trivial.
The difficulty for statistical and machine learning separation methods is that there is a trade-off between a statistical model that describes point sources and the disk, which is necessarily complex, and being able to implement the algorithm at the scale of Planck data.  Hence scalability and modelling are key challenges for statistical and machine learning approaches to source separation.  The usual solution is to simply apply a mask to these sources and so be able to use a practical model.
Thus, masks are usually applied to the full-sky
data and $E$-to-$B$ leakage may occur. This is why mask apodization
is a key step in computing polarization power spectra
both in small sky patches and in full sky maps. In practice the
apodized masks are tested with simulated sky maps in order to
validate the standard procedure \cite{BenoitLevy:2013bc}.
In the case of finite cut-sky CMB maps, the extraction of the
pure  $E$- and pure $B$-mode signals is even more
complex  \cite{Bunn:2002df,Baccigalupi:2002bh}, and different methods have been proposed
\cite{Larson:2006ds,Zhao:2010sza,Bunn:2016lxi,
Rogers:2016rdb,Remazeilles:2017zlb}.
In our program we propose to use machine
learning (ML) methods to extract the
pure $E$ and $B$ modes being motivated to see
if ML can improve component separation.

Understanding the detailed properties of foreground dust and synchrotron emission
polarization
is important to be able to separate dust from the CMB, which
has different statistical properties.
Assuming that the galactic foreground with its highly non-gaussian
statistics is strongest at low galactic latitudes,
we can use advanced ML tools to separate it
from the CMB with its gaussian statistical properties.
Thus, to facilitate this approach, a detailed understanding
of the properties of dust polarization is essential.

As another example of the importance of understanding the properties of dust,
usually parity-odd cross correlations $TB$ and $EB$ are
{\it a priori} assumed to be zero and used to calibrate detectors \cite{Kaufman:2013vbd,Koopman:2016tsg}.
We aim to investigate if this calibration can be justified when accounting
for dust polarization.


{\em The proposed research is multidisciplinary, involving data analysis and
ML, physics of the ISM and MHD turbulence, and the CMB.
The obtained results will improve ISM turbulence modeling, CMB sky
component separation techniques, and the results will advance our
understanding of galactic microwave polarization and potentially will
lead to critical developments in extracting a primary $B$-mode signal.}

Below we summarize the major goals of our proposed research program.

\section{Brief Summary of the Proposed Research Program}

We propose to study the following issues:

\begin{itemize}\itemsep-2pt
\item
Interstellar dust is a key component of the ISM to
probe the interstellar gas mass, \cite{Hildebrand:1983ki}. This is important
both for astrophysical processes in the local Universe, namely for
star formation activity in the Galaxy, and for accurately probing the
cosmic times, as the ISM is an unavoidable filter for primordial
lights coming to us. In the context of studying primordial
GWs and their imprint on the CMB, dust polarized
emission is today becoming a main focus. The amount of polarization
coming from interstellar dust however is still poorly
constrained. Dust polarization depends (i) on intrinsic dust
properties, such as dust chemical composition, shape, and emissivity;
(ii) on the statistical alignment of the dust grains with the local interstellar
magnetic field; and (iii) on the morphology of the galactic magnetic field.
Since the net polarization is the combination of all such
effects, modeling dust polarization is a complex issue with a high
degree of degeneration. Nevertheless, recent observational data will
allow us to further explore dust properties in total  and polarized
emissions and to test the latest state-of-the-art models \cite{Guillet2018,Jones2013}

\item
Because of the degenerate problem described above,
we propose to tackle the dust polarization modeling from two
different perspectives.
First of all, we will focus on investigating the role of interstellar
turbulence, and its impact on the density and magnetic field structures,
and, in turn, their effect on the observed polarization.
Assuming uniform intrinsic dust properties in the ISM
\cite{Ade:2014hna}, we propose to perform large eddy and direct
numerical simulations of MHD turbulence using a
variety of physical conditions, including supernova-driven turbulence
in the interstellar medium as well as artificially forced turbulence
and free decay cases. We consider isotropic and anisotropic turbulence, and compare theoretical
polarization spectra with {\it Planck} dust polarization ones.
Second, we will attempt to add dust properties using the {\sc Polaris} code \cite{polaris},
to disentangle
the effects related to the magnetic field structure from those of dust
grains on the observed polarization.

\item
Given the knowledge acquired through the previous goals, we propose to
elaborate a recipe to properly remove the dust and synchrotron foreground in order to
study new physics related to the CMB $B$-mode polarization.
We use advanced ML algorithms to separate contributions from CMB and
dust. Our approach is based on their systematically different statistical properties.
We also propose to use an ML approach to decompose CMB polarization maps
in order to extract pure $E$- and pure $B$- mode signals.
As explained further below, we will also apply these methods to other
data sets where we know the answer.
This will allow us to test the ML technique, and it enables us to see if
the separation can be improved.
As described below, our ML approach based on an Independent Component
Analysis (ICA) method developed by our group is superior to previous ML
approaches used in this context.

\item
An important application of our work is to determine the influence
of magnetic and kinetic helicity on the polarization signature, and to
find out if helicity can be detected in parity-odd cross correlations
between $E$- and $B$-mode polarization.
We will therefore use advanced tools to analyze the cross correlation between $E$- and $B$-modes ($EB$)
and compare with models and other observational data such as from the solar
surface \cite{Brandenburg:2018khp}, where we can measure magnetic helicity also directly from the
inferred magnetic field \cite{ZBS14,ZBS16,BPS17}.
To isolate different physical effects, we apply the tool of analyzing
$E$- and $B$-mode polarizations in other circumstances such as scattering
emission of polarized light from the magnetized solar surface.
These applications to solar data have the advantage that we have
independent means of determining the underlying magnetic field,
allowing us to validate our approach.
This argument also applies to simulation data discussed below.

\item
One of our approaches to source separation uses a Bayesian algorithm with an emphasis on providing reasonable and principled measures of uncertainty around the estimates of the polarization sources such as dust and CMB. This gives important information on how secure we can be as regards our theory for the causes of the polarization, based on the data that we currently have, and can provide input into what new missions should target.  The algorithm that first appeared in \cite{Wilson:2008wilson} will be used, modified from its initial use for temperature map separation and taking account of other known properties of the sources.
\end{itemize}

\section{The Relevance of the Proposed Work to NASA Objectives and Goals}
Some of the research will use {\it Planck} (and WMAP) data in novel  ways
and help to understand better the galactic microwave polarization and
to remove foreground effects when analyzing $B$-polarization data.
As a result, our investigation aims to extract more information about
and provides constraints on fundamental physics.
The research proposed here will also help to guide the development of
future CMB polarization missions.

