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\section{Introduction}
\label{sec:intro}
The bending of light due to the presence of structures in its path is one very significant method to study the distribution of matter in the universe. The deflection is independent of the nature of the intervening matter, if it is dark or baryonic, and hence, this phenomenon, referred to as gravitational lensing, provides a unique tool to map the dark side of the universe. Under controlled systematics of the experiment, weak gravitational lensing, where the deflection of light rays are not significant enough to observe multiple images of the source but strong enough to deform the shape of the source, is a very powerful probe to study the nature of dark energy \citep{2006astro.ph..9591A}. The future sky surveys, like Euclid \citep{2011arXiv1110.3193L, 2009ExA....23...17R, 2009ExA....23...39C}, are expected to provide maps of the sky with un-precedented accuracy and high resolution like never before \citep{2013LRR....16....6A}. It is an opportunity to employ the advantage of such high quality data to answer the most important questions in cosmology - the energy content of the universe, its dynamics, its evolution and the formation of structure. Weak gravitational lensing can be used as an ideal tool for such high quality data and can deliver, with sub-percent level accuracy, measurements of the main cosmological parameters.
The deformation of the shape of the observed galaxies due to the intervening matter is referred to as {\it shear}. This signal is very small, nearly 1$\%$ of the intrinsic ellipticity of the source galaxies, but can be measured statistically under the assumption that the intrinsic ellipticity of the background galaxies do not have a preferred direction. There are a number of interpretation of the two-point shear statistics based on dark matter only (collision-less) simulations which is a good approximation in the linear regime. However, at non-linear scales baryonic physics becomes important and can introduce a bias of 5 to 20 percent in the interpretation of the measurements, which in turn can introduce a bias in the cosmological constraints. So, in the era of precision cosmology, it is very important to quantify the effect of baryonic physics in the two-point shear statistics or the power spectrum.
Baryons account for nearly 20\% of the matter content of the universe. Its distribution depends on the dark matter potential well, AGN feedback, supernovae, structure formation history and radiative cooling. Further baryonic distribution affects the matter power spectrum at small scales, which to the extension, affects the two point shear statistics. The effect of baryons on several statistics relevant for cosmology has been already studied by various authors. For instance, \cite{2009MNRAS.394L..11S,2012MNRAS.423.2279C,2014MNRAS.440.2290M} and \cite{2014MNRAS.439.2485C} focused on the effects on the halo mass function. The effect of baryonic processes on the power spectrum and on the weak gravitational lensing shear signal has been studied too \citep{2004APh....22..211W, 2004ApJ...616L..75Z, 2006ApJ...640L.119J, 2008ApJ...672...19R, 2010MNRAS.405..525G, 2011MNRAS.417.2020S, 2011MNRAS.415.3649V, 2014ApJ...783..118R, 2014arXiv1407.0060M}.
In most of the previous works (see references above), the approach was based on simulations, which suffer from finite volume and finite resolution effects, are performed using only one
cosmology and baryonic model. They however capture the non-linear physics of gravitational collapse and the associated baryonic effects. In this work, we employ the halo model, an analytical approach, to build two-point shear statistics with and without baryons. This allows one to recover various different realizations of any cosmological models. We also compare our results with simulations at various stages to validate our main assumptions.
The outline of the paper is as follows: In section \ref{sec:theory}, we review the necessary concepts of the halo model and propose our baryonic model as a modification in the radial density profiles of the halos. We compare the model to simulations with AGN feedback models. We also review the modelling of shear power spectrum. We talk about the covariance matrix of the $C_{\ell}$, Gaussian and non-Gaussian parts. In section \ref{sec:comparison}, we make a comparison between the dark-matter-only model (DMO) and our baryonic model (BAR) and shows the behaviour of the baryonic correction as a function of our main AGN-feedback-parameter, $M_{\rm crit}$. We introduce our fiducial model and mock datasets to perform the likelihood analysis in section \ref{sec:fiducial}. In section \ref{sec:cosmology}, we talk about the cosmological implication of these baryonic corrections and the forecasts on the cosmological parameters, its accuracy and precision. Finally in section \ref{sec:discussion} we discuss the implications of our results and propose possible strategies for future works.
\section{Theoretical model - a short review}
\label{sec:theory}
We employ an analytic approach to model the effects of baryonic physics on the matter power spectrum and to the extension, on the shear power spectrum. The model has two broad parts: $(i)$ the dark-matter-only model (DMO), and $(ii)$ the modified model with baryonic physics (BAR). These two approaches modify the density profile of dark matter halos. We used the halo model \citep{2000MNRAS.318..203S,2000MNRAS.318.1144P,2000ApJ...543..503M,2002PhR...372....1C} to construct the matter power spectrum based on the density profiles of halos of mass $M$ and at redshift $z$.
\subsection{The halo model}\label{sec:halomodel}
We employed the halo model \citep{1977ApJ...217..331M,2000MNRAS.318..203S,2000ApJ...543..503M,2000MNRAS.318.1144P,2002PhR...372....1C} approach to calculate the matter power spectrum given the density profile of the halos. The halo model assumes all the matter in the universe to be in spherical halos with mass defined by a threshold density as:
\begin{equation}
M_{\bigtriangleup} = \dfrac{4}{3} \pi R_{\bigtriangleup}^3 \bigtriangleup \bar{\rho}_{m}
\end{equation}
\\
where $M_{\bigtriangleup}$ is the mass of the halo and $R_{\bigtriangleup}$ is the boundary where the density of the halo drop to $\bigtriangleup$ times the mean matter density of the Universe, $\bar{\rho}_{m}$. We use $\bigtriangleup=200$ throughout this paper, unless stated otherwise. We define the virial radius of the halo $R_{\rm vir}$ to be $R_{200}$.
In this framework, the matter power spectrum can be split into two parts:
\begin{equation}
P(k) = P_{1h}(k) + P_{2h}(k),
\end{equation}
\\
where, the two terms on the right hand side correspond to 1-halo term, describing the correlation between dark matter particles within the halo and 2-halo term which describes the halo-halo correlation respectively. These terms are given by
\begin{equation}
P_{1h} = \int d\nu (f_{dm}+f_{gas}(\nu)) f(\nu) \dfrac{M}{\rho} |u(k|\nu)|^2,
\end{equation}
\begin{equation}
P_{2h} = \left(f_0b_0 + \int d\nu (f_{dm}+f_{gas}(\nu)) f(\nu) u(k|\nu) b(\nu) \right)^2 P_{\rm lin}(k),
\end{equation}
\\
where, $M$ is the mass of the halo and $\nu = \delta_c/\sigma(M,z)$ with $\delta_c = 1.686$. The term $f(\nu)$ is the functional form of the mass function and we used the fitting formula from \cite{2008ApJ...688..709T}. The term $b(\nu)$ resembles the bias in the dark matter halos and we used the fitting formula in \citep{2010ApJ...724..878T}. To fulfill the underlying assumptions of the halo model, these two functional forms, $f_{\nu}$ and $b_{\nu}$ have to be expressed as in the following relations:
\begin{equation}
\int_{0}^{\infty} f(\nu)d\nu = 1
\end{equation}
\begin{equation}
\int_{0}^{\infty} f(\nu)b(\nu)d\nu = 1
\end{equation}
\\
However, assuming a lower mass cut corresponding to $\nu_{\rm min}$, we introduce new background factors $f_0$ and $b_0$ such that:
\begin{equation}
f_0 + \int_{\nu_{\rm min}}^{\infty} (f_{dm}+f_{gas}(\nu)) f(\nu)d\nu = 1
\end{equation}
\begin{equation}
f_0b_0 + \int_{\nu_{\rm min}}^{\infty} (f_{dm}+f_{gas}(\nu)) f(\nu)b(\nu)d\nu = 1
\end{equation}
\\
Additionally, the term $f_{dm}+f_{gas} = 1$ for simpler models like no feedback, but for more exotic models, like with AGN feedback or including other baryonic physics, this term may deviate from unity. This will be more useful as explained in section \ref{sec:baryonicmodel}
\begin{figure*}
\centering
\subfigure{\includegraphics[width=0.48\textwidth]{figures/conc_mhalo.eps}}
\subfigure{\includegraphics[width=0.48\textwidth]{figures/bcg_mhalo.eps}}
\subfigure{\includegraphics[width=0.48\textwidth]{figures/fgas_mhalo.eps}}
\subfigure{\includegraphics[width=0.48\textwidth]{figures/rho_nfw_gas_z0.eps}}
\caption{Top left: Concentration parameter as a function of halo mass for variable redshift. Top right: Mass of the central galaxy as a function of halo mass for variable redshift. Bottom left: Gas mass fraction as a function of halo mass for variable $M_{\rm crit}$. Bottom right: Density profile for NFW (solid lines) and intra-cluster gas (dashed lines) for different halo masses at redshift 0.}
\label{fig:halo}
\end{figure*}
We used the \cite{1998ApJ...496..605E, 1999ApJ...511....5E} transfer function calculations to account for the linear matter power spectrum term, $P_{lin}(k)$. The term $u(k|M)$ is the Fourier transform of the normalized density profile and is given by,
\begin{equation}
u(k|M) = \dfrac{4 \pi}{M} \int_0^{R_{\rm vir}}dr\ r^2\ \rho(r|M)\ \dfrac{\sin(kr)}{kr}.
\end{equation}
\\
where, $\rho(r|M)$ is the density profile of the halo of mass $M$. The function $u(k|M)$ is normalised such that $u(k=0|M)=1$ .The dispersion of the smoothed density field, $\sigma(M,z)$, is given by,
\begin{equation}
\sigma^2(M,z) = \dfrac{1}{2 \pi^2} \int P_{\rm lin}(k) k^2 |\tilde{W}(R,k)|^2 dk,
\end{equation}
\\
where, $\tilde{W}(R,k)$ is the Fourier transform of top-hat filtering function and given by,
\begin{equation}
\tilde{W}(R,k) = 3 \dfrac{\textrm{sin}(kR) - kR \textrm{cos}(kR)}{(kR)^3}
\end{equation}
This framework of the halo model is applied to both DMO and BAR model which, differ in the halo density profiles and normalization of the mass function. The following two sections explains the corresponding profiles.
\subsection{Dark matter only}\label{sec:darkmatteronly}
We started with the radial density profile of dark matter halos given by the functional form:
\begin{equation}
\rho(r|M) = \dfrac{\rho_s}{(r/R_s)^\alpha (1+r/R_s)^\beta},
\end{equation}
\\
where, $R_s$ is the characteristic radius given by the concentration parameter ($c$) and the virial radius of the halo ($r_{vir}$) as $c = R_{\rm vir}/R_s$. We used the two parameters $\alpha$ and $\beta$ to be 1 and 2 respectively, corresponding to the Navarro-Frenk-White (NFW) profile \citep{1997ApJ...490..493N}. The characteristic density $\rho_s$ which is strongly degenerate with $R_s$ and also proportional to the critical density of the Universe when the halo was formed. So, the NFW profile for dark matter halos is completely described by its concentration.
The concentration parameter $c$ gives the information about the environment or the mean background density during the formation of the halo. A number of $N$-body simulations \citep{1997ApJ...490..493N, 1999MNRAS.310..527A, 2000ApJ...535...30J, 2001MNRAS.321..559B, 2001ApJ...554..114E, 2003ApJ...597L...9Z, 2007MNRAS.381.1450N, 2007MNRAS.378...55M, 2008MNRAS.390L..64D, 2008MNRAS.387..536G, 2014MNRAS.441.3359D} has prescribed various power laws between mass of the halo ($M$) and its concentration parameter $c$ at redshift $z$. We used the fitting formula given in \citep{2011MNRAS.411..584M}:
\begin{equation}
\log(c) = a(z)\log(M_{\rm vir}/[h^{-1} M_{\bigodot}]) + b(z)
\end{equation}
\\
where,
\begin{equation}
a(z) = \omega z - m
\end{equation}
\\
and
\begin{equation}
b(z) = \dfrac{\alpha}{z+\gamma} + \dfrac{\beta}{(z+\gamma)^2}
\end{equation}
\\
The fitting parameters $\omega,\ m,\ \alpha,\ \beta\ {\rm and}\ \gamma$ are 0.029, 0.097, -110.001, 2469.720 and 16.885 respectively.
Figure \ref{fig:halo} (top-left panel) shows the behaviour of the concentration parameter as function of halo mass at different redshifts. There is an anti-correlation between the mass of the halo and its concentration. Also for a given halo mass, the concentration decreases with redshift. We limit the minimum concentration to 4 (dashed line in figure \ref{fig:halo} upper-left panel). This is because the higher mass halos did not reach there maximum formation efficiency redshift and will reach it in future. So, on an average, there concentration must not be less than a few. A very recent study from \cite{2014MNRAS.441.3359D} shows that this behaviour is consistent and the minimum concentration is very close to 4.
\subsection{A baryonic model}\label{sec:baryonicmodel}
Our baryonic model accounts within each halo for: 1) a central galaxy, the major stellar component whose properties are derived from abundance matching techniques; 2) a hot plasma in hydrostatic equilibrium and 3) an adiabatically-contracted (AC) dark matter component. This analytic approach allows us to compare our model to the DMO case. Apart from the normalization of the mass function, there is only one term that is affected by these baryonic components and is the density profile of the halo, which no longer follows the NFW profile. We can write the modified NFW (BAR) profile as:
\begin{equation}
\rho_{\rm BAR}(r|M) = f_{\rm dm}\rho_{\rm NFW}^{\rm AC}(r) + \rho_{\rm BCG}(r) + f_{\rm gas}(M)\rho_{\rm gas}(r),
\end{equation}
\\
we discuss each of these terms in more details.
\subsubsection{Stellar component}\label{sec:stellar}
We used the fitting function from \cite{2013MNRAS.428.3121M} based on abundance matching to map the stellar mass of the central galaxy $M_{\rm CentralGalaxy}$ (BCG), which is the major component of stellar mass in a cluster, to the mass of the halo ($M_{\rm{halo}}$). Figure \ref{fig:halo} (top-right panel) shows the mapping between halo mass and stellar mass fraction associated to the central galaxy for a variety of redshifts. The relation has a positive slope for low mass halos, however, at about the size of the Milky way halo, the slope turns negative. At this peak, the central galaxy stellar mass contributes about 4-5 $\%$ of the total mass of the halo. Also this peak shifts to higher masses for higher redshifts but contributes lower fraction.
The actual distribution of stellar mass in galaxy groups and clusters can be quite complex. The total stellar mass budget can be decomposed in 3 components: satellite galaxies, Brightest Cluster Galaxy (BCG, the massive elliptical galaxy dominating the cluster centre) and Intra-cluster Light (ICL, an extended stellar halo surrounding the BCG). The BCG and ICL represent $\sim 40$ \% of the mass in clusters, with this ratio decreasing with total cluster mass \citep{2007ApJ...666..147G}. However, BCG+ICL dominate the inner part of the cluster and constitute $\sim 70\%$ of the total stellar mass within 0.1 $R_{200}$. This fact is particularly relevant for computing the effect of baryon condensation on the dark matter profiles (see Subsection \ref{sec:adcon}). The BCG+ICL component is usually modelled using superimposition of fitting functions, typically multiple Sersic profiles. Given that we are not interested in detailed modelling of the stellar distribution, we consider a simplified model for the BCG+ICL.
we adopted a radial density profile for BCG, where the enclosed mass goes linearly with the radius,
\begin{equation}
M_{\star}(<r) = M_{\rm CentralGalaxy} \dfrac{r}{2R_{1/2}}
\end{equation}
\\
this gives,
\begin{equation}
\rho(r) = \dfrac{M_{\rm CentralGalaxy}}{8 \pi R_{1/2} r^2},\ r<2R_{1/2}
\end{equation}
\\
where, $R_{1/2}$ is the half mass radius. We use $R_{1/2} = 0.015 R_{\rm vir}$ which is a good fit to the observations \citep{2014arXiv1401.7329K}. We forced the density profile to drop exponentially after $2R_{1/2}$.
\subsubsection{Intra-cluster plasma}\label{sec:gas}
\label{sec:icm}
The major component of the baryonic matter in a galaxy cluster is the hot intra-cluster gas. It is mainly ionized hydrogen at very high temperature and low density. This plasma radiates in X-rays and can safely be assumed to be in hydrostatic equilibrium. We assume this gas distribution in the halo according the hydrostatic equilibrium equations given in \cite{2013MNRAS.432.1947M},
\begin{equation}
\rho(x) = \rho_0 \left[\dfrac{\ln(1+x)}{x} \right]^{\dfrac{1}{\Gamma -1}}
\end{equation}
\\
where, $x$ is the distance from the centre of the halo in unit of scale radius $R_s$. The effective polytropic index $\Gamma$ is given by,
\begin{equation}
\Gamma = 1+ \dfrac{(1+x_{eq})\ln(1+x_{eq}) - x_{eq}}{(1+3x_{eq})\ln(1+x_{eq})}
\end{equation}
\\
where, $x_{eq}=c/\sqrt{5}$. Figure \ref{fig:halo} (bottom-right in dashed lines) shows the density profile of the hot gas for variable halo masses at redshift 0 and also shows the comparison to the NFW profile (solid lines). For $x>x_{eq}$, the gas density profiles follows the NFW profile, however, it approaches a nearly constant values near the centre of the halo.
The normalization of the gas density profile, $\rho_0$, is fixed by the gas fraction $f_{\rm{gas}}$. if we assume no feedback from the baryonic component of the halo, this number can be a constant, however, many hydrodynamical simulations \citep{2005MNRAS.356..107R, 2005MNRAS.360..892D, 2006Natur.442..539M, 2012MNRAS.421.3464P, 2013MNRAS.429.3068T, 2013MNRAS.432.1947M} shows signatures of the expulsion of gas from the halo. This expulsion is stronger in low mass halos than the high mass halos. So the low mass halos are generally deficit in this hot plasma component. Following the same physical motivation, we used the gas mass fraction of the halo to be the function of the mass of the halo following the parametric form:
\begin{equation}
f_{\rm{gas}}(M_{\rm halo}) = \dfrac{\Omega_b/\Omega_m}{1 + \left (\dfrac{M_{\rm{crit}}}{M_{\rm{halo}}} \right) ^{\beta}}
\end{equation}
\\
where, $M_{\rm{crit}}$ is a free parameter and $\beta$ is fixed to 2. This parameter controls the gas fraction in halos of different mass. A higher value for $M_{\rm{crit}}$ represents less gas in the halo up to higher halo masses. This parameter can also be interpreted as the control sequence for AGN feedback. Figure \ref{fig:halo} (bottom-left panel) shows the variation of $f_{\rm{gas}}$ with halo mass for variety of $M_{\rm{crit}}$. We chose $M_{\rm crit} = 10^{13} h^{-1} M_{\bigodot}$ as the most realistic model. In this case, all halos with mass lower than $\sim 2\times 10^{12} h^{-1} M_{\bigodot}$ have expelled all their gas to the background (outside the $R_{\rm vir}$) and all halos with mass larger than $\sim 2\times 10^{13} h^{-1} M_{\bigodot}$ have all their gas inside the halo. The intermediate mass halos have a very smooth transition from no gas to all gas inside the halo. This behaviour matches well with recent study from \cite{2014arXiv1409.8617S}. We studied this case in detail for all its cosmological implications at different scales. We also studied one optimistic\footnote{Optimistic in the sense of less AGN feedback that makes the baryonic corrections less troublesome} model, where the feedback is not as strong as in our realistic model, with $M_{\rm crit} = 10^{12} h^{-1} M_{\bigodot}$.