The objectives and goals of our
proposal are directly related to the NASA Science Goal
``Discover how the universe works, explore how the universe began and evolved'', and objectives
``Understand the origin and destiny of the universe, and understand the many
phenomena and processes associated with galaxy, stellar, and planetary system
formation and evolution from the earliest epochs to today''. The science
questions addressed in this proposal are central questions of the Physics of the
Cosmos program.


\section{Methodology and Approach}
In this section we review previous studies to model galactic microwave
polarization (including ISM turbulence) and data analysis tools that we
aim to use.

\subsection{Dust: the Link between Magnetic Fields and Polarization}
Observations of dust polarized emission are today one of the most
efficient indirect probes of magnetic fields in the ISM.
Here we provide a brief description of the main mechanisms that link dust polarization to the magnetized ISM. We begin by summarizing the standard view for how the magnetic field impacts the polarization of radiation. It is observed that starlight is polarized, and the more extinction along the line of sight, the more polarized it is, with maximum polarization in the visual roughly equal to $p_{\rm max} = 0.03 A_V$, where $A_V$ is the extinction coefficient in V-band. The orientation of the polarization tends to follow the magnetic field of the Galaxy, so this suggests that dust is influenced by the magnetic field and in turn impacts the polarization of radiation.
Interstellar dust
permeates the ISM. Large dust grains (from $\mu$m to mm sizes) dominate the visible extinction and the far-infrared/sub-mm
domains of the galactic electromagnetic spectrum \cite{Draine:2008hu}. Since these particles are typically not
spherical, both starlight extinction and dust thermal emission consist
of anisotropic radiation that is linearly polarized.
Magnetic fields in the
ISM are responsible for grain alignment \cite{Fermi1953}.
Large grains in the ISM
spin around their axis of maximal inertia (their shortest axis) at
superthermal velocities due to torques produced by collisions with
atoms and molecules, ejection of $H_{2}$ molecules formed on their
surfaces, or radiation field (radiative torques, see Ref.~\cite{Lazarian2011}).
Since dust grains are generally charged particles, their rotation contributes to their total magnetic moment ${\bf M}$. Thus, in the
presence of an external magnetic field ${\bf B}$, a magnetic torque ${\bf M} \times {\bf B}$ is
exerted on the grains. If the angular momentum ${\bf J}$ of the grains
is not aligned with the field, the torque forces the grains to precess
around the field lines. Paramagnetic relaxation \cite{Davis1951}, or radiative torques \cite{Lazarian:2007fn}, yield
then a global alignment of the grains with the ambient magnetic
field by dissipating the kinetic energy associated with the precession
motion. The amount of observed polarization is therefore tightly
linked to the degree of alignment of the grains
with the field.
Given a certain degree of grain alignment, which is however still poorly constrained
by observations (\cite{Guillet2018} and references therein), one may
consider dust grains as compasses in the ISM, so that, by observing
dust polarized emission, it is possible to infer the orientation of the magnetic field in the ISM. In particular,
the observed polarization from dust is maximal when the magnetic field is
perpendicular to the line of sight, and minimal when the magnetic field is pointing
toward the observer. Hence, dust polarization represents a suitable tracer for
the orientation of the magnetic field projected on the plane of the
sky.


\subsection{Modeling ISM Turbulence}

The ISM is multiphased, consisting of hot, warm, and cold phases,
which are being segregated by heating through supernova driving and the
temperature-dependence of the cooling function.
Such simulations will be performed in the shearing box approximation
to incorporate the effects of galactic shear, which leads to a strong
toroidal component of the magnetic field, \cite{BKM07}.
They incorporate density stratification and the different statistics of
supernovae I and II, which have different rates and scale heights.
This was first done in our work of nearly 20 years ago \cite{KBSTN99}.
On smaller length scales, it is customary to ignore shear and density
stratification, and focus in highly supersonic turbulence that is just
driven by volume forcing \cite{KBSTN99,BKM07,Gent13a,Gent13b}.
This is also the case in the recent work of \cite{Kritsuk:2017aab}.

To make contact with observations, we integrate the intrinsic
polarization along the line of sight.
This has been done extensively also in other applications, including
also a number of cases studied in the {\sc Pencil Code} community
(for example see Refs.~\cite{Bhat2013,Brandenburg:2014zia}).

To characterize dust polarization, two are the main observables: the
polarization fraction $p$ and the polarization angle $\psi$. The former carries
information about the fraction of detected intensity that
is polarized compared to the total intensity, and it depends on dust
properties (composition and grain alignment) and on the geometry of the
field with respect to the observer. The polarization angle,
instead, allows us to infer the orientation of the magnetic field
projected on the plane of the sky.
In practice, polarization observations provide the three Stokes
parameters $I$, $Q$, and $U$ of linear polarization. While the Stokes $I$
parameter represents the total intensity of the incoming radiation,
and is defined as a positive quantity, Stokes $Q$ and $U$ include the geometry of the polarization plane and can assume both
positive and negative values. The two aforementioned observables are
$\psi={1\over2}\arctan{(U/Q)}$. Since sub-mm dust thermal emission is
optically thin, these quantities are representative of the whole line
of sight. For dust polarization one can write:
\begin{equation}
I=\int S_\nu\,e^{-\tau_\nu}\left[1-p_{\rm dust}R\left(\cos^2\!\gamma-\textstyle{\frac{2}{3}}\right)\right]\mathrm{d}\tau_\nu,\quad
Q+\ii U=\int p_{\rm dust}R\,S_\nu\,e^{-\tau_\nu+2\ii\psi}\cos^2\!\gamma\,\mathrm{d}\tau_\nu,
\end{equation}
where $S_\nu$ is the source function of dust emission (usually
considered as a modified black-body spectrum), $\tau_\nu$ is the dust
optical depth, $\gamma$ is the angle between the magnetic field and
the plane of the sky, $p_{\rm dust}$ is the intrinsic polarization fraction of dust
(corresponding to the ratio $C_{\rm pol}/C_{\rm avg}$, where $C_{\rm pol}$ and $C_{\rm avg}$ are the polarization
and the average cross-sections of dust, as defined in Ref.~\cite{Ade:2014hna}), and $R$ is the Rayleigh reduction factor, which
characterizes the degree of dust grain alignment with the local
magnetic field \cite{Lee1985,Hildebrand1988}.