\subsubsection{Adiabatic contraction}\label{sec:ac}
\label{sec:adcon}
In the DMO model, we adopted the NFW profile for the distribution of dark matter in the halo which is nearly scale-free and completely described by the concentration parameter. However, in the presence of baryons, the dark matter component follows NFW only in the outskirts of the halo, but in the very centre the dark matter profile becomes steeper and deviates from pure a NFW profile. This is because the baryons, which are dominant in the centre of the halo, drag some extra matter from the surrounding towards the centre making the dark matter profile steeper towards the centre. The total distribution of matter is expected to dynamically respond to the condensation of baryons at the centre of the halo in a way that approximately conserves the value the adiabatic ``invariant'' $R\times M(R)$, where $R$ is the distance from the halo centre and $M(R)$ is the mass enclosed in a sphere of radius R \citep{1986ApJ...301...27B, 2004ApJ...616...16G}. We adopted a simplified model for this effect following the appendix of \cite{2011MNRAS.414..195T} where this adiabatic contraction (AC) of the dark matter profile is solely governed by the central galactic disk.
\subsection{Comparison with simulations}\label{sec:simulations}
\begin{figure*}
\includegraphics[width=0.48\textwidth]{figures/AGN00001.eps}
\includegraphics[width=0.48\textwidth]{figures/AGN00003.eps}\\
\includegraphics[width=0.48\textwidth]{figures/AGN00004.eps}
\includegraphics[width=0.48\textwidth]{figures/AGN00005.eps}\\
\includegraphics[width=0.48\textwidth]{figures/AGN00006.eps}
\includegraphics[width=0.48\textwidth]{figures/AGN00008.eps}
\caption{A comparison of our model density profiles (dashed lines) with hydrodynamical simulations of Martizzi et. al. 2014 (solid lines). There is a remarkable agreement, except at the very centre of the halo.}
\label{fig:compare}
\end{figure*}
\begin{table}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
\hline
Type & $H_0$ & $\sigma_{\rm 8}$ & $n_{\rm s} $ & $\Omega_\Lambda$ & $\Omega_{\rm m}$ & $\Omega_{\rm b}$ \\
\hline
\hline
DMO & 70.4 & 0.809 & 0.963 & 0.728 & 0.272 & - \\
BAR & 70.4 & 0.809 & 0.963 & 0.728 & 0.272 & 0.045 \\
\hline
\hline
\end{tabular}
\caption{Cosmological parameters adopted in our simulations. }\label{tab:cosm_par}
\end{center}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{|l|c|c|c|}
\hline
\hline
{\itshape Type} & $m_{\rm cdm}$& $m_{\rm gas}$ & $\Delta x_{\rm min}$ \\
& $[10^{8}$ M$_\odot$/h] & $[10^{7}$ M$_\odot$/h] & [kpc/h] \\
\hline
\hline
Original box & $ 15.5 $ & n.a. & $2.14$ \\
DMO zoom-in & $1.94$ & n.a. & $1.07$ \\
BAR zoom-in & $1.62$ & $3.22$ & $1.07$ \\
\hline
\hline
\end{tabular}
\end{center}
\caption{Mass resolution for dark matter particles, gas cells and star particles, and spatial resolution (in physical units) for our simulations. }\label{tab:mass_par}
\end{table}
We consider data from a set of cosmological re-simulations performed with the {\scshape ramses} code \cite{2002A&A...385..337T}. These simulations are part of a larger set recently used
by \cite{2014MNRAS.440.2290M} to study the baryonic effects on the halo mass function. Thanks to the adaptive mesh refinement capability of the {\scshape ramses} code, the resolution
achieved in these simulations is sufficient to study the properties of low redshift BCGs.
In these calculations, the cosmological parameters are: matter density parameter $\Omega_{\rm m}=0.272$, cosmological constant density parameter $\Omega_\Lambda=0.728$, baryonic matter
density parameter $\Omega_{\rm b}=0.045$, power spectrum normalization $\sigma_{\rm 8}=0.809$, primordial power spectrum index $n_{\rm s}=0.963 $ and Hubble constant $H_0=70.4$ km/s/Mpc
(Table~\ref{tab:cosm_par}). We generated initial conditions for the simulations using the \cite{1998ApJ...496..605E} transfer function and the {\scshape grafic++} code\footnote{http://sourceforge.net/projects/grafic/}, based on the original {\scshape grafic} code \citep{2001ApJS..137....1B}. These simulations come in two flavours: DMO (dark matter only)
which only follow the evolution of dark matter, BAR which include baryons and galaxy formation prescriptions.
The technique we adopted to perform the zoom-ins is described in the following. First, we ran a dark matter only simulation with particle mass $m_{\rm cdm}=1.55\times 10^9$~M$_\odot$/h
and box size $144$~Mpc/h. The initial level of refinement was $\ell=9$ ($512^3$), but as the simulation evolved more levels of refinement were allowed. At redshift $z=0$ the grid was
refined down to a maximum level $\ell_{\rm max}=16$. Subsequently, we ran apply the AdaptaHOP algorithm \cite{2004MNRAS.352..376A} to identify the position and masses of dark
matter halos. We selected 51 halos whose {\it total} masses lie $M_{\rm tot}>10^{14}$~M$_\odot$ and whose neighbouring halos do not have masses larger than $M/2$ within a spherical
region of five times their virial radius. We determined that only 25 of these clusters are relaxed. High resolution initial conditions were extracted for each of the 51 halos and were used
to run zoom-in re-simulations. Three different re-simulations per halo have been performed: (I) including dark matter and neglecting baryons, (II) including dark matter, baryons
and stellar feedback, (III) including baryons, stellar feedback and AGN feedback. In this paper we focus on cases (I) and (III), labelled DMO and BAR, respectively.
In the DMO re-simulation, the dark matter particle mass is $m_{\rm cdm}=1.94\times 10^{8}$~M$_\odot$/h. In the BAR re-simulations, the dark matter particle mass is
$m_{\rm cdm}=1.62\times 10^{8}$~M$_\odot$/h, while the baryon resolution element has a mass of $m_{\rm gas}=3.22\times 10^{7}$~M$_\odot$. The maximum refinement level was set to $\ell=17$,
corresponding to a minimum cell size $\Delta x_{\rm min} = L/2^{\ell_{\rm max}}\simeq 1.07$ kpc/h. The grid was dynamically refined using a quasi-Lagrangian approach:
when the dark matter or baryonic mass in a cell reaches 8 times the initial mass resolution, it is split into 8 children cells. Table~\ref{tab:mass_par} summarizes the particle mass and
spatial resolution achieved in the simulations.
The physical prescription implemented in the code to perform the BAR simulations is here briefly described. In {\scshape ramses} gas dynamics is solved via a second-order unsplit Godunov
scheme \citep{2002A&A...385..337T} based on different Riemann solvers (we adopted the HLLC solver) and the MinMod slope limiter. The gas is described by perfect gas equation of state
(EOS) with polytropic index $\gamma=5/3$. Gas cooling is modelled with the \cite{1993ApJS...88..253S} cooling function which accounts for H, He and metals. Star formation and supernovae
feedback ("delayed cooling" scheme, \cite{2006MNRAS.373.1074S}) and metal enrichment have been included in the calculations. AGN feedback has been included too, using a method inspired by
the \cite{2009MNRAS.398...53B} model. In this scheme, super-massive black holes (SMBHs) are modeled as sink particles and AGN feedback is provided in form of thermal energy injected in a
sphere surrounding each SMBH. More details about the AGN feedback scheme and about the tuning of the galaxy formation prescriptions can be found in \cite{2011MNRAS.414..195T} and \cite{2012MNRAS.422.3081M}.
Figure~\ref{fig:compare} shows the comparison between the dark matter, gas, stellar and total mass density profiles of 6 halos in the \cite{2014MNRAS.440.2290M} catalogue and the mass model described in Section \ref{sec:halomodel}. The model for the adiabatically contracted dark matter profile (red dashed lines) fits well the simulations down to scales $\sim 10$ kpc. The model for the Intra-cluster plasma (green dashed lines) fits well the results of the simulations down to scales $\sim 50$ kpc. The relation between mass of the central galaxy and that of the halo has a lot of scatter. So, to compare with simulations we use the stellar mass from the simulation itself for the given halo, which define the normalisation of our stellar model. The model (blue dashed lines) is a good fit to the results of the simulations except in the outskirts. This is expected since the data from the simulations include BCG, ICL and satellite galaxies. However, the model is constructed in such a way that the stellar mass expected from abundance matching is associated to the central regions of the halos. The overall result is that the model for the total mass (black dashed lines) provides an excellent match to the results of cosmological simulations down to a scale of $\sim 10$ kpc. Therefore we conclude that the mass model is good enough to be adopted for the purposes of this paper.
\subsection{From $P(k)$ to $C(\ell)$}
\label{sec:pk2cl}
In this section we develop the mapping from 3D matter power spectrum $P(k,z)$ to the 2D projected shear angular power spectrum $C_{\ell}$ following the theoretical framework explained in \cite{2009MNRAS.395.2065T}.
The distortion of the source shape due to weak gravitational lensing can be quantified with two quantities: shear $\gamma$ and convergence $\kappa$. The convergence $\kappa$ is the local isotropic part of the deformation matrix and can be expressed as:
\begin{equation}
\kappa(\vec{\theta}) = \dfrac{1}{2} \vec{\bigtriangledown}.\vec{\alpha}(\vec{\theta})
\end{equation}
\\
where, $\alpha$ is the deflection angle. If we know the redshift of the source galaxies, additional information can be gained by dividing the sources in different redshift bins. This process is referred to as lensing tomography and is very useful to gain extra constraints on cosmology from the evolution of the weak lensing power spectrum \citep{1999ApJ...522L..21H, 2002PhRvD..65f3001H, 2004MNRAS.348..897T}. In cosmological context, the convergence field can be expressed as the weighted projection of the mass distribution integrated along the line of sight in the $i$th redshift bin,
\begin{equation}
\kappa_i(\vec{\theta}) = \int_0^{\chi_H} g_i(\chi) \delta(\chi \vec{\theta},\chi)d\chi,
\end{equation}
\\
where, $\delta$ is the total 3 dimensional matter overdensity, $\chi$ is the comoving distance and $\chi_H$ is the comoving distance to the horizon. For a complete review see \cite{1999ARA&A..37..127M, 2001PhR...340..291B, 2006glsw.conf..269S}. The lensing weights $g_i(\chi)$ in the $i$th redshift bin with comoving distance range between $\chi_i$ and $\chi_{i+1}$ are given by:
\begin{equation}
g_i(\chi) = \begin{cases} \dfrac{g_0}{\bar{n}_i} \dfrac{\chi}{a(\chi)} \int_{\chi_i}^{\chi_{i+1}} n_s(\chi^{\prime})\dfrac{dz}{d\chi^{\prime}} \dfrac{(\chi^{\prime} - \chi) }{\chi^{\prime}} d\chi^{\prime}, & \chi \le \chi_{i+1} \\
0, & \chi > \chi_{i+1} \end{cases}
\end{equation}
\\
where, $a(\chi)$ is the scale factor at comoving distance $\chi$. Also,
\begin{equation}
g_0 = \dfrac{3}{2} \dfrac{\Omega_m}{H_0^2}
\end{equation}
\\
and,
\begin{equation}
\bar{n}_i = \int_{\chi_i}^{\chi_{i+1}} n_s(\chi(z)) \dfrac{dz}{d\chi^{\prime}} d\chi^{\prime}.
\end{equation}
\\
where, $n_s(\chi(z))$ is the distribution of sources in redshift. We assume a source distribution along the line of sight of the form:
\begin{equation}
n_s(z) = n_0 \times 4z^2 \exp\left(-\dfrac{z}{z_0} \right)
\end{equation}
\\
with $n_0 = 1.18 \times 10^{9} $ per unit steradian and $z_0$ is fixed such that the corresponding projected source density $n_g$ resembles the experiment, like Euclid etc.
\begin{equation}
\int_0^{\infty} n_s(z)dz = \bar{n}_g.
\end{equation}
\\
For Euclid like survey, we choose $z_0$ such that $\bar{n}_g=50$ sources per arcmin$^{-2}$ \citep{2008ARNPS..58...99H}.
Finally the shear power spectrum between redshift bins $i$ and $j$ can be computed as:
\begin{equation}
C_{ij}(\ell) = \int_0^{\chi_H} \dfrac{g_i(\chi) g_j(\chi)}{ \chi^2} P\left(\dfrac{\ell}{\chi},\chi \right)d \chi
\end{equation}
\\
where, $P$ is the 3D matter power spectrum calculated using the halo model framework as described in section \ref{sec:halomodel}. Larger $\ell$ corresponds to the smaller scale and the large contribution of $C_{\ell}$ at higher $\ell$ comes from non-linear clustering.
We divided the big cosmological volume into 3 redshift bins with boundaries: 0.01, 0.8, 1.5 and 4.0; so we calculated total 6 convergence cross-spectra (3 auto-spectra and 3 cross-spectra).
The auto-spectra is contaminated by the intrinsic ellipticity noise and assuming its distribution to be completely uncorrelated to different source galaxies, the observed power spectrum $C_{ij}^{\rm{obs}}(\ell)$ is given by,
\begin{equation}
C_{ij}^{\rm{obs}}(\ell) = C_{ij}(\ell) + \delta_{ij}\dfrac{\sigma_{\epsilon}^2}{\bar{n}_i},
\end{equation}
\\
we choose $\sigma_{\epsilon}=0.33$ which is the RMS intrinsic ellipticity. The cross spectra is not contaminated by shot noise.
The covariance matrix of $C_{\ell}$ has two contributions: Gaussian and non-Gaussian (NG). In this work we only consider the Gaussian contribution to the covariance matrix which is given by the following expression,
\begin{align}
{\rm Cov}_{ij,mn}(\ell,\ell^{\prime})& = \dfrac{\delta_{\ell\ell^{\prime}}}
{\Delta\ell(2\ell+1) \rm{f_{sky}}} \times \nonumber\\
&\qquad \left(C_{im}^{\rm{obs}}(\ell)C_{jn}^{\rm{obs}}(\ell)+
C_{in}^{\rm{obs}}(\ell)C_{jm}^{\rm{obs}}(\ell) \right),
\end{align}
\\
where, $\Delta\ell$ is the bin width of the $\ell$ and $f_{sky}$ is the sky fraction for the targeted experiment. This term is dominated by cosmic variance for lower $\ell$ and shot noise for higher $\ell$, however, for large number of sources, as in case of Euclid, and larger size of bins $(\Delta \ell)$ towards higher end of $\ell$, the shot noise can be significantly reduced.
The NG contribution to the covariance matrix of $C_{\ell}$ is rather complicated to calculate. It gives the correlation between different $\ell$. At the matter power spectrum level, this term depends on the matter trispectrum. To compute the NG covariance to lensing, we need to integrate the trispectrum in redshift and angle on the sky and then compute this quantity for various $\ell$ and $\ell^{\prime}$. So this is a 4D calculation of trispectrum which is computationally very expensive. \cite{2012PhRvD..86h3504Y} shows that these NG correction to the covariance becomes significant for $\ell$ of few thousand and \cite{2002PhR...372....1C} shows that neglecting this will introduce the bias in the cosmological parameters up to 20 $\%$. In this work, we are not taking into account these corrections and we are doing our analysis for different $\ell_{max}$: 1000, 2000, 3000, 4000, 5000, 6000, 8000, 10000 and 20000. We will discuss more about the NG covariance in section \ref{sec:NG}.
\section{Comparing BAR and DMO model}
\label{sec:comparison}
\begin{figure*}
\includegraphics[width=0.48\textwidth]{figures/pk_ratio_z0.0.eps}
\includegraphics[width=0.48\textwidth]{figures/pk_ratio_Mcrit_1d13.eps}
\includegraphics[width=0.48\textwidth]{figures/cl_ratio_nobin.eps}
\includegraphics[width=0.48\textwidth]{figures/cl_ratio_bins_Mcrit_1e13.eps}
\caption{Top row: Relative deviation of the matter power spectrum predicted by the BAR model from the DMO model predictions as a function of $k$ for different $M_{\rm crit}$ at redshift zero (left) and for fixed $M_{\rm crit} = 10^{13} h^{-1} M_{\bigodot}$ and different redshifts (right). Bottom row: Relative deviation of the shear power spectrum ($C_{\ell}$) predicted by the BAR model from DMO model predictions for different $M_{\rm crit}$ in one big redshift bin (left) and for three tomographic redshift bins and fixed $M_{\rm crit} = 10^{13} h^{-1} M_{\bigodot}$ (right). Dashed lines are the calculations without adiabatic contraction (AC) and solid lines with adiabatic contraction (AC). The horizontal dashed line shows the cosmic baryon fraction.}
\label{fig:pkandcl}
\end{figure*}
In this section we try to draw a comparison between the baryonic model (BAR) and the dark-matter only (DMO) model. We would like to establish an understanding of the scales where the baryonic corrections become important and how these scales changes with redshift and the only free parameter, $M_{\rm crit}$.
Figure \ref{fig:pkandcl} (top-left panel) shows the relative differences between the BAR and DMO predictions for the matter power spectrum, also referred as {\it boost} in this article. There is only one free parameter of the baryonic model, $M_{\rm crit}$ which regulates the amount of AGN feedback and which is introduced in section \ref{sec:icm}. The overall shape of the deviation is similar in all cases for various $M_{\rm crit}$ and redshifts: the BAR model follows the DMO model for large scales, suffers a deficit in power at intermediate scales due to flatter gas profile compared to the NFW profile and finally the power shoots up due to the central stellar component. Also without adiabatic contraction (AC) the raise in the matter power spectrum occurs at very small scales, but including AC effect this raise can be seen at comparatively lower $k$ or larger scales. This is because AC makes the profile steeper in the centre and shallower in the outskirts.