\subsection{${\boldsymbol E}$ and ${\boldsymbol B}$-Modes Polarization Analysis}
\label{PolarizationAnalysis}

As detailed above, linear polarization is measured in terms of the two Stokes parameters
$Q$ and $U$ in the plane of the sky. For our purpose, however, in order to statistically characterize the
polarization sky and compare it with numerical simulations, it is
useful to introduce two rotational invariants, which can be directly
derived from the Stokes parameters.
In the simulations, on the other hand, we extract the magnetic field
components perpendicular to the plane of the sky, $b_x$ and $b_y$, say.
We then compute the linear polarization as complex polarization
$P=Q+\ii U=(b_y-\ii b_x)^2$, which assumes that the electric field
is $90^\circ$ rotated relative to the magnetic field.
In both observations and simulations, we compute the
E and B polarizations in Fourier space, denoted by tildes,
as functions of the wavevector $(k_x,k_y)$ as
$\tilde{E}+\ii\tilde{B}=(k_x-\ii k_y)^2\tilde{P}$.
This approach is valid for short enough separations in the sky.
Next, we compute spectra, averaged over concentric shells in
$\kk$ space, for example
$C_{ EB}(k)=\mbox{Re}\,\bra{\tilde{E}(\kk)\tilde{B}^\ast(\kk)}_{\phi}$.
As said in the beginning, the ratio $C_{EE}(k)/C_{BB}(k)$ is found
to be around $2$ in interstellar dust polarization and the correlation
$C_{ EB}(k)$ might reflect helicity.
We prefer to characterize these in terms of quantities that are between
$-1$ and $+1$, namely
\begin{equation}
c_S(k)\equiv[C_{EE}(k)-C_{BB}(k)]/[C_{EE}(k)+C_{BB}(k)],\quad
c_A(k)\equiv2C_{EB}(k)/[C_{EE}(k)+C_{BB}(k)].
\end{equation}
Thus, $C_{EE}(k)/C_{BB}(k)=2$ corresponds to $c_S=1/3$,
and $c_A(k)$ characterizes parity breaking.

In the solar context, there is a large-scale modulation across the solar disk that affects
the sign of the helicity between the northern and southern hemispheres \cite{BPS17},
or, in the context of cross helicity, over the scale of sunspot pairs \cite{Singh_etal18}.
or over the scale of sunspot pairs.
In both cases, we are able to account for this variation characterized
by the wavevector $\KK$ using the two-scale method
\cite{BPS17,ZB18,Singh_etal18} by computing
$C_{ EB}(k;\KK)=-\mbox{Im}\,\bra{\tilde{E}(\kk+\KK/2)
\tilde{B}^\ast(\kk-\KK/2)}_{\phi}$.
We have already extended this method to measuring $c_A(k)$ by taking the
north/south antisymmetry into account; see below.

\subsection{Machine Learning -- Component Separation}\label{seq:MLCompSep}

The observations from the {\em Planck} dataset contain a convoluted
mixture of various sources including CMB, galactic synchrotron,
dust emissions, Sunyaev-Zeldovich effect in clusters, etc. The pixel
intensity distribution of the foreground source objects are expected
to be non-gaussian, while the CMB is expected to be close to gaussian.
In this task our goal is to separate the CMB component from the other
components. We are not the first trying to accomplish this goal; there
have been numerous attempts proposed to this problems with varying
success. Below we briefly summarize some of these methods, and propose
an alternate method that has the potential to achieve better results.

\paragraph{Internal Linear Combination.} One of the  simplest component separation method is the \emph{Internal Linear Combination (ILC)} \cite{Delabrouille2009}.
In this setting, for any available directions $\xi$ in the sky, $M$
channels of the {\em Planck} dataset at frequencies $\nu_1,\ldots\nu_M$
provide an $M$-dimensional observational data vector
$Y(\xi)=[y_1(\xi),\ldots,y_M(\xi)]^T\in\mathbb{R}^M$.
The ILC method assumes that the template of emission of the component of interest
is the same at all frequencies of observation and for any available directions $\xi$ in the sky, and for each frequency channel $i=1,\ldots, M$ the following equation holds:
$$
y_i(\xi)=s(\xi)+f_i(\xi)+n_i(\xi),
$$
where $f_i(\xi)$ and $n_i(\xi)$ are foregrounds and noise contributions respectively in
channel $i$. Our goal is to estimate the CMB component $s(\xi)$ from the observations $y_i(\xi)$. To accomplish this goal, ILC defines an estimator, $\hat{s}(\xi)=\sum_{i=1}^M w_iy_i(\xi)$,
and the weights $w_i$ are chosen such that $\sum_{i=1}^M w_i=1$ and the variance of the estimated component $\hat{s}(\xi)$ is minimal.


\paragraph{Linear Mixture Model.} The ILC approach makes very strong assumptions and hence the quality of the results it can provide is quite limited. A more general model for component separation is the \emph{Linear Mixture Model}. This model assumes that
for each frequency channel $i$ the observations are the results of a linear mixture of several sources $s_j$ \cite{2008ISTSP...2..735C}:
\begin{align}
y_i(\xi)=\sum_{j=1}^K A_{ij}s_j(\xi)+n_i(\xi), \quad (i=1,\ldots,M). \label{eq:LinModel}
\end{align}

Here each entry $s_j(\xi)$ contains a sky-pattern for the emission of a given component at sky location $\xi$. There have been numerous prior work on this problem \cite{2008ISTSP...2..735C,
2003MNRAS.346.1089D,2009AA...503..691B,2008AA...491..597L,Delabrouille2009,Baccigalupi00MNRAS,vansyngel:tel-01145843}.
Some methods assume that the mixing matrix $A\in\mathbb{R}^{M\times K}$
is known, and some try to estimate this matrix from the observations
$Y(\xi)$.
These methods usually are based on Wiener filtering, wavelet analysis,
or some kind of algorithms for solving the Independent Component Analysis
(ICA) problem. The ICA based methods also assume that the components
$s_1,s_2,\ldots, s_K$ are statistically independent.
They either use simulated data or older released versions of the
{\em Planck} data.