At redshift 0 (top-left panel of figure \ref{fig:pkandcl}), the baryonic correction starts showing up (more than 1\%) at $k \sim 5\ h/Mpc$ for models with negligible AGN feedback (lower $M_{\rm crit}$), whereas for more extreme AGN feedback models (higher $M_{\rm crit}$) this correction is important at much larger scales like $k \sim 0.1\ h/Mpc$. In our fiducial BAR model with $M_{\rm crit} = 10^{13} h^{-1} M_{\bigodot}$, the baryonic effects become significant, i.e., more than 1 percent, at $k \sim 0.5\ h/Mpc$. The maximum dip in the intermediate scales vary for different $M_{\rm crit}$; for the most extreme models where AGN feedback can push all the gas out of the halo, this dip is nearly the cosmic baryon fraction, $\Omega_b/\Omega_m$. However, for more a realistic model ($M_{\rm crit} = 10^{13} h^{-1} M_{\bigodot}$) this dip is nearly 7-8\%. For more optimistic models like $M_{\rm crit} = 10^{12} h^{-1} M_{\bigodot}$, this dip is even smaller, nearly 4-5\%. Therefore, we can conclude the more extreme AGN feedback models triggers the deviation of matter power spectrum from DMO model at larger scales and also the dip in the power at intermediate scales can be as large as the cosmic baryon fraction in case where all the gas are pulled out by the AGN feedback, however, for more realistic and optimistic models, the deviation starts at relatively small scales and also the maximum dip is comparatively smaller.
Figure \ref{fig:pkandcl} (top-right panel) shows the same quantity for a fixed $M_{\rm crit} = 10^{13} h^{-1} M_{\bigodot}$ at different redshifts. If we go to higher redshift, the overall shape of the deviation of the BAR matter power spectrum from the prediction of the DMO model (boost) is nearly the same as at redshift zero, however, the scales and the maximum dip amplitude at various redshifts change. We see that at higher redshifts, the dip starts to trigger at larger scales and also the maximum dip converge to the cosmic baryon fraction.
In figure \ref{fig:pkandcl} (bottom-left panel), the baryonic correction to $C_{\ell}$ is shown in one big redshift bin ($z=0.01-4.0$). Here, the shear power spectrum starts to deviate from DMO predictions at about $\ell =100$ for the most extreme AGN feedback models and at $\ell$ of about several thousands for models with weak AGN feedback. For our realistic model (green curve), this deviation occurs at about $\ell \sim 700$. The maximum dip in power is very similar to that of the matter power spectrum explained above. It is worth noticing that for $\ell=10000$ the deviation is very significant for the realistic model ($M_{\rm crit} = 10^{13} h^{-1} M_{\bigodot}$), however, it is negligible for the optimistic model ($M_{\rm crit} = 10^{12} h^{-1} M_{\bigodot}$). Because these are the cases that we study in our likelihood analysis, we will show in section \ref{sec:cosmology} that this behaviour is consistent with the cosmological parameter estimation with these models.
\section{Fiducial model and mock datasets}
\label{sec:fiducial}
In this section, we would like to mention two factors that are quite important for our experiments - fiducial parameters and mock datasets. The fiducial parameters assumed in this work, particularly about cosmology, baryonic model and Euclid mission, are very standard. Also the mock datasets generated are correctly contaminated with random noise. Following are the key numbers and information about the fiducial model assumed and mock datasets:
\begin{enumerate}
\item We used WMAP - 5th year cosmology as our fiducial model with $[\Omega_m, \Omega_b, h, n_s, \sigma_8, w_0, w_a]$ as $[0.279, 0.0462, 0.701, 0.96, 0.817, -1.0, 0.0]$. We assume the equation of state of dark-energy is redshift dependent as \citep{2001IJMPD..10..213C,2003PhRvL..90i1301L},
\begin{equation}
w(a) = w_0 + (1-a)w_a
\end{equation}
\\
where, $a = 1/(1+z)$ is the scale factor at redshift $z$.
\item We used three redshift bins to do the tomographic analysis with boundaries $[0.01, 0.8, 1.5, 4.0]$. So we calculated a total of six spectra - three auto-spectra between bins 1-1, 2-2 and 3-3 and three cross-spectra between bins 1-2, 1-3 and 2-3.
\item We perform the likelihood analysis for different $\ell_{max}$ with $\ell_{min}=10$ and 100 equally spaced logarithmic bins. So the bin sizes for the likelihood analysis with different $\ell_{max}$ are different.
\item We assumed that the mean redshift of the source distribution to be nearly 1.0 which gives approximately 50 galaxies per arc min$^2$ and $f_{\rm sky}=0.55$ which resembles Euclid like survey.
\item For the baryonic model, we used the realistic AGN feedback model $M_{\rm crit}= 10^{13} h^{-1}M_{\bigodot}$ as the fiducial value for total nine $\ell_{max}$ (1000, 2000, 3000, 4000, 5000, 6000, 8000, 10000, 20000). We also performed one case with more optimistic model $M_{\rm crit} = 10^{12} h^{-1}M_{\bigodot}$ for $\ell_{max}=10000$. So there are ten cases in total.
\item We used our fiducial model stated above to generate shear power spectrum $C_{\ell}$ for these ten cases and perturbed all $C_{\ell}$ with normally distributed multi-variate random numbers drawn from a distribution with mean $C_{\ell}$ and the corresponding covariance matrix. These $C_{\ell}$ are catalogued and constitute the mock data sets. So, there are total ten mock data sets. In figure \ref{fig:bestfit} we show the mock datasets up to $\ell_{max}=20000$ for the six spectra and the best fits (which will be discussed in section \ref{subsec:goodness}).
\end{enumerate}
\begin{figure*}
\includegraphics[width=0.48\textwidth]{figures/bestfit_1-1.eps}
\includegraphics[width=0.48\textwidth]{figures/bestfit_1-2.eps}
\includegraphics[width=0.48\textwidth]{figures/bestfit_2-2.eps}
\includegraphics[width=0.48\textwidth]{figures/bestfit_1-3.eps}
\includegraphics[width=0.48\textwidth]{figures/bestfit_3-3.eps}
\includegraphics[width=0.48\textwidth]{figures/bestfit_2-3.eps}
\caption{Mock datasets (including random noise) for $\ell_{max}=20000$ in all six spectra (in black). The left column shows the three auto-spectra and the right column shows the three cross-spectra. Solid lines show the best fit for the DMO (red) and BAR (green) models.}
\label{fig:bestfit}
\end{figure*}
For each bin combination (1-1,1-2 etc), the length of the data vector ($\ell$ or $C_{\ell}$) is 100. Therefore, the total number of data points in each data set is 600. However, the two cross-spectra, 1-2 and 1-3, are highly correlated which actually leads us to have only 5 degree of freedom for each $\ell$. Therefore, the total number of degree of freedom in each data set is about 492 (500 - 8 free parameters). Hence, the best fit to each dataset can have a $\chi^2$ in the range 492 $\pm \sqrt{(2\times 492)}$ which is between 470 and 514.
In figure \ref{fig:pkandcl} (bottom-right panel), we show the boost for the unperturbed (without random noise) mock datasets up to very high $\ell_{max}$ with the corresponding DMO model. In all six curves of this figure, we kept $M_{\rm crit}=10^{13} M_{\bigodot}$. The auto-spectra in the first bin (1,1), starts deviating (more than 1\%) from the DMO model at about $\ell=300$ whereas the auto-spectra of the third bin (3,3) starts showing deviation at nearly $\ell=800$. All other auto-spectra and cross-spectra are between these two extremes. This behaviour is justified by looking at the same figure in upper-right panel, which shows the redshift evolution of the correction for the same $M_{\rm crit}$. It can be seen that at higher redshifts, the BAR matter power spectrum starts to deviate from DMO at smaller scales but also induces a larger dip at intermediate scales due to gas expulsion. This behaviour can be seen in the bottom-right panel. The $C_{\ell}$ in the lower redshift bin (1-1) starts deviating from DMO at larger scales as compared to the higher redshift bin (3-3), but the maximum dip in the two cases can be seen in the higher redshift bin (3-3). If we compare this to the bottom-left panel of the same figure, one can notice that the baryonic correction becomes even important when binning in redshift rather than using one big redshift bin. This provides additional constraints on $M_{\rm crit}$ while performing the analysis in tomographic bins compared to poorer constraints when only one bin is used.
\begin{figure*}
\centering
\includegraphics[width=0.8\textwidth]{figures/errors_euclid.eps}
\caption{Relative 1$\sigma$ errors on different cosmological parameters as a function of $\ell_{max}$ for $M_{\rm crit}=10^{13} h^{-1} M_{\bigodot}$. Solid lines are for the BAR model and dashed curves are for the DMO model. Horizontal black dashed lines mark the $\pm 1$ and vertical black dashed lines shows important scales.}
\label{fig:errors}
\end{figure*}
\section{Likelihood analysis and cosmological implications}
\label{sec:cosmology}
We performed a likelihood analysis using MCMC to explore the cosmological parameter space for nine different $\ell_{max}$ (1000, 2000, 3000, 4000, 5000, 6000, 8000, 10000, 20000) using $M_{\rm crit}=10^{13} h^{-1} M_{\bigodot}$, which is our most realistic model, and for $\ell_{max}=10000$ using $M_{\rm crit}=10^{12} h^{-1} M_{\bigodot}$ which is our optimistic model.
We run MCMC on the ten mock datasets obtained adopting both the DMO and BAR models, therefore we run a total of 20 MCMC. Each MCMC is performed using the publicly available code COSMOMC \citep{2002PhRvD..66j3511L}, with 16 chains in each case. So total 320 CPUs are used for nearly 10 days to reach the desired convergence. The whole analysis required about 76800 hours.
We demonstrate the results of the MCMC and the interpretation in the following two sections, targeting particularly the precision and accuracy in predicting the cosmological parameters.
\begin{figure*}
\centering
\includegraphics[width=0.48\textwidth]{figures/Om_s8.eps}
\includegraphics[width=0.48\textwidth]{figures/w0_wa.eps}\\
\includegraphics[width=0.48\textwidth]{figures/Om_s8_1d12.eps}
\includegraphics[width=0.48\textwidth]{figures/w0_wa_1d12.eps}
\caption{Top row: 1$\sigma$ 2D error ellipses for different cosmological parameters using mock datasets with $M_{\rm crit}=10^{13} h^{-1} M_{\bigodot}$ and different $\ell_{max}=$ 3000 (red), 5000 (blue), 8000 (green), 10000 (magenta), 20000 (cyan). Bottom row: 1$\sigma$ 2D error ellipses using mock datasets with $M_{\rm crit}=10^{12} h^{-1} M_{\bigodot}$ for $\ell_{max}=$ 10000. All solid curves are for the BAR model and dashed curves are for the DMO model.}
\label{fig:mcmc}
\end{figure*}
\subsection{Precision in cosmology}
Future experiments, like Euclid, are expected to provide very tight constraints on cosmological parameters. Here we show the constraints expected from using the weak lensing shear power spectrum as a function $\ell_{max}$. Figure \ref{fig:errors} shows the relative variance of four cosmological parameters and one baryonic parameter using both models, BAR (solid curves) and DMO (dashed curves). The matter density of the Universe ($\Omega_m$) and the amplitude of fluctuations ($\sigma_8$) are the most constrained parameters, however, other parameters like the equation-of-state of dark-energy today ($w_0$) are relatively less constrained. The overall behaviour of all parameters is the same, weak constraints for small $\ell_{max}$, better constraints with increasing $\ell_{max}$ and a flattening beyond $\ell_{max}\sim 8000$. The constraints derived from the BAR model are relatively weaker than the constraints derived from DMO model, which is the consequence of the extra parameter, $M_{\rm crit}$.
The normalized matter density of the Universe $\Omega_m$ can already be determined up to 5\% at $\ell_{max}$=1000 which improves as good as 2-3\% at $\ell_{max}$=8000 whereas the amplitude of fluctuations $\sigma_8$ can be determined much better at corresponding scales. At $\ell_{max}$=1000, $\sigma_8$ can be known up to 3\% and these constraints improves better than 1\% at $\ell_{max}$=8000. After $\ell_{max}$=8000, the variance of both the parameters remains the same and no further constraints can be drawn by going up to lower scales or higher $\ell_{max}$. There is a certain degeneracy in these two parameters which can be seen in figure \ref{fig:mcmc} upper-left panel, where different colours represent different $\ell_{max}$.
The constraints on the two parameters describing the redshift evolution of the equation of state of dark energy, $w_0, w_a$ can also be improved with this kind of experiments. At $\ell_{max}$=1000 $w_0$ can only be determined as good as 12\%, whereas for $\ell_{max}$=8000 it can be constrained up to 6-7\% and with the same precision for higher $\ell_{max}$. However, the constraints on $w_a$ are much weaker. The absolute error on $w_0$ is nearly 0.35 for $\ell_{max}$=1000, $\sim$0.18 for $\ell_{max}$ = 8000 and the same afterwards.
The flattening of the relative errors of the parameters indicates that there is no gain in precision of cosmological parameters estimation after a certain threshold $\ell_{max}\sim 8000$. In practice, an experiment like Euclid may provide us with very high quality data to even resolve and measure the shear power spectrum at $\ell_{max}$ as high as $10^5$, but our analysis shows that the constraints becomes constant after $\ell_{max}\sim 8000$ and no further improvement can be achieved.
This forecast suggests that by measuring $C_{\ell}$s up to $\ell_{max}\sim 8000$, one can constrain $\Omega_m$ to about 2\% precision and $\sigma_8$ to about 0.5\% precision without any loss of information from high $\ell$s and including baryonic physics. However, $w_0$ can only be constraints up to 6-7\% with some information about $w_a$, the time derivative of the equation of state of dark energy.
\begin{figure*}
\centering
\includegraphics[width=0.8\textwidth]{figures/bias_lmax.eps}
\caption{The ratio of bias and 1$\sigma$ error of various cosmological parameters as a function of $\ell_{max}$ for $M_{\rm crit}=10^{13} h^{-1} M_{\bigodot}$. Solid lines are for the BAR model and dashed curves are for the DMO model. Horizontal black dashed lines mark the $\pm 1$ and vertical black dashed lines shows important scales.}
\label{fig:bias}
\end{figure*}
\subsection{Accuracy in cosmology}
When precision cosmology is the goal, one should also take into account the ability to recover the cosmological parameter accurately. If there are systematic errors in the model, one can still derive very tight constraints from the wrong model, but the recovered parameters will be wrong or biased as compared to the true values. In this section we will present the results from our analysis of the bias in the cosmological parameters due to the lack of baryonic physics in DMO models and we will assess if these biases are significant. We define bias as the difference between the mean value of the parameter in MCMC and its fiducial or true value.
Figure \ref{fig:mcmc} shows the 1$\sigma$ error ellipses of cosmological parameters when the model is BAR (solid curves) and DMO (dashed curves). For small $\ell_{max}$, the two models are indistinguishable, a consequence of the fact that baryonic physics becomes more important only at smaller scales. But as we go higher and higher in $\ell_{max}$, the target density of the DMO model shifts further from the true target density, however, the BAR model remains at the correct location. We find that for all BAR models this bias is smaller than the 1$\sigma$ error of the parameter, however, the bias in the parameters obtained fitting for the DMO model increases with increasing $\ell_{max}$.
Figure \ref{fig:bias} shows the ratio of these biases and the 1$\sigma$ error on the cosmological parameters as a function of $\ell_{max}$ for the two models, BAR (solid curves) and DMO (dashed curves). The bias never exceeds the 1$\sigma$ error for the BAR models, however, it does for the DMO models only after $\ell_{max} \sim 4000$. This is again a consequence of the fact that baryonic physics is only important at smaller scales. This indicates that if we only perform our experiment up to $\ell_{max}$=4000, no baryonic physics needs to be taken into account, however, if one is interested in $\ell_{max}>4000$ baryonic physics becomes very important. After $\ell_{max}$=4000 the bias increases with $\ell_{max}$ and goes as big as 10$\sigma$ at $\ell_{max}$=10000 and remain flat after that. We see in the previous section that constraints on cosmological parameter can still be improved up to $\ell_{max}$=8000, but considering the wrong model, DMO, the cosmological parameters will be 5-10$\sigma$ away from the true values. So, in order to gain the best constraints on cosmology, baryonic physics must be taken into account.
\subsection{An optimistic model}
We analysed the $\ell_{max}=10000$ case for our optimistic model with $M_{\rm crit}=10^{12} h^{-1} M_{\bigodot}$. As in our previous analysis, we performed two MCMC in this case too, fitting for the BAR model and for the DMO model. Figure \ref{fig:mcmc} (bottom row) shows the 1$\sigma$ error ellipses of cosmological parameters. In this case the bias in the cosmological parameters does not exceed the $1\sigma$ error and hence is not a very troubling case. This was expected, as for lower $M_{\rm crit}$, baryonic physics is less important even at comparatively small scales as compared to cases where $M_{\rm crit}$ is higher. For example if we look at figure \ref{fig:pkandcl} (bottom-left panel), we can see that for $M_{\rm crit}=10^{12} h^{-1} M_{\bigodot}$, the deviation of $C_{\ell}$ from the DMO model is negligible at $\ell=10000$. Hence, we actually expect smaller or no bias.
\section{Discussion and conclusions}
\label{sec:discussion}
In this work we first review the important theoretical framework necessary to calculate the matter power spectrum using the halo model and to compute the shear angular power spectrum in different redshift bins. We presented an analytic prescription to distribute baryons into two components -- the intra-cluster plasma in hydrostatic equilibrium within the halo, and the BCG, which dominates the mass distribution in the centre of the halo, and whose properties are well measured using abundance matching techniques. We also take into account the adiabatic contraction of the dark matter particles due to the central condensation of baryons. We also compared these analytic density profiles to the simulations of \cite{2014MNRAS.440.2290M}, both dark-matter-only and baryonic with AGN feedback, and found a remarkable agreement.
We model the shear power spectrum in the two models, BAR and DMO, and found that baryonic corrections are important after $k \sim 0.5\ h/Mpc$ in the matter power spectrum at redshift 0 for our most realistic AGN feedback model, which translates into $\ell \sim 800$ for the shear power spectrum in one big redshift bin. However, if binned in redshift space (lensing tomography), these corrections become larger in each bin and for each auto- and cross-correlation function. These baryonic corrections have one free parameter, $M_{\rm crit}$, which regulates AGN feedback, i.e., it controls how much gas will be inside the halo as a function of the halo mass. We believe the most realistic value of this parameter is near $10^{13} h^{-1}M_{\bigodot}$, which sets the most likely magnitude of baryonic corrections.
We perform the likelihood analysis using MCMC for total ten different datasets. Nine of them assume our realistic model for the AGN feedback with $M_{\rm crit} = 10^{13} h^{-1}M_{\bigodot}$ but different $\ell_{max}$, and one assumes a less extreme (optimistic) model with $M_{\rm crit} = 10^{12} h^{-1}M_{\bigodot}$. For each mock dataset, we perform MCMC to fit for both models, BAR and DMO.