\paragraph{Bayesian ICA Model.} Given the large uncertainties that most likely will exist in the estimation of the $B$-mode and its interactions with other modes, and the availability of a large amount of \textit{a priori} information about it properties, a Bayesian learning approach has some attractions. From a model that originally appeared in \cite{Wilson:2008wilson}, a Bayesian source separation algorithm will be modified so that it can be applied to polarization data. This makes use of the linear mixture model, but with a set of options for modelling prior properties of the sources such as their distribution (Gaussian vs.\ non-Gaussian) and degree of spatial smoothness. For example, in \cite{Wilson:2012quiros}, dependencies between sources were modelled. Spatial smoothness can be modelled by an intrinsic Gaussian prior that constrains neighboring pixels in a source map to differ by a zero-mean Gaussian:
\[ s_j(\xi_k) - s_j(\xi_{k^{\prime}}) \: \sim \: N(0,\tau_j), \]
for 'neighboring' directions in the pixel discretization $\xi_k$ and $\xi_{k^{\prime}}$. The Bayesian approach's strength is in correctly propagating the uncertainties due to measurement error, and the uncertainty in other information that we have, to give a principled estimate of the uncertainty of the $B$-mode estimate given the information that we have, through the posterior distribution.  Implementation of the inference at the scale of Planck data is certainly an issue and to this end a method based on the INLA approach \cite{Wilson:2009rue} has been used.



\section{Proposed Research Program}

\subsection{Dust Polarization and ISM Turbulence Modeling}

Our aim is to statistically compare observational data with
synthetic data produced with numerical simulations (see next
sections). Thus, the data analysis will consist in applying
the same methodology to the two datasets.
With respect to the observational data, we will make use of the
latest full-sky observations made with the {\it Planck} satellite.
The {\it Planck} data consist of maps
in intensity and polarization between
$30$ GHz and $353$ GHz \cite{Ade:2013sjv}, allowing
us to constrain the CMB polarization after accounting for
galactic synchrotron and dust and synchrotron foreground emissions.
We will mainly focus on the latter two, which is today the
least characterized foreground signal
\cite{Kandel:2017xjx,Krachmalnicoff:2018imw}.
In order to compare observations and simulations, we will develop ML algorithms
(see next section) and use standard approaches
with statistical tools already in hands, such as power
spectra both in intensity and in polarization (i.e., Ref. \cite{Adam:2014bub}). Since we have access to the whole
sky, we will be able to perform both statistics on the
sphere and on cut-out images of the full sky,
allowing us to study angular scales larger than a few
degrees, which are those interesting for the detected
$E$-$B$ modes signals in the {\it Planck} data
(i.e., see Figs.~2--5 of Ref.~\cite{Akrami:2018wkt}).
Before proceeding with the comparison between
observations and numerical simulations, both at
the power spectrum level and at the map level, we will
test our algorithms using toy models of the polarized sky
\cite{Vansyngel:2016fbn}, in order to validate
our methodology without the complexity of the real data.

\begin{figure*}[t!]\begin{center}
\includegraphics[width=.76\textwidth]{peb4.ps}
\end{center}\caption[]{\small{
$E$-mode (left) and $B$-mode (right) polarization for isotropic
fully helical magnetohydrodynamic turbulence using data from
\cite{Brandenburg:2016odr} (their Fig.~4d--f).
Dark (light) shades indicate negative (positive) velocity.
The right panel shows that $c_S(k)$ exceeds 1/3, so the ratio of $EE$ to $BB$
exceeds the factor of 2 as $k$ increases.
Adapted from Ref.~\cite{Brandenburg:2018khp}.}
}\label{peb4}\end{figure*}

\subsubsection{Turbulence Simulations}

Astrophysical turbulence comes in a multitude of different forms:
it can be helical or nonhelical, it can be magnetically dominated or
subdominant, it can possess cross helicity, with a systematic alignment
or anti-alignment between magnetic field and velocity.
These turbulence simulations provide a means of performing experiments in
a variety of circumstances and environments.
This allows us to isolate specific physical effects or even just the
mathematical terms in the underlying equations and associate them with
distinct features in the synthesized diagnostics which, in this case,
are correlations in the $E$- and $B$-modes; see \Fig{peb4}.
Of particular interest for the present proposal is our ability to
consider turbulence that can be strongly anisotropic, with multiple
preferred directions, and lacking statistically mirror symmetry, i.e.,
being helical.

In the context of early universe turbulence, we have studied decaying
MHD turbulence which is magnetically dominated, i.e., the
magnetic energy exceeds the kinetic energy by typically a factor of ten
\cite{Brandenburg:2017rnt}.
The turbulence is then mostly driven by the Lorentz force.
The resulting $E$- and $B$-mode polarizations are shown in \Fig{peb4}
for a particular example.
We find that $c_S(k)>1/3$ for large $k$, i.e., $C_{EE}/C_{BB}>2$;
see the right-most panel of \Fig{peb4}.
Our recent work \cite{Brandenburg:2018khp} now suggests that for both
dust and synchrotron emission, the probability density function of the
$E$ polarization is strongly skewed, which contributes to an enhanced
$C_{EE}/C_{BB}$ ratio.
This is comparable with the observed foreground polarization detected
with {\em Planck} \cite{Adam:2014bub,Akrami:2018wkt}, clearly more
than what is theoretically expect \cite{Caldwell:2016xkd} and may
be explicable with Alfv\'en waves \citep{Kandel:2017xjx}.

\begin{figure*}[t!]\begin{center}
\includegraphics[width=.98\textwidth]{pslice}
\end{center}\caption[]{\small{
Magnetic field vectors and line-of-sight component (color-coded; left)
as well as $E$-mode (middle) and $B$-mode polarization (red)
in rotating convection viewed from the top onto the convecting layer.
Adapted from Ref.~\cite{Brandenburg:2018khp}.}
}\label{pslice}\end{figure*}

\begin{figure*}[t!]\begin{center}
\includegraphics[width=.98\textwidth]{pvarEBmodes_comp_hor}
\end{center}\caption[]{\small{
The functions $c_A(k)$ and $c_S(k)$ for rotating convection
against $k$ for two runs A and B.
Red (blue) lines and symbols are for north (south), demonstrating
opposite signs of $c_A(k)$, but the same sign of $c_S(k)$.
Adapted from Ref.~\cite{Brandenburg:2018khp}.
}}\label{pspec_conv}\end{figure*}

It is interesting to note that, even though the turbulence is
nearly fully helical with a fractional helicity of about 98\%,
the $EB$ correlation, $C_{ EB}(k)$, is actually zero.
In hindsight, this is not surprising because the parity-odd
polarization cross correlation $EB$
characterizes the shape of two-dimensional structures on a surface,
e.g., it can distinguish between the letters $p$ and $q$, which are mirror
images of one another.
In three dimensions, however we can flip any structure and view it from
the backside.

There can be certain physical circumstance, however, that give preference
of one over the other side of a plane.
The CMB may be one such example.
Rotating stratified convection is another quite intuitive example; see
\Fig{pslice} for the results of such a simulation.
There one sees cyclonic convection in the northern hemisphere viewed from
the top, so all converging inflows attain a counterclockwise swirl,
and so do all diverging outflows.
It is only when inspecting this pattern from beneath that we will
see a mirror image of the original pattern and therefore the opposite
sign of the $B$ polarization.
This can be seen in \Fig{pspec_conv}, where we plot $c_S(k)$, $c_A(k)$, and
$c_S$ versus $c_A$ for north (red) and south (blue) in a parametric representation.
There is now a systematic $EB$ correlation, so $c_A(k)$ is positive in the north
and negative in the south; see Refs.~{Brandenburg:2018khp,Bracco:2018vyi} for our recent work on this.
This is very promising and agrees with our intuition.
Surprisingly, however, $c_S(k)$ is now negative, so the $BB$ correlation
now exceeds the $EE$ correlation.
This can be explained by a strong rotational influence, which leads to a
strong parity-invariant signal and therefore to a large $BB$ correlation.
In addition, the magnetic field is here subdominant, in contrast to the
situation shown in \Fig{peb4}.


\subsubsection{Simulations of the Interstellar Medium}

Using shock diffusivities in the {\sc Pencil Code}, we can run high Mach
numbers of around ten, as was done in \cite{Gustafsson:2006},
or around 25, as done in \cite{Gent13a}.
Turbulence will be driven by supernova explosions, as done in \cite{Gent13a,Gent13b}.
We use a parameterized cooling function, as done in \cite{BKM07}.
We will simulate a domain of size $1\times1\times2$ kpc$^3$ over about one
Gyr, to accumulate sufficient statistics.

\subsection{ML Algorithms for Component Separation }\label{seq:MLAlgsForCompSep}

We will apply ICA to separate signals of different origins in sky maps. i)
Instead of the previously used ICA methods, we propose to  use more
robust ICA algorithms that are more tolerant to noise and outliers. ii)
The previously used ICA based methods assumed linear mixing of the
hidden components. We will investigate the application of nonlinear ICA
methods that are capable of dealing with nonlinear mixing instead of
linear. iii) The previously used component separation methods used older
versions of the Planck dataset. We will demonstrate the applicability
and effectiveness of these methods on the new {\em Planck} data.

\subsubsection{Robust ICA}\label{seq:robustICA}

Using vector and matrix notation, the above described Linear
Mixture Model of Eq.~(\ref{eq:LinModel}) can be formulated as $Y=AS+N$,
where $Y=[y_1,\ldots,y_M]\in \mathbb{R}^M$ is our observation, $A\in
\mathbb{R}^{M\times K}$ is the mixing matrix, $S\in \mathbb{R}^K$ denotes
the hidden components we want to estimate, and $N\in  \mathbb{R}^M$
is a noise vector. We assume that the coordinates of $S(\xi)$, that is
$s_1(\xi),\ldots, s_K(\xi)$ are statistically independent, and therefore
their mutual information is zero.

Most ICA algorithms seek for the solution in the form of $\hat{S} = WY$, where $W\in\mathbb{R}^{K\times M}$. Since according to our assumptions, the coordinates of $S(\xi)$ are statistically independent, a natural cost function is to minimize their mutual information (MI) of the coordinates of $\hat{S}$:
$\min_{W} I(\hat{s}_1,\ldots,\hat{s}_K)$. The quality of the solution thus depends on
how accurate is the corresponding MI estimation. The
problem with using MI is that without a parametric
assumption on the functional form of the joint distribution,
MI cannot be evaluated exactly, and numerical
estimation can be both inaccurate and computationally expensive.
Estimating the mutual information is usually quite challenging,
therefore many approximations have been proposed. The FastICA algorithm \cite{Hyvarinen99Fast,Hyvarinen01ICABook} for example, uses kurtosis or
other higher order moments to achieve this goal.


Higher order moment estimation, however, can be very sensitive to noise and outliers. As an extreme case, for example, one simple large outlier can completely ruin the statistics for mean, variance, or other higher-order moment estimation.
Instead, we will use ICA methods developed by our team that are more robust to noise and outliers \cite{Kirshner:2008:IIU:1390156.1390215}.

A key component our method is to use rank statistics and the copula of the multivariate distribution $S(\xi)\in\mathbb{R}^K$ to estimate the statistical dependence of its coordinates. For simplicity, we explain the main ideas with two statistically dependent one-dimensional random variables $X$ and $Y$, but the method can be naturally generalized to higher-dimensions. Let $(X, Y ) \in \mathbb{R}^2$ be distributed
according to a bivariate probability distribution $P$. Assume we are given $N$ samples of $(X, Y )$,
$D = \{(x_1, y_1), \ldots,(x_N , y_N )\}$. Let the rank $r_x (x)$ be
the number of $x_i$, $i = 1, \ldots, N$ such that $x > x_i$, and
let $r_y (y)$ be defined similarly.

Many nonlinear dependence measures are based on ranks. Among most commonly used are Kendall's $\tau$ and Spearman's $\rho$ rank correlation coefficients.
Both $\tau$ and $\rho$ have a range of $[-1, 1]$
and are equal to $0$ if $X$ and $Y$ are
independent. However, the converse is not true;
both $\tau$ and $\rho$ can be 0 even if $X$ and $Y$ are not independent. Therefore, while they are robust to outliers, neither $\rho$
nor $\tau$ is a good ICA objective function. Rank correlations can be extended from samples to distributions with the help of copulas, distributions over continuous multivariate ranks. Below we will propose
an effective robust contrast for ICA using a measure of dependence for copulas.