The main results of the likelihood analysis are summarized in figure \ref{fig:mcmc}, \ref{fig:errors} and \ref{fig:bias}. The results are very interesting in two aspects: first, we found that the constraints on all cosmological parameters improve with increasing $\ell_{max}$, but after $\ell_{max} \sim 8000$, the variance of each parameter becomes nearly constant. This indicates that even if we go to higher $\ell_{max}$ (or smaller scales), no additional constraints on the cosmological parameters can be gained. Second, if the wrong model, in this case DMO, is fitted to the data, after $\ell_{max}=4000$ the mean recovered value of the parameters starts moving away from its true value. We refer to the difference between the true value and recovered mean value as bias in the cosmological parameter. The bias in the parameters becomes more than 1$\sigma$ after $\ell_{max}=5000$ and goes up to 10$\sigma$ for $\ell_{max}=10000$, remaining flat afterwards. So, there is a very interesting window from $\ell_{max}=4000-8000$ which is useful for improving the constraints on cosmology, but if wrong model like DMO is chosen, the recovered cosmology can be highly biased from few to 10-$\sigma$.
\subsection{Goodness of fit}
In the previous sections we see that for $\ell_{max}<4000$, there is no significant bias added to the determination of the cosmological parameters in our analysis, however, for $\ell_{max}>5000$ the bias exceeds 1$\sigma$ and keep increasing up to 10$\sigma$ with increasing $\ell_{max}$. The question here is: can we discard these biased models by looking at the goodness of fit? The answer to this question lies in figure \ref{fig:chi2} where we show the ratio between the best fit $\chi^2$ in the DMO model and that in the BAR case as a function of $\ell_{max}$. This ratio is as little as 5-10\% up to $\ell_{max} \sim 5000$ but after that it only goes up to 25\% at $\ell_{max}=20000$ where bias is more than 10$\sigma$. Now, the reduced $\chi^2=1.25$ does not appear as such a bad fit for our cosmological measurements. So, by looking at the $\chi^2$ only, it is not really possible to discard a model. The same conclusion can be drawn from figure \ref{fig:bestfit}, where we show the mock datasets of the six spectra (between different bins) for $\ell_{max}=20000$. In this figure, we also show the two best fit from the DMO model (in red) and the BAR model (green). As we expect, the green curve is a better fit to the data than the red curve. But if the green curve is not present in this figure, the red curve does not appear to be a very bad fit. So, when deriving constraints on cosmology from this kind of experiments, one should be extremely careful about the possible magnitude of baryonic effects at small scales, because, although the results obtained with the wrong model may appear as a good fit, the corresponding bias can be in fact as high as many $\sigma$. Also, the recovered parameters from the wrong model (DMO) move away from the true value with increasing $\ell_{max}$. This suggests a potential test for a given model, the cosmological parameter space should not move significantly when analysing up to different scales, the difference should only be seen in the variance of the parameters and not in its mean value.
\subsection{$M_{\rm crit}$ parameter}
The only free parameter in our BAR model, $M_{\rm crit}$, regulates the amount of gas inside the halo as a function of halo mass. We explore the consequences of what we believe to be a realistic model ($M_{\rm crit} = 10^{13} h^{-1}M_{\bigodot}$) in details considering nine different $\ell_{max}$. At $\ell_{max}=1000$, there is hardly any constraint drawn from weak lensing on this parameter, but as we increase $\ell_{max}$, baryonic physics become more and more important and thus constraints can be put on $M_{\rm crit}$. In fact, the constraints on this parameter increase rapidly from 15\% at $\ell_{max}=1000$ to 1-2\% at $\ell_{max}=4000$. After this, no significant improvement on the constraints can be gained on this parameter. The variance of $M_{\rm crit}$ becomes constant after nearly $\ell_{max}=8000$, which is what happens for the other cosmological parameters. So, with this kind of weak lensing experiment, $M_{\rm crit}$ (or $\log(M_{\rm crit})$) could be constrained up to 1-2\%, which is quite impressive.
\label{subsec:goodness}
\begin{figure}
\includegraphics[width=0.48\textwidth]{figures/chi2.eps}
\caption{Showing the ratio of the best fit $\chi^2$ in DMO model and BAR model at different $\ell_{max}$.}
\label{fig:chi2}
\end{figure}
\subsection{Non-Gaussian covariance vs baryonic corrections}
\label{sec:NG}
Being able to extract cosmological information from clustering data down to a few percent accuracy can be considered very optimistic. It can be jeopardized by many unresolved issues. The two most important issues are (i) baryonic physics at small scales, and (ii) non-Gaussian effects in the covariance matrix of the power spectrum. These two issues can be quantified in projected weak-lensing statistics, like the shear power spectrum. In this work, we primarily talk about the effect of baryonic physics at small scales on the shear power spectrum and its cosmological implications. However, we ignore the effect of non-Gaussianity (NG) on the covariance matrix.
The NG contribution to the covariance becomes more important at small scales, like baryonic physics \citep{2009MNRAS.395.2065T, 2013PhRvD..87l3504T}. Now the question is, which one is more important to deal with and which one appears first when going towards smaller scales? This question does not have a very straightforward answer. Ignoring both of these contributions may result in highly biased cosmological parameters estimations.
\cite{2012PhRvD..86h3504Y} (figure 9, right panel) shows the constraints on the amplitude of fluctuations ($\sigma_8$) as we go to smaller scales. If one considers only Gaussian errors, the constraints continue to improve until the instrumental shot noise kicks in. However, NG contribution are likely to dominate over Gaussian errors after $\ell=700$. But we cannot directly compare to this plots as the constraints depend on many other details. We can still compare the ratios of the NG and Gaussian contributions. At $\ell_{max}=10000$, the NG covariance is six times the Gaussian covariance. On the other hand, in figure \ref{fig:bias} the bias in cosmology becomes close to 10$\sigma$ for $\sigma_8$ at $\ell=10000$. This means that the NG corrections are sub-dominant than the baryonic effects. However, our analogy is very hand-wavy and requires further study.
\subsection{The ideal configuration}
We explore the baryonic effects on the cosmological parameter estimation and found big bias in cosmological parameters if the analysis include $\ell>4000$. After this limit, the cosmological parameters start to become biased and mislead the constraints. However, the constraints keep improving up to $\ell=10000$. So the question arises, what is the ideal configuration to perform weak-lensing power spectrum analysis to put useful constraints on cosmology with Euclid-like surveys?
We explore this answer in our analysis and stated our results in the previous sections. To summarize, the ideal configuration is to go as high as $\ell=8000$, including baryonic physics and marginalize over the baryonic parameters, in our case $M_{\rm crit}$. In this configuration, one can find unbiased estimates of the cosmological parameters. Having the unbiased estimates, we can also constrain the cosmological parameter space with much better accuracy than before. In this configuration, $\Omega_m$ and $\sigma_8$ can be estimated with nearly 2\% and 0.5\% respectively. The variance of the two parameters defining the redshift evolution of the equation of state of dark energy, $w_0$ and $w_a$ are 0.07 and 0.15 respectively. Along with cosmological parameters, the baryonic parameter $M_{\rm crit}$ can also be estimated to very high accuracy, as good as 1-2\%.
When dealing with real clustering datasets, we are also able to use independent constraints on the baryonic parameters, such as abundance matching data and/or X-ray data on individual halos, providing a solid understanding of the overall signal and the underlying baryonic effects.
\section{Acknowledgement}
I.M. would like to thank Prasenjit Saha, Uros Seljak and Ravi Sheth for useful discussions about the topic and their suggestions.
\bibliographystyle{mn2e}
\defApJ{ApJ}
\defApJL{ApJL}
\defApJS{ApJS}
\defAJ{AJ}
\defPRD{PRD}
\defMNRAS{MNRAS}
\defA\&A{A\&A}
\defPhysicsReports{PhysicsReports}
\defNature{Nature}
\defARAA{ARAA}
|
\section{Introduction}
The first detection of a baryon $\Xi^{++}_{cc}$ containing two charm quarks (i.e., having the structure {\it ccu}) has been made at CERN by LHCb collaboration last summer \cite{01}. Its mass was measured to be $3621$ MeV. The existence of such a baryon may be considered is an inevitable consequence of the existence of the $c$-quark itself. Calculations of the $\Xi^{++}_{cc}$-baryon mass have started almost thirty years ago \cite{02,03,04}. Probably the first review paper on this subject dates back to 2002 \cite{05}. A reliable prediction on the double charmed baryon mass and properties turned out to be a difficult talk -- see a list of references in \cite{01}. This is not surprising since the theoretical description of the QCD ``hydrogen atom'' -- charmonium, to a great extent relies on the phenomenological input rather than on the first principles of QCD. Connection between different approaches to the double charmed baryons, e.g., the quark-diquark model, the potential model, and the QCD sum rules, their interreletion and relation to the fundamental QCD Lagrangian are obscure. The aim of the present note is to investigate the $\Xi^{++}_{cc}$ mass and the structure of the wave function taking our works \cite{03,04} as a starting point for a discussion. In \cite{03,04} the spin-averaged masses and the wave functions of multiquark systems made up to 12 quarks were calculated in potential model through Green Function Monte Carlo, or random walks method -- see below.
Despite the fact that QCD gives no sound arguments in favor of the effective potential with only two-body forces, the constituent quark model has given results which are in surprisingly good agreement with experimental hadron spectroscopy. Attempts to apply this approach to multiquark systems encounter serious difficulties even if one ignores problems like complicated QCD structure in the infrared region. With growing number of particles, especially with non-equal masses, and for the complicated form of the potential, the traditional methods -- variational, integral equations, hyper-spherical functions, face difficulties. The accuracy, in particular that of the many-body wave function determination, becomes uncontrollable and the computation time increases catastrophically. This is true even for the three quark system. The convergence for the wave function is much slower than for the binding energy both in the harmonic oscillation expansion and in the hyperspherical formalism \cite{02}. In \cite{03,04} the spectrum and the wave functions of multiquark systems were investigated using the Green Function Monte Carlo method (GFMC). The GFMC is based on the idea attributed to Fermi that the imaginary time Schrodinger equation is equivalent to the diffusion equation with branching (sources and sinks). {The GFMC allows to calculate easily and with high accuracy the spectrum and, what is most important, the wave function of a multiquark system in various models.} Originally GFMC has been used to calculate the ground-state properties of a variety of systems in statistical and atomic physics \cite{06}. At present it is the most powerful method to solve the many-body problem \cite{07}. Similar methods have been used in Quantum Field Theory under the names Projector Monte Carlo \cite{08} and Guided Random Walks \cite{09}.
The description of the GFMC is beyond the scope of this paper. We only stress that the method does not require to solve differential or integral equations for the wave function. There is no need even to write down such equations. In GFMC the exact many-body Schrodinger equation is represented by a random walk in the many-dimensional space in such a way that physical averages are exactly calculated given sufficient computational resources. {At this point we emphasize the need to disentangle the accuracy of computations from possibly large uncertainty related to the use of a particular model.} When we tried to apply GFMC to multiquark system, i.e., to a system of fermions, we encounter a difficulty. For the fermionic system the kernel of the Green function may take a negative value at a certain step of the sampling procedure. A recipe to circumvent this difficulty was proposed in \cite{03}. We note in passing that similar problem does not allow to perform lattice Monte Carlo investigation of quark matter properties at finite density.
Now we come to the description of the interquark interaction chosen for the calculation of the ground state mass of \textit{(ccu)} system and evaluation of the wave function. For a system of $N$ quarks the interaction between quarks is taken in the form
\begin{equation}
V_{ij}(r_{ij})=\lambda_i\lambda_j V_{8}(r_{ij}),
\label{eq:01}
\end{equation}
{
\noindent with $\lambda_i$ being the Gell-Mann matrices. Solving the quark problem with $N \geq 4$ with the interaction (\ref{eq:01}) one encounters a problem that the coupling of colors of the constituents into a color singlet is not unique -- see \cite{04} for a discussion. For $V_{8}(r)$ we have taken the well-known Cornell potential \cite{10}
\begin{equation} V_{8}(r)=-\frac{3}{16}\left(-\frac{\varkappa}{r}+\frac{r}{a^2}+C_f \right), \label{eq:02} \end{equation}
where $\varkappa=0.52$ and $a=2.34$ GeV$^{-1}$ which corresponds to the string tension $\sigma = a^{-2} = 0.18$ GeV$^2$. For the baryon $\left<\lambda_i\lambda_j\right>=-8/3$. We would like to present two more arguments in addition to the well known ones \cite{10} in support of the Cornell potential. Slightly varying its parameters one may obtain a good fit of the lattice simulation of the quark-antiquark static potential \cite{11}. Cornell like behavior arises also from AdS/QFT correspondence \cite{12}.}
In Ref. \cite{13} an excellent fit of both meson and baryon sectors has been obtained under the assumption that the constant $C_f$ is weakly flavor dependent. Following \cite{13} we have chosen the input parameters for \textit{(ccu)} baryon (all values in GeV units)
\begin{equation}
m_u=0.33,\,\,\,m_c=1.84,\,\,\,C_{uc}=-0.92.
\label{eq:03}
\end{equation}
The value of $m_c$ might look too high but it corresponds to the Classical Cornell set of parameters \cite{10}.
Next one has to evaluate the contribution from spin-spin splitting \cite{14}
\begin{equation}
V_{s_i s_j} = \frac{16\pi}{9}\,\alpha_S\,\frac{\bm{s}_i\bm{s}_j}{m_i m_j}\,\delta^{(3)}(\bm{r}_{ij}).
\label{eq:04}
\end{equation}
We have calculated the ground-state expectation values $\delta_{ij} = \left<\delta(\bm{r}_{ij})\right>$, where $\left<\, \ldots \,\right>$ means the average over the ground-state wave function. To obtain $\delta_{ij}$ we made a smearing over the small sphere around the origin and then averaged over a sequence of such spheres. For \textit{(ccu)} baryon we obtain the following results for $10^3\,\delta_{ij}$ (GeV$^3$)
\begin{equation}
\delta_{cc}=45.25 \pm 2.90,\quad \delta_{cu}=11.28 \pm 1.94.
\label{eq:05}
\end{equation}
The GFMC method allows to obtain the wave function and its arbitrary moments with the accuracy restricted only by the computational resources. In particular, there is no need to introduce the quark-diquark structure as a forced anzatz. The system will form such configuration by itself if it corresponds to the physical picture. This turned out to be the case for \textit{(ccu)} baryon. Indeed, the ground-state expectation values of $\left< r_{ij}^{2} \right>^{1/2}$ in GeV$^{-1}$ are
\begin{equation}
\left< r_{cc}^{2} \right>^{1/2} = 2.322\pm 0.024,\quad
\left< r_{cu}^{2} \right>^{1/2} = 3.407 \pm 0.035.
\label{eq:06}
\end{equation}
If we identify $\left< r_{cc}^{2} \right>^{1/2} \simeq 0.46$ fm with the size of a diquark, then it is more compact one than a diquark with $r_d=0.6$ fm introduced ad hoc in \cite{15}. {It is instructive to look at the interquark distances (\ref{eq:06}) from an angle of the $\Xi^{++}_{cc} - \Xi^{+}_{cc}$ isospin splitting. This is a long standing puzzle. More than a decade ago SELEX Collaboration reported \cite{16} the observation of the $\Xi^{+}_{cc}$ \textit{ccd} baryon with a mass $3519$ MeV. However, this result was not confirmed by other experiments (see \cite{01} for references). The isospin splitting of about $100$ MeV between \textit{ccu} and \textit{ccd} states and its sign are hardly possible to explain. The $d$-quark is heavier than the $u$-quark and to overcome the ``wrong'' sign of the splitting by about $100$ MeV the electromagnetic mass difference should be very large. This is turn requires the $\Xi^{++}_{cc}$ baryon to be very compact \cite{17}. To obtain $9$ MeV splitting the radius should satisfy $\sqrt{\left< r^2 \right>} < 0.26$ fm \cite{17}. From (\ref{eq:06}) we see that the diquark size is $0.46$ fm. If we identify the baryon radius with three quarks hyperradius $\rho = \sqrt{\bm{\eta}^2 + \bm{\xi}^2}$ with $\bm{\eta}$ and $\bm{\xi}$ being the Jacobi coordinates \cite{02}, the result is $\rho=(0.53-0.57)$ fm depending on the value of the Delves angle $\tan \varphi = {\xi}/{\rho}$. Therefore our result on the $\Xi^{++}_{cc}$ wave function strongly contradicts the abnormal value and the sign of the conjectural isospin splitting.}
{
Our result for the center of gravity (spin-averaged) mass $\Xi^{++}_{cc}$ baryon is \begin{equation} m\left[ \Xi^{++}_{cc} \right] = 3632.8 \pm 2.4 \text{ MeV}.\label{eq:07} \end{equation}
The error characterizes the accuracy of the GFMC calculations with the Cornell potential parameters specified above. We did not vary these parameters since they were fitted to a great body of observables. Next we take into account the hyperfine interaction (\ref{eq:04}) which induces the splitting between the lowest state with $S=1/2$ and its $S=3/2$ partner.
Taking into account that statistics requires the \textit{cc} pair to be in a spin $1$ state, we can write the following equation for the hyperfine energy shift
\begin{equation}
\Delta E_{hf} = \frac{4\pi}{9}\,\alpha_S\,\left\{\frac{\delta_{cc}}{m_{c}^2} + \frac{2\delta_{cu}}{m_{c}m_{u}}\left[ S(S+1) - \frac{11}{4}\right]\right\}.
\label{eq:08}
\end{equation}
We used the Cornell value of $\alpha_S=\frac34\,\varkappa=0.39.$ With this value of $\alpha_S$ baryon magnetic moments were successfully described \cite{18}. Equation (\ref{eq:08}) yields
\begin{equation} \Delta E_{hf}\left(S=1/2\right)=-32.2\text{ MeV}, \qquad \Delta E_{hf}\left(S=3/2\right)=26.9\text{ MeV}. \label{eq:09} \end{equation} and \begin{equation}
m\left[ \Xi^{1/2\,++}_{cc} \right] = 3601 \text{ MeV},\qquad m\left[ \Xi^{3/2\,++}_{cc} \right] = 3660 \text{ MeV}.\label{eq:10} \end{equation}
In (\ref{eq:10}) the uncertainty of GFMC calculations which are about $(2-3)$ MeV are not presented since as repeatedly stated above the model-dependent theoretical uncertainty may be much larger as one can see from theoretical predictions presented in the reference list in \cite{01}.
Another doubly charmed baryon which may be observed soon is $\Omega_{cc}$ with quark content {(ccs)}. Our result for its c.o.g. is \begin{equation} m\left[ \Omega_{cc} \right] = 3760.7 \pm 2.4 \text{ MeV}.\label{eq:11} \end{equation}
}
\noindent\makebox[\linewidth]{\resizebox{0.4\linewidth}{1.2pt}{$\bullet$}}\bigskip
The author is grateful to Yu.S.~Kalashnikova for enlightening discussions, to M.~Karliner, V.~Novikov, J.-M.~Richard, and M.I.~Vysotsky for questions and remarks. The interest to the work fron V.Yu.~Egorychev is gratefully acknowledged. The work was supported by the grant from the Russian Science Foundation project number \#16-12-10414.
|
\section{Introduction}
In the context of complex networks, the Laplacian formalism can be used to find many useful properties of the underlying graph
\cite{MERRIS1994143, Chung97, benzi, biggs93, MR1324340, Mitrovic2009}.
In particular, the idea of spectral clustering is to extract some important information on the network structure from the
matrices associated with the network, by considering one or few of the leading eigenvectors \cite{Boccaletti2006175}. \\
According to the Fiedler theory, a bipartition of a graph can be obtained from the second eigenvector both of the
Laplacian matrix \cite{fiedler73, fiedler75, donath1973lower}, and of the Normalized Laplacian matrix \cite{Chung97}.
More precisely, one can obtain a good ratio cut of the graph from any vector orthogonal to the all-ones vector, with a small Rayleigh quotient \cite{conf/focs/Mihail89}.\\
In general, a different number of clusters can obtained by means of the following strategies:
\begin{description}
\item[a)] by a Recursive Spectral Bisection (RSB) \cite{journals/concurrency/BarnardS94, I91partitioningof, CPE:CPE4330070103}: after using the Fiedler
eigenvector to split the graph into two subgraphs, one can find the Fiedler eigenvector in each of these subgraphs, and continue recursively until
some a-priori criterion is satisfied;
\item[b)] by using the first $k$ eigenvectors related to the smallest eigenvalues, to induce further partitions through clustering algorithms applied
to the corresponding invariant subspace \cite{journals/dam/AlpertKY99, journals/tcad/ChanSZ94}.
\end{description}
We consider the second approach, recalling that the optimal number $k$ of clusters is often indicated by a large gap between the $k$ and the $k+1$ eigenvalues
for both the Laplacian and Normalized Laplacian matrices \cite{Lee:2012:MSP:2213977.2214078}.\\
Within this framework, we are interested consider the algebraic multiplicity of Laplacian eigenvalues, since the corresponding eigenvectors
can be considered equivalent in a partition procedure of graphs. In presence of multiple eigenvalues, we investigate the possibility of reducing the dimensionality
of the original graph (i.e. of removing some of its nodes) keeping fixed its spectral properties
\cite{onred11, DBLP:conf/cdc/BeckLLW09, DBLP:conf/cdc/DengMM09,sonin1999state, Sadiq:2000:APM:344358.344369}.
After some preliminary remarks (section \ref{sec:2}), in section \ref{sec:3} we define two classes of graphs, by giving conditions on the graph structure
which implies the presence of multiple eigenvalues. Then we propose a reduction on the number of nodes, such that it is possible to get an identical spectrum
for the Laplacian matrices of the original and the reduced graphs (up to the eigenvalue multiplicity) with respect to a suitable diagonal
\textit{mass matrix}, that changes the link weights a plays the role of metric matrix.
Furthermore, we get a connection between the primary and the reduced graph eigenvectors.
Thanks to these results it is possible to perform a partition of the primary and the reduced graphs using the same procedure.
Finally, in section \ref{sec:4} we draw some conclusions and give an outlooks on future developments.\\
\section{Premises}\label{sec:2}
We consider an undirected weighted connected graph $\mathcal G:=(\mathcal V, \mathcal E, w)$, where the $n$ vertices $\mathcal V$ are connected by
the $\mathcal E$ edges with $w$ the weight function: $w:\mathcal E\rightarrow \mathbb R^{+}.$
Let $A$ be the weighted adjacency matrix, which is symmetric since the graph is undirected ($A\in Sym_n(\mathbb R^+)$),
$$A_{ij} = \begin{cases} w(i,j), & \mbox{if $i$ is connected to $j$ } (i\sim j) \\
0 & \mbox{otherwise } \end{cases}$$
where $i,j\in\mathcal V,$,
the Laplacian matrix $L\in Sym_n (\mathbb R)$ and normalized Laplacian matrix $\hat L\in Sym_n (\mathbb R)$ are respectively defined
$$L_{ij} = \begin{cases} -w(i,j), & \mbox{if } i\sim j \\
\sum_{k=1}^n w(i,k), & \mbox{if } i= j\\
0 & \mbox{otherwise } \end{cases}$$
$$\hat L_{ij} = \begin{cases} -\displaystyle \frac{w(i,j)}{\sqrt{\sum_{k=1}^n w(i,k)\sum_{k=1}^n w(k,j)}}, & \mbox{if } i\sim j \\
1, & \mbox{if } i= j\\
0 & \mbox{otherwise }. \end{cases}$$
Whenever we refer to the $k$-th eigenvalue of a Laplacian matrix, we will refer to the $k$-th nonzero eigenvalue according to a increasing order.
For the classical results on Laplacian matrices theory, one may refer to \cite{Chung97,doi:10.1080/03081088508817681, MERRIS1994143}.
\section{Eigenvalues multiplicity theorems}\label{sec:3}
The first result is an extension of Theorem {(4)} in \cite{grone94} to weighted graphs: by defining the weighted $(m,k)$-stars in a graph,
we are able to give a condition on both the structure and edge weights of graphs in order to get the eigenvalue multiplicity.
{As we will see later, an $(m,k)-$star is nothing else that the union of a $k$-cluster of order $m$ and its $k$ neighbours.}\\
The second result, that is the main results of this work, is a further extension of the previous Theorem to understand the relation between
eigenvalue multiplicity and the structure of the weights of graphs.\\
The third result concerns the reduction of graphs with one or more $(m,k)$-stars under some conditions, and possible applications
on spectral graphs partitioning.
\subsection{$(m,k)$-star and $l$-dependent: eigenvalues multiplicity}
We recall that a vertex of a graph is said \textit{pendant} if it has exactly one neighbour, and \textit{quasi pendant}
if it is adjacent to a pendant vertex. It is possible to prove that the multiplicity $m_{L}(1)$ of the eigenvalue $\lambda=1$ of the Laplacian of an
unweighted graph, is greater or equal than the number of pendant vertices less the number of quasi pendant vertices of the graph \cite{FARIA1985255}.\\
To extend these definitions to vertices with $k$ neighbours, we define a $(m,k)$-star:
\begin{figure}[!!h]
\begin{subfigure}{}
\includegraphics[width=6cm]{S63.png}\end{subfigure}
\begin{subfigure}{}
\includegraphics[width=6cm]{S36_2.png}\end{subfigure}
\caption{In the left a $S_{6,3}$ graph, in the right a $S_{3,6}$ graph.}
\end{figure}
\begin{defn}[$(m,k)$-star: $S_{m,k}$ ]
A $(m,k)$-star is a graph $\mathcal G=(\mathcal V, \mathcal E,w)$ whose vertex set $\mathcal V$ has a bipartition $(\mathcal V_1,\mathcal V_2)$
of cardinalities $m$ and $k$ respectively, such that the vertices in $\mathcal V_1$ have no connections among them, and each of these vertices is connected
with all the vertices in $\mathcal V_2$: i.e
$$\forall i\in \mathcal V_1,\forall j\in \mathcal V_2,\quad (i,j)\in \mathcal E$$
$$\forall i,j\in \mathcal V_1, \quad (i,j)\notin \mathcal E$$
We denote a $(m,k)$-star graph with partitions of cardinatilty $|\mathcal V_1|=m$ and $|\mathcal V_2|=k$ by $S_{m,k}.$
\end{defn}
We define a \textit{$(m,k)$-star of a graph} $\mathcal G=(\mathcal V, \mathcal E,w)$ as the $(m,k)$-star of partitions $\mathcal V_1$, $\mathcal V_2\subset \mathcal V$,
both of them univocally determined, such that the vertices in $\mathcal V_1$ have no connection with vertices
in $\mathcal V\setminus \mathcal V_2$ in $\mathcal G., $: i.e.\\
$$\forall i\in \mathcal V_1,\forall j\in \mathcal V_2,\quad (i,j)\in \mathcal E$$
$$\forall i \in \mathcal V_1, \forall j\in \mathcal V\setminus\mathcal V_2 \quad (i,j)\notin \mathcal E$$
{ \begin{obs} In \cite{grone94} is defined a $k$-cluster of $\mathcal G$ to be an independent set of $m$ vertices of $\mathcal G$, $m>1$,
each of which with the same set of neighbours.
The order of a $k$-cluster is the number of vertices in $k$-cluster.
Therefore, the set $\mathcal V_1$ of the $(m,k)$-star is a $k$-cluster of order $m$ and the set $\mathcal V_2$ is the set of the $k$ neighbour vertices.
An $(m,k)-$star of a graph $\mathcal G$ is the union of a $k$-cluster (i.e. $\mathcal V_1$) and its neighbour vertices (i.e. $\mathcal V_2$).
\end{obs}}
By defining the degree and weight of a $(m,k)$-star we simplify the stating of the theorems on eigenvalues multiplicity.
\begin{defn}[Degree of a $(m,k)$-star: $deg(S_{m,k})$]
The \textit{degree} of a $(m,k)$- star is $deg(S_{m,k}):=m-1$ and the degree of a set $\mathcal S$ of $(m,k)$-stars, as $m$ and $k$ vary in $\mathbb N$ ,
such that $|\mathcal S|=l,$ is defined as the sum over each $(m,k)$-star degree, i.e.
$$deg(\mathcal S):=\sum_{i=1}^l deg(S_{m_i,k_i}).$$
\end{defn}
\begin{defn}[Weight of a $(m,k)$-star: $w(S_{m,k})$]
The \textit{weight} of a $(m,k)$-star of vertices set $\mathcal V_1\cup\mathcal V_2$ is defined as the strength of the vertices in $\mathcal V_1$,
provided that the following condition holds:\\
let $\{i_1,...,i_m\}=\mathcal V_1$, then $w(i_1,j)=...=w(i_m,j), \forall j\in\mathcal V_2.$. More precisely the
weight of a $(m,k)$-star: $w(S_{m,k})$ is $$w(S_{m,k}):=\sum_{j\in \mathcal V_2}w(i,j)\mbox{ for any }i\in\mathcal V_1.$$
+\end{defn}
We are ready to enunciate the first theorem, that is an extension to weighted graph of the theorem in \cite{grone94}.
Given a graph $\mathcal G=(\mathcal V,\mathcal E,w)$ associated with the Laplacian matrix $L$, and denoting $\sigma(L)$ the set of the
eigenvalues of $L$ and $m_L(\lambda)$ the algebraic multiplicity of the eigenvalue $\lambda$ in $L$, the following theorem holds\\
\begin{thm}\label{th:one}
Let
\begin{itemize}
\item $s$ be the number of all the $S_{m,k}$ as $m$ and $k$ vary in $\mathbb N$ and $m+k\leq n,$ of $\mathcal G$;
\item $r$ be the number of $S_{m,k}$ with different weight, $w_1,...,w_r$, i.e. $w_i\neq w_j$ for each $i\neq j,$ where $ i,j\in\{1,...,r\};$\\
\end{itemize}
then for any $i\in\{1,...,r\},$
$$\exists \lambda\in{\sigma(L)} \mbox{ such that } \lambda=w_i \mbox{ and } m_{L}(\lambda)\geq deg(\mathcal S_{w_i})$$
where $\mathcal S_{w_i}:=\{S_{m,k}\in \mathcal G | w(S_{m,k})=w_i\}$.
\end{thm}
{
Before proving Theorem \ref{th:one}, we introduce some useful definitions.
\begin{defn}[$k$-pendant vertex]
A vertex of a graph is said to be \textit{$k$-pendant} if its neighborhood contains exactly $k$ vertices.
\end{defn}
\begin{defn}[$k$-quasi pendant vertex]
A vertex of a graph is said to be \textit{$k$-quasi pendant} if it is adjacent to a $k$-pendant vertex.
\end{defn}
We remark that in the definition of an $(m,k)-$star, the vertices in $\mathcal V_1$ are $k-$pendant vertices, and
vertices in $\mathcal V_2$ are $k-$quasi pendant vertices.\\
\begin{proof}\ref{th:one} \\
We consider connected graphs; indeed if a graph is not connected the same result holds, since the $(m,k)$-star degree
of the graph is the sum of the star degrees of the connected components and
the characteristic polynomial of $L$ is the product of the characteristic polynomials of the connected components. \\
Let a $(m,k)$-star of the graph $\mathcal G$.\\
Under a suitable permutation of the rows and columns of weighted adjacency matrix $A$, we can label the $k$-pendant vertices with the indices $1,...,m$,
and with $m+1,...,m+k$ the indices of the $k$-quasi pendant vertices.\\
We call $v_1,...,v_m$ the rows corresponding to $k$-pendant vertices, then the adjacency matrix has the following form
\[ A = \left( \begin{array}{ccc|ccccccc}
0 & ... & 0 & w(1,m+1) & w(1,m+2) &...& w(1,m+k)& 0&...&0\\
\vdots & ... & \vdots & \vdots & \vdots &...& \vdots & 0&...&0\\
0 & ... & 0 & w(m,m+1) & w(m,m+2) &...& w(m,m+k)& 0&...&0\\
\hline
w(1,m+1)& ... & w(m,m+1) & & & & & & &\\
\vdots & ... & \vdots & & & & & & &\\
w(1,m+k)& ... & w(m,m+k) & & & & & & & \\
0& ... & 0 & & & & & & & \\
\vdots& ... & \vdots & & & A_{22} & & & \\
0& ... & 0 & & & & & & & \\
\end{array} \right)\]
where the block $A_{22}$ is any $(n-m)\times(n-m)$ symmetric matrix.\\
The $m$ rows (and $m$ columns) $v_1,...,v_m$ are linearly dependent such that $v_1=...=v_m$, then $v_1,...,v_{m-1}\in ker(A)$.\\
Hence
$$\exists \mu_1,...,\mu_{m-1}\in\sigma(A)\quad \mbox{ such that }\quad \mu_1=...=\mu_{m-1}=0.$$
By considering the Laplacian matrix $L$, it has at least $m$ diagonal entries with value $\sum_{j=1}^k w(1,m+j)=w(S_{m,k}):=w_1$.\\
Then also in the matrix $(L-w_1 I)$ there are the linearly dependent vectors $v_i, \ i\in\{1,...,m\}$, hence $v_1,...,v_{m-1}\in ker(L-w_1 I)$ and
$$\exists \mu_1,...,\mu_{m-1}\in\sigma(L-w_1 I)\quad \mbox{ such that }\quad \mu_1=...=\mu_{m-1}=0.$$
Let $\mu_i$ be one of these eigenvalues, then
$$0=det((L-w_1 I)-\mu_i I)=det(L-(w_1 +\mu_i )I)$$
so that $\lambda:=w_1\in\sigma (L)$ with multiplicity greater or equal to $deg(S_{m,k})$.\\
Let us now consider a number $s$ of $S_{m,k}$ in $\mathcal G$, namely $S_{m_1,k_1},...,S_{m_s,k_s}$.
Denoting $w_1,...,w_r$ the different weights of such a $(m,k)$-stars, and $r\leq s$,
we prove that for any $i\in\{1,...,r\},$
$$\exists \lambda\in{\sigma(L)} \mbox{ such that } \lambda=w_i \mbox{ and the multiplicity of } \lambda\geq deg(\mathcal S_{w_i})=
\sum_{S_{m_j,k_j}\in\mathcal S_{w_i}} deg(S_{m_j,k_j}),$$
where $\mathcal S_{w_i}:=\{S_{m,k}\in \mathcal G | w(S_{m,k})=w_i\}$.
Let $i\in\{1,...,r\}$ and let $R_i\leq r$ be the number of $(m,k)$-stars in $\mathcal S_{w_i}$, and $\sum_{i=1}^r R_r=s$,
we assume that the first $R_1$ indexes refer to the $(m,k)$-stars in $\mathcal S_{w_1}$, whereas the indexes $R_1+1,...,R_1+R_2$ refer to the $(m,k)$-stars in
$\mathcal S_{w_2}$, and so on.
We focus on the $R_i$ $(m,k)$-stars in $\mathcal S_{w_i}$.
The rows in $A$ corresponding to the $k_j$-pendant vertices with$j\in\{\sum_{q=1}^{i-1} R_q+1,...,\sum_{q=1}^{i} R_q\}$,
are $m_j$ vectors $(v^{(j)}_{j_1},...,v^{(j)}_{j_{m_j}})$, linearly dependent and such that $v^{(j)}_{j_1}=...=v^{(j)}_{j_{m_j}}$,
whose indexes are
$$j_1=\sum_{p=1}^{j-1} m_{p}+1,...,{j_{m_j}}=\sum_{p=1}^{j-1} m_{p}+m_j$$
when $j>1$, or
$$j_1=1,...,{j_{m_j}}=m_j$$
when $j=1$.\\
Then we get
$$v^{(j)}_{j_1},...,v^{(j)}_{j_{{m_j}-1}}\in ker(A),\quad \forall j\in\{\sum_{q=1}^{j-1} R_q+1,...,\sum_{q=1}^{j} R_q\}.$$
and
$$\exists \mu_{j_1},...,\mu_{j_{{m_j}-1}}\in\sigma(A)\quad \mbox{ such that }\quad \mu_{j_1}=...=\mu_{j_{{m_j}-1}}=0.$$
This is true for each $j\in\{\sum_{q=1}^{j-1} R_q+1,...,\sum_{q=1}^{j} R_q\}$, so that
$$\exists \mu_1,...,\mu_{deg(\mathcal S_{w_i})} \in\sigma(A)\quad \mbox{ such that }\quad \mu_1=...=\mu_{deg(\mathcal S_{w_i})}=0.$$
and the Laplacian matrix $L$ has at least $deg(\mathcal S_{w_i})+R_i$ diagonal entries with value $w_i$.\\
In the matrix $(L-w_i I)$ there are $v^{(j)}_{j_q}, \ q\in\{1,...,m_j\}$ vectors linearly dependent for each $j$, as a consequence
$v^{(j)}_{j_1},...,v^{(j)}_{j_{m_j-1}}\in ker(L-w_i I)$ and
$$\exists \mu_1,...,\mu_{deg(\mathcal S_{w_i})} \in\sigma(L-w_i I)\quad \mbox{ such that }\quad \mu_1=...=\mu_{deg(\mathcal S_{w_i})} =0.$$
Finally, let $\mu_p$ be one of these eigenvalues, then
$$0=det((L-w_i I)-\mu_p I)=det(L-(w_i +\mu_p )I)$$
and $\lambda:=w_i\in\sigma (L)$ with multiplicity greater or equal to $deg(\mathcal S_{w_i})$.\\
\end{proof}
}
Some corollaries on the signless and normalized Laplacian matrices can be obtained by using similar proofs.