A bivariate copula $C$ is a probability function (cdf) defined on a unit
square, $C : [0,1]\times[0,1] \to [0,1]$ such that its univariate marginals
are uniform, i.e., $C (u, 1) = u$, $C (1, v) = v$, $\forall u, v, \in  [0,1]$.
Let $U = P_x (X)$ and $V = P_y (Y )$ denote the corresponding
cdfs for the previously defined random variables
$X$ and $Y$.
Then, for $(u, v) \in [0,1]\times[0,1]$, define $C$ as
$C (u, v) = P(P^{-1}_x(u), P^{-1}_y(v))$.
It is easy to verify that $C$ is a copula. Sklar's theorem \cite{Sklar59}
 states that such copula exists for any
distribution $P$, and that it is unique on the range of
values of the marginal distributions. Copulas can also be viewed as a canonical form of multivariate distributions as they preserve multivariate dependence properties of the corresponding families
of distribution. Therefore, if we want to measure the dependence between two random variables $X$ and $Y$, it is enough to measure the dependence between the marginal variables of their copula distributions.

As the true distribution $P$ and its copula $C$ are
not known, rank statistics can be used to estimate the copula with the so-called empirical copula \cite{Deheuvels79}. For a data set $\{(x_1, y_1), \ldots,(x_N , y_N )\}$,
an empirical copula $C_N$ is given by
$$
C_N\left(\frac{i}{N},\frac{j}{N}\right)=\frac{\# \textrm{ of } (x_k,y_k) \textrm{ s.t. } x_k\leq x_i \textrm{ and }y_k\leq y_j}{N}.
$$
As we can see from this expression, the copula distribution can be estimated with rank statistics only, and hence it is robust to outliers. Well-known sample versions of several nonlinear dependence measures, such as Kendall's $\tau$ and Spearman's
$\rho$,  can be obtained using the empirical copula (e.g., \cite{nelsen06}).
As we discussed, part of the problem with
Kendall's $\tau$ and Spearman's
$\rho$
as a dependence measure for ICA is a property that their
value may be 0 even though the corresponding variables
$X$ and $Y$ are not independent. Instead, we suggest
using Schweizer-Wolff $\sigma$, a measure of dependence
between two continuous random variables \cite{Schweizer81}:
$$
\sigma = 12\int_{[0,1]^2}
|C (u, v) - uv| dudv.$$
Here $\sigma$ can be viewed as an $L_1$ norm between a copula for
the distribution and a product of uniform independent variables. It has a range
of [0, 1], with an important property that $\sigma = 0$ if and
only if the corresponding variables are mutually independent. A sample version of $\sigma$ can be derived as:
\begin{align}\label{eq:SWICA}
\hat\sigma =
\frac{12}{N^2 - 1}
\sum_{i=1}^N
\sum_{j=1}^N \left|
C_N\left(\frac{i}{N},\frac{j}{N}\right) -
\frac{i}{N}\times \frac{j}{N}
\right|
\end{align}
To use this dependence measure in an ICA algorithm, we just simply have
to minimize the pairwise dependencies measured by the
Schweizer-Wolff dependence criterion \eqref{eq:SWICA}. We call this ICA method Schweizer-Wolff ICA (SWICA). We note that other measures of dependence can also be potentially used as an ICA contrast. We
will investigate the $L_\infty$ version of $\sigma$ too: $4 \sup_{[0,1]^2} |C (u, v)-uv|,$
a dependence measure similar
to the Kolmogorov-Smirnov univariate statistic (\cite{Schweizer81}).

As our preliminary results indicate (see below), these dependence measures can be used to derive new efficient ICA methods. During the dependence estimation, they only use rank statistics, and hence they are robust to outliers.

\subsubsection{Post-nonlinear ICA}\label{seq:nonlinearICA}

The previous methods in the Linear Mixture Model, $Y(\xi)=AS(\xi)+N(\xi)$, assumed that the observations $Y(\xi)$ are results of linear mixtures of statistically independent hidden components. Using nonlinear ICA methods developed by our group \cite{szabo07post} and others, we propose to investigate the more general case, when this mixing can be still unknown but nonlinear. In particular, efficient algorithms exist for the post-nonlinear mixing case when
$$
y_i(\xi)=f\sum_{j=1}^K (A_{ij}s_j(\xi))+n_i(\xi), \quad (i=1,\ldots,M).
$$
Here $f$ is an unknown monotone function, and $A$ is an unknown mixing matrix. Our goal is to estimate the unknown independent components ($s_j(\xi)$) from the observations ($y_i(\xi)$).

\subsubsection{${\boldsymbol E}$- and ${\boldsymbol B}$-Modes  Separation with Convolutional Neural Networks}\label{seq:convNN}

Recently, Isola et al. \cite{Isola17ICCV} proposed a new neural
network architecture based on Conditional Generative Adversarial Neural
Networks. This architecture is capable of learning a map from a wide
range of input images to output images. Hence, the name of this network
is Pix2Pix. As particular applications, it has been demonstrated that
Pix2Pix can be used to colorize grayscale images, remove background from
pictures, predict Aerial photos from their maps, or predict the original
images from their sketched versions. The algorithm assumes that we have
a set of input-output pair images and this training set can be
used to learn the map between the input and output images.

Our goal in this task is to investigate how well these Pix2Pix neural networks (or their modifications) can be used to learn a map between an image patch of the {\it Planck} data and its $E$ and $B$ components. We assume that a dataset of simulated images with their ground truth E and B components are available to us for training the neural networks. The only difference between our setting and Pix2Pix is that in our task the output consists of two images, the $E$ and $B$ components, instead of one single component. We are planning to use both real and simulated datasets in this task. Neural networks usually require large amounts of data, but as it has been reported by \cite{Isola17ICCV} in some application Pix2Pix can provide satisfactory results with using only 400 images. We will have access to much larger real datasets and we can also generate large amounts of simulated data that can be also used for training these neural networks. Recently we have successfully used Generative Adversarial Neural Networks for generating realistic looking fake galaxy images \cite{Lanusse17deep,Ravanbakhsh17AAAI}.

\subsubsection{Preliminary Results: Robustness of Component Separation}\label{seq:robustICAResults}
\begin{wrapfigure}{r}{.4\textwidth}\begin{center}
\vspace{-2em}
\includegraphics[width=0.4\textwidth]{SWICA.eps}
\vspace{-3em}
\end{center}\caption[]{\small
Box plot of Amari errors (multiplied by 100) for
the mixed sounds with outliers. Plot was computed over 100 experiments.
\vspace{-1em}
}\label{fig:swica1}\end{wrapfigure}



Here on a synthetic example we demonstrate the robustness of our component separation method to outliers.
We mixed $N = 1000$ samples from 8 sound pieces of
an ICA benchmark \cite{aalto}
mixing matrix. Then we added 20 outliers to this mixture and demixed them using SWICA and 4 state-of-the-art ICA algorithms, \cite{Hyvarinen99Fast,elm03JMLR,bach02JMLR,Cardoso99JADE}.
Figure~\ref{fig:swica1} shows that SWICA outperforms its competitor methods
on this task. The separation error is measured by the Amari-error \cite{Hyvarinen01ICABook}. The closer this number is to zero, the better the separation is.