Let $B$ and $\hat L$ be the signless and normalized Laplacian matrices of $\mathcal G=(\mathcal V,\mathcal E,w)$ respectively
and let $\sigma(B)$, $\sigma(\hat L)$ the eigenvalues of $B$ and $\hat L$ with algebraic multiplicity
$m_B(\lambda)$, $m_{\hat L}(\lambda)$ for the eigenvalue $\lambda$ in $B$ and $\hat L$ respectively.\\
\begin{cor}
If
\begin{itemize}
\item $s$ is the number of all the $S_{m,k}$ as $m$ and $k$ vary in $\mathbb N$ and $m+k\leq n,$ of $\mathcal G$,
\item $r$ is the number of $S_{m,k}$ with different weights, $w_1,...,w_r$,\\
\end{itemize}
then for any $i\in\{1,...,r\},$
$$\exists \lambda\in{\sigma(B)} \mbox{ such that } \lambda=w_i \mbox{ and } m_B(\lambda)\geq deg(\mathcal S_{w_i})$$
where $\mathcal S_{w_i}:=\{S_{m,k}\in \mathcal G | w(S_{m,k})=w_i\}$.
\end{cor}
\begin{cor}
If
\begin{itemize}
\item $s$ is the number of all the $S_{m,k}$ as $m$ and $k$ vary in $\mathbb N$ and $m+k\leq n,$ of $\mathcal G$,
\item $r$ is the number of $S_{m,k}$ with different weights, $w_1,...,w_r$,\\
\end{itemize}
then for any $i\in\{1,...,r\},$
$$\exists \lambda\in{\sigma(\hat L)} \mbox{ such that } \lambda=1 \mbox{ and } m_{\hat L}(\lambda)\geq \sum_{i=1}^r deg(\mathcal S_{w_i})$$
where $\mathcal S_{w_i}:=\{S_{m,k}\in \mathcal G | w(S_{m,k})=w_i\}$.
\end{cor}
A wider class of graphs for which the previous results can be extended is the class of the $l$-dependent graphs, defined as follows:
\begin{defn}[$l$-dependent graph: $D^l$]
A $l$-dependent graph is a graph $(\mathcal V,\mathcal E, w)$ whose vertices can be partitioned into four subsets:
the independent set $\mathcal V_1$, the central set $\mathcal V_2$, the independent set $\mathcal V_3$ and the set
$\mathcal V\setminus (\mathcal V_1\cup\mathcal V_2 \cup\mathcal V_3)$ such that
\begin{enumerate}
\item each vertex in $\mathcal V_1$ has at least one edge in $\mathcal V_2$ and vice versa, i.e.
$$\forall i\in \mathcal V_1,\exists j\in \mathcal V_2\ \mbox{ such that } \ (i,j)\in \mathcal E$$
$$\forall j\in \mathcal V_2,\exists i\in \mathcal V_1\ \mbox{ such that } \ (i,j)\in \mathcal E$$
\item vertices in $\mathcal V_1$ and $\mathcal V_3$ have edges only in $\mathcal V_2$, i.e.
$$\forall i\in \mathcal V_1\cup\mathcal V_3,\forall j\in \mathcal V\setminus\mathcal V_2, \quad (i,j)\notin \mathcal E$$
\item vertices in $\mathcal V_3$ have only edges that are a linear combination of all the edges of some vertices in $\mathcal V_1$, i.e.
$$\forall i\in\mathcal V_3, \exists j_1,...,j_{l_i}\in \mathcal V_1 \mbox{ such that }$$
$$\forall j\in\{j_1,...,j_{l_i}\}, \ \forall z \ \mbox{ such that } \ (j,z)\in\mathcal E, z\in\mathcal V_2\Rightarrow $$
$$\exists a(j)\in \mathbb R^{> 0} \mbox{ and } (i,z)\in\mathcal E, \ \mbox{ such that }\ w(i,z)=a(j)w(j,z).$$
\item $\mathcal V_1, \mathcal V_2, \mathcal V_3\subseteq \mathcal V$ are kept in order to satisfy the following
condition $$l:=\max_{\mathcal V_1,\mathcal V_2, \mathcal V_3\subseteq \mathcal V}|\mathcal V_3|.$$
\end{enumerate}
A $l$-dependent graph with $|\mathcal V_3|=l$, is denoted $D^l.$
\end{defn}
\begin{figure}[!!h
\centering
\includegraphics[width=6cm]{dependent_6.png}\caption{$D^l(\tilde w)$ graph, where the subsets $\mathcal V_1$
(for example the green vertices), $\mathcal V_2$ (the yellow vertices), $\mathcal V_3$ (for example the red vertex) and
$\mathcal V\setminus (\mathcal V_1\cup\mathcal V_2 \cup\mathcal V_3)$ are respectively with cardinality $\bar m=\underline m=2$,
$\bar k=\underline k=3$, $l=1$ and $|\mathcal V\setminus (\mathcal V_1\cup\mathcal V_2 \cup\mathcal V_3)|=0$. In the Laplacian matrix
there is the eigenvalue $\lambda=\tilde w=6$ with multiplicity 1.}
\end{figure}
\begin{obs}
First of all, we remark that neither the uniqueness of partition nor the cardinality of both $\mathcal V_1$ and $\mathcal V_2$ sets is guaranteed.
If we require the uniqueness of the cardinality further conditions are necessary: for instance
\begin{enumerate}
\item[5.*] maximum cardinality of the sets $\mathcal V_1,\mathcal V_2$
$$\bar{m}:=\max_{\mathcal V_1,\mathcal V_2 \subseteq \mathcal V\setminus\mathcal V_3}|\mathcal V_1|$$
$$\bar k:=\max_{\mathcal V_1,\mathcal V_2 \subseteq \mathcal V\setminus\mathcal V_3}|\mathcal V_2|$$
\item[5.**] minimum cardinality of the sets $\mathcal V_1,\mathcal V_2$
$$\underline m:=\min_{\mathcal V_1,\mathcal V_2 \subseteq \mathcal V\setminus\mathcal V_3}|\mathcal V_1|$$
$$\underline k:=\min_{\mathcal V_1,\mathcal V_2 \subseteq \mathcal V\setminus\mathcal V_3}|\mathcal V_2|.$$
\end{enumerate}
Even by requiring the maximum or minimum cardinality of both $\mathcal V_1$ and $\mathcal V_2$ sets, the uniqueness of the partition is not univocally determined.\\
The uniqueness of the set $\mathcal V_2$ is satisfied whenever one of the conditions 5. holds.
We notice that according to 5.**, the set $\mathcal V_2$ is defined as the set of all the
vertices $i\in\mathcal V$ such that $(i,j)\in\mathcal E,\ j\in\mathcal V_3.$
\end{obs}
\begin{figure}[!!h
\centering
\includegraphics[width=6cm]{dependent_10.png}
\includegraphics[width=6cm]{dependent_10b.png}
\caption{$D^l(\tilde w)$ graph, where the subsets $\mathcal V_1$ (green vertices) and $\mathcal V_3$ (red vertices) can be chosen differently.
The cardinalities of the sets are respectively $\bar m=\underline m=3$, $\bar k=\underline k=4$, $l=3$ and
$|\mathcal V\setminus (\mathcal V_1\cup\mathcal V_2 \cup\mathcal V_3)|=0$.
In the Laplacian matrix there is the eigenvalue $\lambda=\tilde w=4$ with multiplicity 3.}
\end{figure}
\begin{obs}
Whenever in the condition [3.] the set $\{j_1,...,j_{l_i}\}$ coincides with the set $\mathcal V_1$,
then the $l$-dependent graph is also a graph with an (m,k)-star, with m=l+1.
\end{obs}
We define an $l$-dependent graph of weight $\tilde w$, $D^l(\tilde w)$ as the $l$-dependent graph such that each vertex $i\in\mathcal V_1\cup\mathcal V_3$
has strength $\tilde w$.\\
Now we can extend the Theorem \ref{th:one} on graphs with $(m,k)$-star to $l$-dependent graphs, that is one of the main results of this work.\\
Let $\mathcal G=(\mathcal V,\mathcal E,w)$ be a graph, and $L$ the Laplacian matrix of $\mathcal G$.
\begin{thm}\label{th:main}
If $\mathcal G$ be a $D^l(\tilde w)$ graph, with $\tilde w\in\mathbb(R^{>0})$ and $l\in \mathbb N$,\\
then
$$\exists \lambda\in{\sigma(L)} \mbox{ such that } \lambda=\tilde w \mbox{ and } m_L(\lambda)\geq l.$$
\end{thm}
\begin{proof}
The proof is similar to Theorem \ref{th:one}. By definition of $D^l(\tilde w)$, each vertex $i\in\mathcal V_3$ has a corresponding
row in the adjacency matrix $A$, that is a linear combination of the rows of some vertices $j_1,...,j_{l_i}\in\mathcal V_1$.
Therefore the adjacency matrix $A$ has an eigenvalue $\mu=0$ of multiplicity at least $l$.
Since each vertex $i\in\mathcal V_1\cup\mathcal V_3$ has strength $\tilde w$ we can conclude the proof.
\end{proof}
\begin{obs}
The previous result does not require the conditions 5.
\end{obs}
We observe that a $D^{l}(\tilde w)$ graph, with $l\in\mathbb{N},\ \tilde w\in\mathbb{R}^+$, could be also a $D^{l_*}(\tilde w_*)$ graph,
for any $\ l_*\in\mathbb{N},\ \tilde w_*\in\mathbb{R}^+$.
As for the Theorem \ref{th:one}, some corollaries on the signless and normalized Laplacian matrices can be obtained by means of similar proofs.
Let $\mathcal G=(\mathcal V,\mathcal E,w)$ be a graph, and $B$ and $\hat L$ the signless and normalized Laplacian matrices respectively.
\begin{cor}\label{cor:main}
If
$\tilde w_1,...,\tilde w_m\in\mathbb(R^{>0})$ and $l_1,...,l_m\in \mathbb N$ such that $\mathcal G$ is a $D^l_i(\tilde w_i)$ graph, $ i\in\{1,...,m\}$;\\
then
$$\exists \lambda \in{\sigma(\hat L)} \mbox{ such that } \lambda=1 \mbox{ and } m_{\hat L}(\lambda)\geq \sum_{i=1}^m l_i.$$
\end{cor}
\subsection{(m,k)-star graph reduction}
According to the previous results, we have defined a class of graphs whose Laplacian matrices have an eigenvalues spectrum with known multiplicities and values.
Now, our aim is to simplify the study of such graphs by collapsing these vertices into a single vertex replacing the original graph with a reduced graph.
\\
At this purpose, the following definitions are useful:
\begin{defn}[$(m,k)$-star $q$-reduced: $S^q_{m,k}$]
A $q$-reduced $(m,k)$-star is a $(m,k)$-star of vertex sets $\{\mathcal V_1,\mathcal V_2\}$, such that $q$ of its vertices in $\mathcal V_1$ are removed.
Hence the order and degree of the $S^q_{m,k}$ are $m+k-q$ and $m-q-1$ respectively.\\
\end{defn}
\begin{defn}[$q$-reduced graph: $\mathcal G^q$]
A $q$-reduced graph $\mathcal G^q$ is obtained from a graph $\mathcal G$ with some $(m,k)$-stars removing $q$ of the vertices in the set $\mathcal V_1$
of $\mathcal G.$
\end{defn}
We derive a spectrum correspondence between graphs $\mathcal G$ and $\mathcal G^q$
\begin{defn}[Mass matrix of a $S_{m,k}^q$]
Let $\mathcal V_1$ and $\mathcal V_2$ be the vertex sets of the graph $S_{m,k}^q,\ q< m$.\\
Let $B$ be the adjacency matrix of $S_{m,k}^q$. The mass matrix of a $S_{m,k}^q$, $M$ is a diagonal matrix of order $m+k-q$ such that
\begin{equation}\label{eq:diagM}
M_{ii} = \begin{cases} \frac{m}{m-q}, & \mbox{if } i\in\mathcal V_1 \\ 1 & \mbox{otherwise } \end{cases},
\end{equation}
\end{defn}
The mass matrix $M$ can be defined in the same way also for a graph $\mathcal G^q$, with one (or more) $S_{m,k}^q$ by means
of a matrix of order $n-q$, whenever the graph $\mathcal G^q$ is composed by $n-q$ vertices.\\
Now we state the second main result of this paper.
\begin{thm}[$(m,k)$-star adjacency matrix reduction theorem]\label{th:reduction1}
Let
\begin{itemize}
\item $\mathcal G$ be a graph, of n vertices, with a $S_{m,k}, \ m+q\leq n$,
\item $\mathcal G^q$ be the reduced graph with a $S_{m,k}^q$ instead of $S_{m,k}$, of $n-q$ vertices,\\
\item $A$ be the adjacency matrix of $\mathcal G$,
\item $B$ be the adjacency matrix of $\mathcal G^q$,
\item $M$ be the diagonal mass matrix of $\mathcal G^q$,
\end{itemize}
then
\begin{enumerate}
\item $\sigma(A)=\sigma(MB),$
\item There exists a matrix $K\in\mathbb R^{n\times (n-q)}$ such that $M^{1/2}BM^{1/2}=K^TAK$ and $K^TK=I$. Therefore, if $v$ is an eigenvector of $M^{1/2}BM^{1/2}$ for an
eigenvalue $\mu$, then Kv is an eigenvector of A for the same eigenvalue $\mu$.
\end{enumerate}
\end{thm}
{ Before proving Theorem \ref{th:reduction1}, we recall the well known result for eigenvalues of
symmetric matrices, \cite{Hwang2004}.
\begin{lem}[Interlacing theorem]
Let $A\in Sym_n(\mathbb R)$ with eigenvalues $\mu_1(A)\geq...\geq \mu_n(A).$ For $m<n$, let $S\in\mathbb R^{n,m}$ be a
matrix with orthonormal columns, $K^TK=I$, and consider the $B=K^TAK$ matrix, with eigenvalues $\mu_1(B)\geq...\geq \mu_m(B).$
If
\begin{itemize}
\item the eigenvalues of $B$ interlace those of $A$, that is,
$$\mu_i(A)\geq\mu_i(B)\geq\mu_{n_A-n_B+i}(A), \quad i=1,...,n_B,$$
\item if the interlacing is tight, that is, for some $0\leq k\leq n_B,$
$$\mu_i(A)=\mu_i(B), \ i=1,...,k\ \mbox{ and } \ \mu_i(B)=\mu_{n_A-n_B+i}(A), \ i=k+1,...,n_B$$
then $KB=AK.$
\end{itemize}
\end{lem}
\begin{proof}
First we prove the existence of the $K$ matrix:\\
let $\mathcal P=\{P_1,...,P_{n_B}\}$ be a partition of the vertex set $\{1,...,n_A\}$, where $n_B=n_A-q.$
The \textit{ characteristic matrix H } is defined as the matrix where the $j$-th column is the characteristic vector of $P_j$ ($j=1,...,n_B$).\\
Let A be partitioned according to $\mathcal P$
\[
A=\left(
\begin{array}{ccc}
A_{1,1} & \dots & A_{1,n_B} \\
\vdots & & \vdots \\
A_{n_B,1} & \dots & A_{n_B,n_B}
\end{array}
\right),\]
where $A_{i,j}$ denotes the block with rows in $P_i$ and columns in $P_j$.
The matrix $B=(b_{ij})$ whose entries $b_{ij}$ are the averages of the $A_{i,j}$ rows, is called the \textit{quotient matrix} of $A$ with respect $\mathcal P$,
i.e. $b_{ij}$ denote the average number of neighbours in $P_j$ of the vertices in $P_i$.\\
The partition is equitable if for each $i,j$, any vertex in $P_i$ has exactly $b_{ij}$ neighbours in $P_j$.
In such a case, the eigenvalues of the quotient matrix $B$ belong to the spectrum of $A$ ($\sigma(B)\subset\sigma(A)$) and the spectral radius
of $B$ equals the spectral radius of $A$: for more details cfr. \cite{brouwer12}, chapter 2.\\
Then we have the relations
$$MB=H^TAH, \quad H^TH=M.$$
Considering a $q$-reduced $(m,k)-$star with adjacency matrix $B$, we weight it by a diagonal mass matrix $M$ whose diagonal entries
are one except for the $m-q$ entries of the vertices in $\mathcal V_1$,
\begin{equation}\label{eq:diagM}
M_{ii} = \begin{cases} \frac{m}{m-q}, & \mbox{if } i\in\mathcal V_1 \\ 1 & \mbox{otherwise } \end{cases},
\end{equation}
and we get
$$MB\sim M^{1/2}BM^{1/2}=K^TAK, \quad K^TK=I,$$
where $K:=HM^{1/2}.$
In addition to the th.(\ref{th:one}), the eigenvalues of $MB$ are a subset of the eigenvalues of $A$,
the adjacency matrix of the corresponding $S_{m,k}$ graph
$$
\sigma(MB)\subset\sigma(A).
$$
Whenever $q<m-1$, we get $\sigma(MB)=\sigma(A)$, up to the multiplicity of the eigenvalue $\mu=0$.\\
Finally, if $v$ is an eigenvector of $M^{1/2}BM^{1/2}$ with eigenvalue $\mu$, then $Kv$ is an eigenvector of $A$ with the same eigenvalue $\mu$.\\
Indeed form the equation
$$\tilde Bv=\mu v$$
an taking into account that the partition is equitable, we have $K\tilde B=AK,$ and
$$AKv=K\tilde Bv=\mu Kv.$$
\end{proof}}
We obtain a similar result for the Laplacian matrix.
\begin{thm}[$(m,k)$-star Laplacian matrix reduction theorem]\label{th:reduction2}
If
\begin{itemize}
\item $\mathcal G$ be a graph, of n vertices, with a $S_{m,k}, \ m+q\leq n$,
\item $\mathcal G^q$ be the reduced graph with a $S_{m,k}^q$ instead of $S_{m,k}$, of $n-q$ vertices,
\item $L(A)$ be the Laplacian matrix of $\mathcal G$,
\item $L(B)$ be the Laplacian matrix of $\mathcal G^q$. Let $M$ the diagonal mass matrix of $\mathcal G^q$,
\end{itemize}
then
\begin{enumerate}
\item $\sigma(L(A)=\sigma(L(MB))$
\item There exists a matrix $K\in\mathbb R^{n\times (n-q)}$ such that $M^{1/2}BM^{1/2}=K^TAK$ and $K^TK=I$.
Therefore, if $v$ is an eigenvector of $\tilde L(M B):=diag(MB)-M^{1/2}BM^{1/2}$ for an eigenvalue $\lambda$, then Kv is
an eigenvector of L(A) for the same eigenvalue $\lambda$.
\end{enumerate}
\end{thm}
{The proof for the Laplacian version of the Reduction Theorem \ref{th:reduction1} is similar to that for the adjacency matrix,
in fact using the same arguments as in the proof of
\ref{th:reduction1}, we can say that 1. is true and that the $K$ matrix exists. So we prove directly only the second part of point 2. of the theorem.
\begin{proof}
Let $v$ be an eigenvector of $L(\tilde B):=diag(MB)-M^{1/2}BM^{1/2}$ for an eigenvalue $\lambda$, then
$$L(\tilde B)v=\lambda v.$$
Because of $K\tilde B=AK$ and $diag(A)K=Kdiag(MB)$, we obtain
$$L(A)Kv=diag(A)Kv-AKv=Kdiag(MB)v-K\tilde B v=\lambda Kv.$$
\end{proof}}
According to the previous results, graphs with $(m,k)$-stars and graphs $q$-reduced can be partitioned in the same way, up to the removed vertices.\\
\begin{cor}\label{cor:reduction}
Under the hypothesis of theorem \ref{th:reduction2}, if $v$ is a (left or right) eigenvector of $L(MB)$ with eigenvalue $\lambda$,
then its entries have the same signs of the entries of the eigenvector $u$ of $L(A)$ with the same eigenvalue $\lambda$. \end{cor}
Indeed, the matrices $L(MB)$ and $\tilde L(MB)$ are similar, by means of the non singular matrix $M^{1/2}$.
Furthermore, since the similarity matrix $M^{1/2}$ is diagonal with all positive elements on the diagonal,
then both left and right eigenvectors of $ L(MB)$ preserve the sign of the eigenvectors of $\tilde L(MB)$. { We formally prove the Corollary.
\begin{proof}
$\tilde L(MB)$ and $L(MB)$ are similar by means of the matrix $M^{1/2}$, in fact
\begin{eqnarray}
M^{-1/2}L(MB)M^{1/2}&=&M^{-1/2}diag(MB)M^{1/2}-M^{-1/2}MBM^{1/2}\nonumber\\
&=&diag(MB)-M^{1/2}BM^{1/2}\nonumber\\
&=&\tilde L(MB). \nonumber
\end{eqnarray}
$L(MB)$ preserves the sign of the eigenvectors of $\tilde L(MB)$.\\
If $\tilde v$ an eigenvector of $\tilde L(MB)$ of the eigenvalue $\lambda\in\sigma(\tilde L(MB))$, then
\begin{eqnarray}
\tilde L(MB) \tilde v=\lambda \tilde v & \Leftrightarrow & M^{-1/2}L(MB)M^{1/2} \tilde v=\lambda \tilde v\nonumber\\
& \Leftrightarrow & L(MB)M^{1/2} \tilde v=\lambda M^{1/2} \tilde v\nonumber
\end{eqnarray}
As a consequence $v:=M^{1/2} \tilde v$ is the eigenvector of $L(MB)$ of the eigenvalue $\lambda,$ and $v_i=(M\tilde v)_i$,
$$v_i=\sum_{r=1}^{n-q} M_{ir}\tilde v_r=M_{ii}\tilde v_i.$$
\end{proof}}
Thanks to the previous result, we can partition the primary graph $\mathcal G$ containing the $(m,k)$-star and the $q$-reduced graph $\mathcal G^q$, weighted
by the matrix $M$, in the same way except for the removed vertices.\\
\section{Concluding remarks}\label{sec:4}
In this work we considered the problem of spectral partitioning of weighted graphs that contain $(m,k)$-stars.
We showed that, under some hypotheses on edge weights, the Laplacian matrix of graphs with $(m,k)$-stars has eigenvalues of multiplicity at least $m-1$
and computable values.\\
We proved that it is possible to reduce the node cardinality of these graphs by a suitable equivalence relation, keeping the same eigenvalues
on the adjacency and Laplacian matrices up to their multiplicity.\\
Furthermore, we have shown that Laplacian matrices of both the original and reduced graphs have the same signs of the eigenvectors entries,
so that it is possible to partition both graphs in the same way, up to removed vertices.\\
According to these results, whenever a weighted graph is composed by one or more $(m,k)$-star subgraphs, it is possible to collapse some of its vertices
into one, and to reduce the dimension of the matrices associated to these graphs, preserving the spectral properties.\\
These results can be relevant for applications to the network partitioning problems, or whenever a sort of node summarization is sought, merging nodes
with similar spectral properties.
These nodes could share similar functional properties, e. g. in the case of proteins with a similar neighborhood structure in interactome networks\cite{bioplex15},
with implications on biomedical and Systems Biology applications \cite{menche15}.
Moreover, the possibility to reduce network dimensionality by an equivalence relation among nodes can possibly be extended in a perturbative approach,
performing network reduction whenever the conditions of our theorems are 'almost satisfied', that is if some eigenvalues are sufficiently close.
\section{Acknowledgments}
The authors thank Nicola Guglielmi (University of L'Aquila, Italy), and E. A. also thanks
Domenico Felice (Max Planck Institute of Leipzig, Germany) and Carmela Scalone (University of L'Aquila, Italy) for useful discussions.
|
\section{Introduction}
In this paper we study numerical integration of smooth functions defined over the $s$-dimensional unit cube.
For an integrable function $f\colon [0,1)^s\to \mathbb{R}$, we denote the integral of $f$ by
\[ I(f) := \int_{[0,1)^s}f(\boldsymbol{x})\, \mathrm{d} \boldsymbol{x}. \]
We approximate $I(f)$ by a linear algorithm of the form
\[ I(f;P_N,W_N) = \sum_{n=0}^{N-1}w_nf(\boldsymbol{x}_n), \]
where $P_N=\{\boldsymbol{x}_n\colon 0\leq n<N\}\subset [0,1)^s$ is the set of quadrature nodes and $W_N=\{w_n\colon 0\leq n<N\}\subset \mathbb{R}$ is the set of associated weights.
A quasi-Monte Carlo (QMC) rule is an equal-weight quadrature rule where the weights sum up to 1, i.e., a linear algorithm with the special choice $w_n=1/N$ for all $n$.
Thus, $I(f)$ is simply approximated by
\[ I(f;P_N) = \frac{1}{N}\sum_{n=0}^{N-1}f(\boldsymbol{x}_n). \]
We refer to \cite{DKS13,DPbook,Nbook,SJbook} for comprehensive information on QMC integration.
The quality of a given quadrature rule is often measured by the worst-case error, that is, the worst absolute integration error in the unit ball of a normed function space $V$:
\[ e^{\mathrm{wor}}(V; P_N,W_N) := \sup_{\substack{f\in V\\ \|f\|_V\leq 1}}|I(f;P_N,W_N)-I(f)|, \]
for a general linear algorithm, and
\[ e^{\mathrm{wor}}(V; P_N) := \sup_{\substack{f\in V\\ \|f\|_V\leq 1}}|I(f;P_N)-I(f)|, \]
for a QMC algorithm.
In this paper, we consider weighted unanchored Sobolev spaces with dominating mixed smoothness $\alpha\geq 2$ as introduced in \cite{DKLNS14}, see Section~\ref{subsec;sobolev} for the details.
For such function spaces consisting of smooth functions, it is possible to construct good QMC integration rules achieving the almost optimal order of convergence $O(N^{-\alpha+\epsilon})$ with arbitrarily small $\epsilon>0$, see for instance \cite{BD09,BDGP11,BDLNP12,D08,G15,GD15,GSY16}.
In particular, so-called interlaced polynomial lattice rules have been recently applied in the context of partial differential equations with random coefficients, see for instance \cite{DKLNS14,DLS16}, due to their low construction cost and weak dependence of the worst-case error on the dimension.
In this paper, we propose an alternative QMC-based quadrature rule, named \emph{extrapolated polynomial lattice rule}, which achieves the almost optimal order of convergence with weak dependence on the dimension and can be constructed at a low computational cost.
Roughly speaking, extrapolated polynomial lattice rules are given by constructing classical polynomial lattice rules with consecutive sizes of nodes and then applying Richardson extrapolation in a recursive way.
Therefore, the resulting quadrature rule is a linear algorithm but not equally weighted.
Our motivation behind introduction of extrapolated polynomial lattice rules lies in so-called fast QMC matrix-vector multiplication which is briefly explained below.
Recently in \cite{DKLS15}, Dick et al.\ consider the problem of approximating integrals of the form
\[ \int_{[0,1)^s}f(\boldsymbol{x} A) \, \mathrm{d} \boldsymbol{x},\]
where $\boldsymbol{x}$ is an $1\times s$ row vector, and $A$ is an $s\times t$ real matrix.
They design QMC quadrature nodes $\boldsymbol{x}_0,\ldots,\boldsymbol{x}_{N-1}\in [0,1)^s$ suitably such that the matrix-vector product $XA$, where $X=(\boldsymbol{x}_0^{\top},\ldots,\boldsymbol{x}_{N-1}^{\top})^{\top}$, can be computed in $O(N\log N)$ arithmetic operations by using the fast Fourier transform without requiring any structure in the matrix $A$. This is done by choosing the quadrature nodes such that $X = CP$, where $C$ is a circulant matrix and the matrix $P$ reorders and extends the vector $\mathbf{a}$ when multiplied with $P$.
The resulting vector $XA=Y=(\boldsymbol{y}_0^{\top},\ldots,\boldsymbol{y}_{N-1}^{\top})^{\top}$ is used to approximate $I(f)$ by
\[ \frac{1}{N}\sum_{n=0}^{N-1}f(\boldsymbol{y}_n). \]
Their proposed method can be applied to classical polynomial lattice rules, but not to interlaced polynomial lattice rules, since the interlacing destroys the circulant structure.
In fact, it has been an open question whether it is possible to design QMC quadrature nodes which achieve higher order of convergence of the integration error for sufficiently smooth functions, and at the same time, can be used in fast QMC matrix-vector multiplication.
Since extrapolated polynomial lattice rules are just given by a linear combination of classical polynomial lattice rules, we can apply fast QMC matrix-vector multiplication to extrapolated polynomial lattice rules in a straightforward manner, which gives an affirmative solution to the above question.
The remainder of this paper is organized as follows.
In the next section we describe the necessary background and notation, namely, weighted unanchored Sobolev spaces with dominating mixed smoothness, Walsh functions, polynomial lattice rules, and Richardson extrapolation.
In Section~\ref{sec:explr}, we first give the key ingredient for introducing extrapolated polynomial lattice rules, and then study their worst-case error in Sobolev spaces with general weights as well as their dependence on the worst-case error bound on the dimension.
Here we prove the existence of good extrapolated polynomial lattice rules achieving the almost optimal order of convergence.
In Section~\ref{sec:cbc}, we restrict ourselves to the case of product weights and show that the so-called fast component-by-component construction algorithm works for finding good extrapolated polynomial lattice rules.
We conclude this paper with numerical experiments in Section~\ref{sec:experiment}.
\section{Preliminaries}
Throughout this paper, let $\mathbb{N}$ denote the set of positive integers and $\mathbb{N}_0=\mathbb{N}\cup \{0\}$.
Let $b$ be a prime, and $\mathbb{F}_b$ be the finite field with $b$ elements which is identified with the set $\{0,1,\ldots,b-1\}\subset \mathbb{Z}$ equipped with addition and multiplication modulo $b$.
Further, we denote by $\mathbb{F}_b[x]$ the set of all polynomials over $\mathbb{F}_b$ and by $\mathbb{F}_b((x^{-1}))$ the field of formal Laurent series over $\mathbb{F}_b$.
For $m\in \mathbb{N}$, we write
\[ G_{b,m} = \{q\in \mathbb{F}_b[x]\colon \deg(m)<m \}\quad \text{and}\quad G^*_{b,m}=G_{b,m}\setminus \{0\}. \]
It is obvious that $|G_{b,m}|=b^m$ and $|G^*_{b,m}|=b^m-1$.
With a slight abuse of notation, we often identify $n\in \mathbb{N}_0$, whose finite $b$-adic expansion is given by $n=\nu_0+\nu_1b+\cdots$, with the polynomial over $\mathbb{F}_b$ given by $n(x)=\nu_0+\nu_1x+\cdots$.
\subsection{Sobolev spaces with dominating mixed smoothness}\label{subsec;sobolev}
We give the definition of weighted Sobolev spaces with dominating mixed smoothness.
Let $\alpha,s\in \mathbb{N}$, $\alpha\geq 2$, $1\leq q,r\leq \infty$, and let $\boldsymbol{\gamma}=(\gamma_u)_{u\subset \mathbb{N}}$ be a set of non-negative real numbers called weights, which plays a role in moderating the importance of different variables or groups of variables in the function space \cite{SW98}.
Assume that $f\colon [0,1)^s\to \mathbb{R}$ has partial mixed derivatives up to order $\alpha$ in each variable.
We define the norm on the weighted unanchored Sobolev space with dominating mixed smoothness $\alpha$ by
\begin{align*}
\|f\|_{s,\alpha,\boldsymbol{\gamma},q,r} & := \Bigg( \sum_{u\subseteq \{1,\ldots,s\}}\Bigg( \gamma_u^{-q}\sum_{v\subseteq u}\sum_{\boldsymbol{\tau}_{u\setminus v}\in \{1,\ldots,\alpha\}^{|u\setminus v|}} \\
& \qquad \qquad \int_{[0,1)^{|v|}}\left| \int_{[0,1)^{s-|v|}}f^{(\boldsymbol{\tau}_{u\setminus v},\boldsymbol{\alpha}_v,\boldsymbol{0})}(\boldsymbol{x})\, \mathrm{d} \boldsymbol{x}_{-v}\right|^q \, \mathrm{d} \boldsymbol{x}_v\Bigg)^{r/q}\Bigg)^{1/r},
\end{align*}
with the obvious modifications if $q$ or $r$ is infinite.
Here $(\boldsymbol{\tau}_{u\setminus v},\boldsymbol{\alpha}_v,\boldsymbol{0})$ denotes a sequence $\boldsymbol{\beta}=(\beta_j)_{1\leq j\leq s}$ with $\beta_j=\tau_j$ if $j\in u\setminus v$, $\beta_j=\alpha$ if $j\in v$, and $\beta_j=0$ if $j\notin u$.
Further, $f^{(\boldsymbol{\tau}_{u\setminus v},\boldsymbol{\alpha}_v,\boldsymbol{0})}$ denotes the partial mixed derivative of order $(\boldsymbol{\tau}_{u\setminus v},\boldsymbol{\alpha}_v,\boldsymbol{0})$ of $f$, and we write $\boldsymbol{x}_v=(x_j)_{j\in v}$ and $\boldsymbol{x}_{-v}=(x_j)_{j\in \{1,\ldots,s\}\setminus v}$.
We denote the Banach-Sobolev space of all such functions with finite norm $\|\cdot\|_{s,\alpha,\boldsymbol{\gamma},q,r}$ by $W_{s,\alpha,\boldsymbol{\gamma},q,r}$.
In what follows, let $B_{\tau}(\cdot)$ denote the Bernoulli polynomial of degree $\tau$.
We put $b_{\tau}(\cdot)=B_{\tau}(\cdot)/\tau!$ and $b_{\tau}=b_{\tau}(0)$.
Further, let $\tilde{b}_{\tau}(\cdot)\colon \mathbb{R}\to \mathbb{R}$ denote the one-periodic extension of the polynomial $b_{\tau}(\cdot)\colon [0,1)\to \mathbb{R}$.
Then, as shown in the proof of \cite[Theorem~3.5]{DKLNS14} we have the following.
\begin{lemma}\label{lem:func_represent}
For any $f\in W_{s,\alpha,\boldsymbol{\gamma},q,r}$, we have a pointwise representation
\[ f(\boldsymbol{y}) = \sum_{u\subseteq \{1,\ldots,s\}}f_u(\boldsymbol{y}_u), \]
where each function $f_u$ depends only on a subset of variables $\boldsymbol{y}_u=(y_j)_{j\in u}$ and is explicitly given by
\begin{align*}
f_u(\boldsymbol{y}_u) & = \sum_{v\subseteq u}\sum_{\boldsymbol{\tau}_{u\setminus v}\in \{1,\ldots,\alpha\}^{|u\setminus v|}}\prod_{j\in u\setminus v}b_{\tau_j}(y_j) \\
& \qquad \times (-1)^{(\alpha+1)|v|}\int_{[0,1)^s} f^{(\boldsymbol{\tau}_{u\setminus v},\boldsymbol{\alpha}_v,\boldsymbol{0})}(\boldsymbol{x}) \prod_{j\in v}\tilde{b}_{\alpha}(x_j-y_j) \, \mathrm{d} \boldsymbol{x}.
\end{align*}
Furthermore we have
\[ \|f\|_{s,\alpha,\boldsymbol{\gamma},q,r} = \left( \sum_{u\subseteq \{1,\ldots,s\}}\|f_u\|_{s,\alpha,\boldsymbol{\gamma},q,r}^r\right)^{1/r}. \]
\end{lemma}
\subsection{Walsh functions}\label{subsec:walsh}
Here we introduce the definition of Walsh functions and state the result on the decay of Walsh coefficients for functions in $W_{s,\alpha,\boldsymbol{\gamma},q,r}$.
\begin{definition}
For a prime $b$, put $\omega_b=\exp(2\pi i/b)$.
For $k\in \mathbb{N}_0$ with finite $b$-adic expansion $k=\kappa_0+\kappa_1b+\cdots$, the $k$-th Walsh function ${}_{b}\mathrm{wal}_k\colon [0,1)\to \{1,\omega_b,\ldots,\omega_b^{b-1}\}$ is defined by
\[ {}_{b}\mathrm{wal}_k(x) := \omega_{b}^{\kappa_0 \xi_1+\kappa_1 \xi_2+\cdots},\]
for $x\in [0,1)$ with $b$-adic expansion $x=\xi_1/b+\xi_2/b^2+\cdots$, where this expansion is understood to be unique in the sense that infinitely many of the $\xi_i$ are different from $b-1$.
For $s\geq 2$ and $\boldsymbol{k}=(k_1,\ldots,k_s)\in \mathbb{N}_0^s$, the $\boldsymbol{k}$-th Walsh function ${}_{b}\mathrm{wal}_{\boldsymbol{k}}\colon [0,1)^s\to \{1,\omega_b,\ldots,\omega_b^{b-1}\}$ is defined by
\[ {}_{b}\mathrm{wal}_{\boldsymbol{k}}(\boldsymbol{x}) := \prod_{j=1}^{s}{}_{b}\mathrm{wal}_{k_j}(x_j) ,\]
for $\boldsymbol{x}=(x_1,\ldots,x_s)\in [0,1)^s$.