As a second preliminary experiment to demonstrate the robustness of SWICA, we mixed 4 natural images of size $128 \times 256$. The pixel intensities we normalized in the [0, 255] interval.
Each image was considered as a realization of a stochastic variable with 32768 sample points. We mixed these 4 images
by a $4\times 4$ random mixing matrix, resulting
in a matrix of size $4 \times 32768$. Then we added
large outliers to 3\% randomly selected
points of this mixture, and then estimated
the original components. Figure~\ref{fig:swica2} shows that SWICA outperforms its competitor methods
on this task. In this project we will use SWICA for separating the hidden components of {\it Planck} data.

\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{SWICA_mixed_pictures.eps}
\end{center}\caption[]{\small{
Separation of outlier-corrupted mixed images. (a) The original images. (b) the mixed images corrupted with
outliers. (c)-(e) The separated images using SWICA, FastICA, and RADICAL algorithms, respectively. The Amari error
of the SWICA, FastICA, Radical was 0.10, 0.30, 0.29 respectively. The quality of the KernelICA and JADE was similar
to that of FastICA and RADICAL.
}
}\label{fig:swica2}\end{figure*}


\section{General Plan of Work}

As highlighted above, our research program consists of two major parts: modeling of dust polarization based on MHD simulations and CMB component separation.

%\subsection{Proposal Team}

The proposed research requires a range of technical expertise
including aspects of CMB and ISM physics, dust polarization and
MHD turbulence modeling, data analysis, and ML.
The proposers have complementary research expertise and have done
fundamental work in all of the relevant areas of this proposal.
In particular,
\begin{itemize}
\item
{\bf Brandenburg} has led the development of the {\sc Pencil Code},
a public domain tool box (\url{https://github.com/pencil-code})
that will also be used in the present work.
Drawing from his experience with supernova-driven turbulence in the
interstellar medium \citep{KBSTN99}, he will perform new simulations of
interstellar turbulence.
Having previously analyzed synchrotron emission from helical magnetic
fields \cite{BS14}, he will use this expertise to compute the dust
polarization signal from those simulations.
He has analyzed data from NASA's {\em Solar Dynamics Observatory}
\cite{BPS17} and the {\em SOLIS} instrument \cite{Singh_etal18}
and will utilize this experience to analyze the $E$- and $B$-mode
polarization of solar data.

\item
{\bf Dodelson}
has broad experience analyzing data from the South Pole Telescope~\cite{Baxter:2017ixz,Manzotti:2017net,Soergel:2016mce,
Baxter:2016ziy,Kirk:2015dpw,Giannantonio:2015ahz,Baxter:2014frs}. Accounting for foregrounds has been a major effort in much of that work. Further, he and a student considered the impact of multiple dust clouds along the line of sight~\cite{Poh:2016rfa}, and this complexity is something we propose to add into our analysis. He will oversee all aspects of the proposal, coordinating the simulations with the map-making and spectral analysis, as well as implementing the ML algorithms for component separation.

\item
{\bf Kahniashvili} has an expertise in studying non-standard
physics (primordial magnetic fields, Lorentz and parity symmetry
violations, etc.), imprints on CMB anisotropies, including the
off-diagonal correlations \cite{Durrer:1998ya,Kahniashvili:2008sh}
and the parity-odd cross correlations $TB$ and $EB$
\cite{Kahniashvili:2005xe,Kahniashvili:2014dfa,
Caprini:2003vc,Kahniashvili:2016bkp}. These works were considered by the {\it Planck} collaboration as methodology when constraining primordial magnetic fields
\cite{Ade:2015cva}. In the framework of the proposed research,
Kahniashvili will work on further extensions of the ISM
module of the {\sc Pencil code} and she
will lead the effort to interpret the simulated MHD turbulence data.
\item
{\bf Poczos} has been working on developing robust methods for independent component analysis and its generalizations.
He studied the cases when the components are noisy, corrupted by outliers
\cite{Kirshner:2008:IIU:1390156.1390215},
nonlinearly mixed
\cite{szabo07post},
or mixed by convolutions
\cite{szabo07JMLR}.
He also introduced new methods for the setting when not all the components are independent from each others
\cite{poczos05ICML}.
In multi-dimensional ICA and Independent Subspace Analysis it is assumed that the components can be organized into groups, the groups are independent from each other, but the components in the groups can be statistically dependent. These ICA based methods will be used to separate CMB from dust and other components as described in Sections~\ref{seq:MLCompSep}, and
\ref{seq:MLAlgsForCompSep}. Poczos also has a long-term expertise in developing convolutional and adversarial neural networks \cite{Li17ICCV,Li17NIPS} and he has successfully applied them in other cosmology applications \cite{Lanusse17deep,Ravanbakhsh17AAAI,Lanusse17DeepLens,Ravanbakhsh16ICML}. In this project conditional adversarial neural networks that will be used for $E$ and $B$ mode separation (Section~\ref{seq:convNN}).

\item
{\bf Wilson} has developed a scalable Bayesian source separation algorithm that follows the Linear Mixture Model approach \cite{Wilson:2012quiros,Wilson:2008wilson} and also allows for (a) incorporation of prior information about the mixing matrix coefficients and properties of the sources such as their spatial smoothness (b) provides estimates of the uncertainty in the estimated values of CMB as well as the linear mixture model parameters and (c) makes use of an approximation that allows for parallelization of the algorithm so that it can scale to Planck-sized data (following work in \cite{Wilson:2004whiley}). On a modest cluster, an all-sky reconstruction can be completed in under an hour, allowing for multiple runs of the code to explore optimal parameter settings and prior sensitivity. The goal for work in this project is to make the code more accessible to users through a front-end written in Python and adapt it from its current use for temperature map separation to polarization map separation.
\end{itemize}

In addition, the proposers have extensive collaboration experience; see
Refs.~\cite{
Kahniashvili:2010gp,
Kahniashvili:2012vt,
Tevzadze:2012kk,
Kahniashvili:2012uj,
Brandenburg:2014mwa,
Kahniashvili:2015msa,
Kahniashvili:2016bkp,
Brandenburg:2016odr,
Brandenburg:2017rcb,
Brandenburg:2017rnt,
Brandenburg:2017neh,
Brandenburg:2018ptt,
Pol:2018pao,Bracco:2018vyi,
Brandenburg:2018khp,
Kahniashvili:2018mzl}.
The earlier successful collaboration of the team members, as well as their experience in the fields of cosmology, MHD turbulence modeling, data analysis, and ML, provide an excellent base that should ensure that the project goals are successfully accomplished.

\subsection{Proposal Time-line}
The first and second years of the proposed research will be devoted to
the analysis of MHD simulations with dust included and to develop the
ML algorithms for the CMB polarization component separation.
In addition, during the second year, we will test our formalism and
compare numerical synthetic data with observations.
While {\it Planck} data will be utilized during the entire research,
we will devote the third year of the proposed research to the application
of the next techniques developed so far.

In addition to the main participants identified by names above,
two young researchers (graduate students) will be involved in the investigation.
We request financial support for graduate students at CMU.  We also anticipate undergraduate students participation in the proposed research (through different available funds).
We also request partial salary support for Kahniashvili
(research faculty at CMU supported by external research grants),
and partial summer supports for Brandenburg, Dodelson, and Poczos.
Collaborators Bracco and Wilson will participate on the proposal with no
salary support.

\section{Results from Prior NASA Support}

{\bf Dodelson} (PI) and {\bf Kahniashvili} (Co-I) have been supported through the following NASA grants:
\begin{itemize}
\item
{\bf NASA ATP Grant NAG5-10842:}
From 1991 until 2006, { Dodelson} was supported on a large NASA ATP group grant to the Fermilab Astrophysics Group. During the final three years of the grant, he was a PI (NASA Grant NAG5-10842, ``Fundamental Physics from Space''). Various federal regulations made it more difficult for a DOE lab to obtain NASA funding, so that support ended in 2006. Throughout those years, he wrote 88 papers, most of which were focused on the connection between cosmology and particle physics,  and one textbook~\cite{Dodelson:2003ft}. The papers included: the proposal that sterile neutrinos could comprise the dark matter~\cite{Dodelson:1993je}; early analyzes of the Sloan Survey~\cite{Dodelson:2001ux,Scranton:2001xa,Szalay:2001rw,
Zehavi:2001nr,Stoughton:2002ae,Tegmark:2003ud,Abazajian:2003jy}; a well-cited review of CMB anisotropies~\cite{Hu:2001bc}; the first predictions for CMB anisotropies for cosmic strings~\cite{Allen:1997ag}; the signatures of massive neutrinos on the CMB~\cite{Dodelson:1995es}; early models of dark energy~\cite{Coble:1996te,Dodelson:2001fq,Ahmed:2002mj}; an early analysis of CMB anisotropy data that pointed to a flat universe~\cite{Dodelson:1999am}; and proposals for discriminating models of inflation~\cite{Dodelson:1997hr}.

\item
{\bf NASA Mission Concept Study Award NNX08AT71G S01:}
{ Dodelson} also led the effort to build the science case for a CMB Polarization mission to the 2010 Decadal Survey. This work was funded by NASA Mission Concept Study Award NNX08AT71G S01 and culminated in 6 articles posted to the arXiv and then collected in Ref.~\cite{Dodelson:2009zz}.


\item
{\bf NASA ATP Grant NNX10AC85G:}
{ Kahniashvili}
has been awarded by the NASA ATP program grant {\it Probing Cosmological Symmetry Breaking with the Cosmic
Microwave Background} (2010-2012), NNXlOAC85G. Within this project
a number of papers have been published
\cite{
Kahniashvili:2008sh,
Kahniashvili:2009mf,
Kahniashvili:2009qi,
Kahniashvili:2010us,
Kahniashvili:2010wm,
Kahniashvili:2010gp,
Kosowsky:2010jm,
Chitov:2009ph,
Campanelli:2011qd,
Kahniashvili:2012vt,
Tevzadze:2012kk,
Kahniashvili:2012dy,
Kahniashvili:2012uj,
Brandenburg:2014mwa,
Kahniashvili:2014dfa}.
Major results
including the following
important precedents for the current proposal:

(i)
Calculation of the microwave background temperature distortion due to
aberration induced by our proper motion with respect to the microwave background rest frame \cite{Kosowsky:2010jm}. We first pointed out
that non-diagonal multipole correlations from aberration would be
detectable in  {\it Planck} sky maps, and the effect was then detected
at the predicted level by {\it Planck} team in the 2013 results Paper XXVII
\cite{Aghanim:2013suk}, the only new effect detected by
{\it Planck} 2013. We have  used the similar techniques to investigate a possible dipole modulation of the microwave background anisotropies to characterize the hemispherical power asymmetry seen in both WMAP and {\it Planck} sky maps \cite{Aiola:2015rqa}.

(ii)
Computation of the main characteristics of gravitational waves generated by the cosmological magnetic fields induced via bubble collisions during the phase transitions \cite{Kahniashvili:2009mf} using aero-acoustic approximation described above.
In the follow-up study we have computed gravitational wave polarization degree assuming that parity violating sources  were present during cosmological phase transitions  \cite{Kisslinger:2015hua}.

(iii)
Direct numerical simulations of forced and freely decaying 3D
magnetohydrodynamic turbulence in order to model magnetic field evolution
during cosmological phase transitions \cite{Kahniashvili:2010gp}.
We have studied magnetic field decay laws during phase-transition
turbulence and for the first time showed numerically that a magnetic
field with a single initial scale is redistributed over a wide ranges
of wavenumbers having a Batchelor spectrum at large length scales,
and the scaling laws are almost independent on the Prandtl numbers
\cite{Tevzadze:2012kk,Kahniashvili:2012uj}. We also studied the evolution of an inflation-generated magnetic field during cosmological phase transitions \cite{Kahniashvili:2012vt}.

(iv)
Constraints on parity-violating magnetic helicity in the early
universe \cite{Kahniashvili:2014dfa}.
In the follow-up study, \cite{Kahniashvili:2016bkp}, we have shown that if the mean magnetic helicity density of a helical field is measured, it can be used to determine a lower bound on the duration of inflation.
\end{itemize}


\newpage
%%%
\begin{thebibliography}{widest-label}
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%%%%%%%%%%%%%%%%%%%%%%%%



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%AB: added ref

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  %{\it et al.}
  [Planck Collaboration],
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%TK
%PLANCK
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%%
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