\end{definition}
\noindent
Since we shall use Walsh functions in a fixed prime base $b$ in this paper, we omit the subscript and simply write $\mathrm{wal}_k$ or $\mathrm{wal}_{\boldsymbol{k}}$. Note that the system $\{\mathrm{wal}_{\boldsymbol{k}}\colon \boldsymbol{k}\in \mathbb{N}_0^s\}$ is a complete orthonormal system in $L^2([0,1)^s)$, see \cite[Theorem~A.11]{DPbook}. Thus for $f\in L^2([0,1)^s)$, we have the Walsh expansion of $f$:
\[ \sum_{\boldsymbol{k}\in \mathbb{N}_0^s}\hat{f}(\boldsymbol{k})\mathrm{wal}_{\boldsymbol{k}}(\boldsymbol{x}), \]
where $\hat{f}(\boldsymbol{k})$ denotes the $\boldsymbol{k}$-th Walsh coefficient of $f$ defined by
\[ \hat{f}(\boldsymbol{k}):=\int_{[0,1)^s}f(\boldsymbol{x})\overline{\mathrm{wal}_{\boldsymbol{k}}(\boldsymbol{x})}\, \mathrm{d} \boldsymbol{x}. \]
Here we note that the integral of $f$ is given by $I(f)=\hat{f}(\boldsymbol{0})$.
The Walsh coefficients of a function $f\in W_{s,\alpha,\boldsymbol{\gamma},q,r}$ are bounded as follows, see \cite[Theorem~14]{D09} and \cite[Theorem~3.5]{DKLNS14} for the proof.
\begin{lemma}\label{lem:walsh_bound}
For $k\in \mathbb{N}$, we denote the $b$-adic expansion $k$ by $k=\kappa_1b^{a_1-1}+\cdots+\kappa_vb^{a_v-1}$ with $a_1>\cdots>a_v>0$ and $\kappa_1,\ldots,\kappa_v\in \{1,\ldots,b-1\}$.
We define the metric $\mu_{\alpha}:\mathbb{N}_0\to \mathbb{N}_0$ by
\[ \mu_{\alpha}(k):=a_1+\cdots+a_{\min(v,\alpha)}, \]
and $\mu_{\alpha}(0):=0$. In case of a vector $\boldsymbol{k}=(k_1,\ldots,k_s)\in \mathbb{N}_0^s$, we define
\[ \mu_{\alpha}(\boldsymbol{k}) := \sum_{j=1}^{s}\mu_{\alpha}(k_j). \]
For a subset $u\subseteq \{1,\ldots,s\}$ and $\boldsymbol{k}_u\in \mathbb{N}^{|u|}$, the $(\boldsymbol{k}_u,\boldsymbol{0})$-th Walsh coefficient of a function $f\in W_{s,\alpha,\boldsymbol{\gamma},q,r}$ is bounded by
\[ |\hat{f}(\boldsymbol{k}_u,\boldsymbol{0})| \leq \gamma_uC_{\alpha}^{|u|}b^{-\mu_{\alpha}(\boldsymbol{k}_u)}\|f_u\|_{s,\alpha,\boldsymbol{\gamma},q,r}, \]
where
\begin{align*}
C_{\alpha} & = \max\left( \frac{2}{(2\sin\frac{\pi}{b})^{\alpha}}, \max_{1\leq z\leq \alpha-1}\frac{1}{(2\sin\frac{\pi}{b})^{z}}\right) \\
& \qquad \times \left( 1+\frac{1}{b}+\frac{1}{b(b+1)}\right)^{\alpha-2}\left( 3+\frac{2}{b}+\frac{2b+1}{b-1}\right).
\end{align*}
\end{lemma}
\begin{remark}\label{rem:walsh_bound}
For the special but important case $b=2$, Yoshiki \cite{Y15} proved that the constant $C_{\alpha}$ can be improved to $C_{\alpha}=2^{-1/p'}$ where $p'$ denotes the H\"older conjugate of $q$, i.e., $1\leq q'\leq \infty$ which satisfies $1/q+1/q'=1$.
\end{remark}
\subsection{Polynomial lattice rules}\label{subsec:poly_lattice}
Polynomial lattice point sets are a special construction of QMC quadrature nodes introduced by Niederreiter in \cite{N88}, which are defined as follows.
\begin{definition}
Let $p\in \mathbb{F}_{b}[x]$ with $\deg(p)=m$ and $\boldsymbol{q}=(q_1,\ldots,q_s)\in (G^*_{b,m})^s$.
We define the map $v_m\colon \mathbb{F}_b((x^{-1}))\to [0,1)$ by
\[ v_m\left( \sum_{i=w}^{\infty}a_i x^{-i}\right) := \sum_{i=\max\{1,w\}}^{m}a_ib^{-i}. \]
For $0\leq n< b^m$, which is identified with a polynomial over $\mathbb{F}_b$, put
\[ \boldsymbol{x}_n = \left( v_m\left(\frac{nq_1}{p}\right),\ldots, v_m\left(\frac{nq_s}{p}\right)\right)\in [0,1)^s. \]
Then the point set $P(p,\boldsymbol{q})=\{\boldsymbol{x}_0,\ldots,\boldsymbol{x}_{b^m-1}\}$ is called a polynomial lattice point set (with modulus $p$ and generating vector $\boldsymbol{q}$).
A QMC rule using the point set $P(p,\boldsymbol{q})$ as quadrature nodes is called a polynomial lattice rule.
\end{definition}
The concept of dual polynomial lattice plays a key role in the error analysis of polynomial lattice rules.
\begin{definition}
For $k\in \mathbb{N}_0$ with finite $b$-adic expansion $k=\kappa_0+\kappa_1b+\cdots$, we define the map $\mathrm{tr}_m\colon \mathbb{N}_0\to G_{b,m}$ by
\[ \mathrm{tr}_m(k) = \kappa_0+\kappa_1x+\cdots +\kappa_{m-1}x^{m-1}. \]
For $p\in \mathbb{F}_{b}[x]$ with $\deg(p)=m$ and $\boldsymbol{q}=(q_1,\ldots,q_s)\in (G^*_{b,m})^s$, the dual polynomial lattice of $P(p,\boldsymbol{q})$ is defined by
\[ P^{\perp}(p,\boldsymbol{q}) := \left\{\boldsymbol{k}\in \mathbb{N}_0^s\colon \mathrm{tr}_m(\boldsymbol{k})\cdot \boldsymbol{q} \equiv 0 \pmod p \right\}. \]
\end{definition}
\begin{remark}\label{rem:poly_lattice}
For $\boldsymbol{k}\in \mathbb{N}_0^s$ such that $b^m\mid k_j$ for all $j$, we have $\mathrm{tr}_m(\boldsymbol{k})=\boldsymbol{0}$.
Thus, regardless of the choice on $p$ and $\boldsymbol{q}$, such $\boldsymbol{k}$ is always included in the dual polynomial lattice $P^{\perp}(p,\boldsymbol{q})$.
\end{remark}
The following lemma shows the character property of polynomial lattice point sets, see for instance \cite[Lemmas~4.75 and 10.6]{DPbook} for the proof.
\begin{lemma}\label{lem:character}
Let $p\in \mathbb{F}_{b}[x]$ with $\deg(p)=m$ and $\boldsymbol{q}=(q_1,\ldots,q_s)\in (G^*_{b,m})^s$.
For $\boldsymbol{k}\in \mathbb{N}_0^s$, we have
\begin{align*}
\sum_{\boldsymbol{x}\in P(p,\boldsymbol{q})}\mathrm{wal}_{\boldsymbol{k}}(\boldsymbol{x}) = \begin{cases}
b^m & \text{if $\boldsymbol{k}\in P^{\perp}(p,\boldsymbol{q})$}, \\
0 & \text{otherwise.}
\end{cases}
\end{align*}
\end{lemma}
\noindent
By considering the Walsh expansion of a continuous function $f\colon [0,1)^s \to \mathbb{R}$ with $\sum_{\boldsymbol{k}\in \mathbb{N}_0^s}|\hat{f}(\boldsymbol{k})|<\infty$ and using Lemma~\ref{lem:character}, we obtain
\begin{align}
I(f; P(p,\boldsymbol{q})) & = \frac{1}{b^m}\sum_{\boldsymbol{x}\in P(p,\boldsymbol{q})}\sum_{\boldsymbol{k}\in \mathbb{N}_0^s}\hat{f}(\boldsymbol{k})\mathrm{wal}_{\boldsymbol{k}}(\boldsymbol{x}) \nonumber \\
& = \frac{1}{b^m}\sum_{\boldsymbol{k}\in \mathbb{N}_0^s}\hat{f}(\boldsymbol{k})\sum_{\boldsymbol{x}\in P(p,\boldsymbol{q})}\mathrm{wal}_{\boldsymbol{k}}(\boldsymbol{x}) \nonumber \\
& = \sum_{\boldsymbol{k}\in P^{\perp}(p,\boldsymbol{q})}\hat{f}(\boldsymbol{k}) = I(f)+\sum_{\boldsymbol{k}\in P^{\perp}(p,\boldsymbol{q})\setminus \{\boldsymbol{0}\}}\hat{f}(\boldsymbol{k}) .\label{eq:poly_lattice_error}
\end{align}
\subsection{Richardson extrapolation}\label{subsec:extrapolation}
Richardson extrapolation is a classical technique to speed up the convergence of a sequence by exploiting the asymptotic expansion of each term, see for instance \cite[Section~1.4]{DRbook} and \cite[Section~3.2.7]{Gbook}. In our current setting, we may have a sequence of polynomial lattice rules with the consecutive sizes of nodes, $b^1,b^2,\ldots$, which means that each term of the sequence corresponds to the approximate value $I(f;P(p,\boldsymbol{q}))$ for some $m\in \mathbb{N}$.
To simplify the situation, instead of an infinite sequence, let us consider a chain of $\alpha$ reals $(I^{(1)}_n)_{m-\alpha+1\leq n \leq m}$ with each given by
\begin{align}\label{eq:extra_seq}
I^{(1)}_n = c_0 + \frac{c_1}{b^n}+\cdots + \frac{c_{\alpha-1}}{b^{(\alpha -1)n}} + R_{\alpha,b^n},
\end{align}
where $b>1$, $c_0,\ldots,c_{\alpha-1}\in \mathbb{R}$ and $R_{\alpha,n}\in O(b^{-\alpha n})$. As shown later in \eqref{eq:decomp}, $I(f;P(p,\boldsymbol{q}))$ has actually such an expansion. In standard notation for extrapolation methods, the reciprocal $1/b^n$ should be regarded as a so-called admissible value of the step parameter $h$ for each term $I^{(1)}_n$. The aim here is to approximate $c_0$ as precisely as possible from the chain $(I^{(1)}_n)_{m-\alpha+1\leq n \leq m}$ without knowing the coefficients $c_1,\ldots,c_{\alpha-1}$.
To do so, let us consider the following recursive application of Richardson extrapolation of successive orders:
For $1\leq \tau< \alpha$, compute
\begin{align*}
I^{(\tau+1)}_{n} = \frac{b^{\tau} I^{(\tau)}_{n}-I^{(\tau)}_{n-1}}{b^{\tau}-1} \qquad \text{for $m-\alpha+\tau < n\leq m$}.
\end{align*}
Regarding this recursion, the following result holds. Although a similar result has been shown, for instance, in \cite{LP17}, we give the proof for self-containedness.
\begin{lemma}\label{lem:extra}
For a given $1\leq \tau \leq \alpha$, let
\[ a_{\nu}^{(\tau)} := \prod_{j=1}^{\nu-1}\left(\frac{-1}{b^j-1}\right)\prod_{j=1}^{\tau-\nu}\left(\frac{b^j}{b^j-1}\right) \qquad \text{for $1\leq \nu\leq \tau$}, \]
where the empty product is set to $1$.
Then we have
\[ I^{(\tau)}_{n} = \sum_{\nu=1}^{\tau}a_{\nu}^{(\tau)} I_{n+1-\nu}^{(1)} \qquad \text{for $m-\alpha+\tau \leq n\leq m$}. \]
\end{lemma}
\begin{proof}
We prove the lemma by induction on $\tau$.
As $a_1^{(1)}=1$, the case $\tau=1$ is trivial.
Let $1\leq \tau<\alpha$ and suppose that the equality
\[ I^{(\tau)}_{n} = \sum_{\nu=1}^{\tau}a_{\nu}^{(\tau)} I_{n+1-\nu}^{(1)} \]
holds for all $m-\alpha+\tau \leq n\leq m$.
It follows from the definition of $I^{(\tau+1)}_{n}$ that
\begin{align*}
I^{(\tau+1)}_{n} & = \frac{b^{\tau} I^{(\tau)}_{n}-I^{(\tau)}_{n-1}}{b^{\tau}-1} \\
& = \frac{b^\tau}{b^\tau-1} \sum_{\nu=1}^{\tau}a_{\nu}^{(\tau)} I_{n+1-\nu}^{(1)} - \frac{1}{b^\tau-1} \sum_{\nu=1}^{\tau}a_{\nu}^{(\tau)} I_{n-\nu}^{(1)} \\
& = \frac{b^\tau}{b^\tau-1} a_1^{(\tau)} I_{n}^{(1)} - \frac{1}{b^\tau-1} a_{\tau}^{(\tau)} I_{n-\tau}^{(1)} + \sum_{\nu=2}^{\tau}\left(\frac{b^\tau}{b^\tau-1}a_{\nu}^{(\tau)}- \frac{1}{b^\tau-1}a_{\nu-1}^{(\tau)}\right)I_{n+1-\nu}^{(1)},
\end{align*}
for $m-\alpha+\tau < n\leq m$.
For each term on the right-most side above, we have
\begin{align*}
\frac{b^\tau}{b^\tau-1} a_1^{(\tau)} & = \frac{b^\tau}{b^\tau-1}\prod_{j=1}^{\tau-1}\left(\frac{b^j}{b^j-1}\right) = \prod_{j=1}^{\tau}\left(\frac{b^j}{b^j-1}\right) = a_1^{(\tau+1)}, \\
- \frac{1}{b^\tau-1} a_{\tau}^{(\tau)} & = \frac{-1}{b^\tau-1}\prod_{j=1}^{\tau-1}\left(\frac{-1}{b^j-1}\right) = \prod_{j=1}^{\tau}\left(\frac{-1}{b^j-1}\right) = a_{\tau+1}^{(\tau+1)} ,
\end{align*}
and for $2\leq \nu\leq \tau$
\begin{align*}
& \frac{b^\tau}{b^\tau-1}a_{\nu}^{(\tau)}- \frac{1}{b^\tau-1}a_{\nu-1}^{(\tau)} \\
& = \frac{b^\tau}{b^\tau-1} \prod_{j=1}^{\nu-1}\left(\frac{-1}{b^j-1}\right)\prod_{j=1}^{\tau-\nu}\left(\frac{b^j}{b^j-1}\right) - \frac{1}{b^\tau-1} \prod_{j=1}^{\nu-2}\left(\frac{-1}{b^j-1}\right)\prod_{j=1}^{\tau-\nu+1}\left(\frac{b^j}{b^j-1}\right) \\
& = \prod_{j=1}^{\nu-1}\left(\frac{-1}{b^j-1}\right)\prod_{j=1}^{\tau+1-\nu}\left(\frac{b^j}{b^j-1}\right) = a_{\nu}^{(\tau+1)}.
\end{align*}
Thus we have
\begin{align*}
I^{(\tau+1)}_{n} = a_1^{(\tau+1)} I_{n}^{(1)} + a_{\tau+1}^{(\tau+1)} I_{n-\tau}^{(1)} + \sum_{\nu=2}^{\tau}a_{\nu}^{(\tau+1)} I_{n+1-\nu}^{(1)} = \sum_{\nu=1}^{\tau+1}a_{\nu}^{(\tau+1)} I_{n+1-\nu}^{(1)},
\end{align*}
which proves the lemma.
\end{proof}
In particular, this lemma shows that the final value $I^{(\alpha)}_{m}$ is given by
\begin{align}\label{eq:extra_final}
I^{(\alpha)}_{m} = \sum_{\tau=1}^{\alpha}a_{\tau}^{(\alpha)} I_{m-\tau+1}^{(1)}.
\end{align}
Regarding the coefficients $a_{\nu}^{(\tau)}$, the following property holds:
\begin{lemma}\label{lem:extra2}
For any $1\leq \tau \leq \alpha$, we have
\[ \sum_{\nu=1}^{\tau}a_{\nu}^{(\tau)} = 1\qquad \text{and}\qquad \sum_{\nu=1}^{\tau}a_{\nu}^{(\tau)}b^{w(\nu-1)} = 0 \qquad \text{for $1\leq w\leq \tau-1$}. \]
\end{lemma}
\begin{proof}
We prove the lemma by induction on $\tau$.
As $a_1^{(1)}=1$, the case $\tau=1$ is trivial.
Suppose that the claim of this lemma holds for some $1\leq \tau<\alpha$.
Using the recursions appearing in the proof of Lemma~\ref{lem:extra}, we have
\begin{align*}
\sum_{\nu=1}^{\tau+1}a_{\nu}^{(\tau+1)} & = a_{1}^{(\tau+1)}+\sum_{\nu=2}^{\tau}a_{\nu}^{(\tau+1)}+a_{\tau+1}^{(\tau+1)} \\
& = \frac{b^\tau}{b^\tau-1} a_1^{(\tau)}+\sum_{\nu=2}^{\tau}\left( \frac{b^\tau}{b^\tau-1}a_{\nu}^{(\tau)}- \frac{1}{b^\tau-1}a_{\nu-1}^{(\tau)} \right) - \frac{1}{b^\tau-1} a_{\tau}^{(\tau)}\\
& = \sum_{\nu=1}^{\tau}\left( \frac{b^\tau}{b^\tau-1}-\frac{1}{b^\tau-1}\right)a_{\nu}^{(\tau)} = \sum_{\nu=1}^{\tau}a_{\nu}^{(\tau)}=1.
\end{align*}
Similarly, for $1\leq w\leq \tau$, we have
\begin{align*}
\sum_{\nu=1}^{\tau+1}a_{\nu}^{(\tau+1)}b^{w(\nu-1)} & = a_{1}^{(\tau+1)}+\sum_{\nu=2}^{\tau}a_{\nu}^{(\tau+1)}b^{w(\nu-1)}+a_{\tau+1}^{(\tau+1)}b^{w\tau} \\
& = \frac{b^\tau}{b^\tau-1} a_1^{(\tau)}+\sum_{\nu=2}^{\tau}\left( \frac{b^{\tau+w(\nu-1)}}{b^\tau-1}a_{\nu}^{(\tau)}- \frac{b^{w(\nu-1)}}{b^\tau-1}a_{\nu-1}^{(\tau)} \right) - \frac{b^{w\tau}}{b^\tau-1} a_{\tau}^{(\tau)}\\
& = \sum_{\nu=1}^{\tau}\left( \frac{b^{\tau+w(\nu-1)}}{b^\tau-1}-\frac{b^{w\nu}}{b^\tau-1}\right)a_{\nu}^{(\tau)} \\
& = \frac{b^\tau-b^w}{b^\tau-1}\sum_{\nu=1}^{\tau}a_{\nu}^{(\tau)}b^{w(\nu-1)}=0,
\end{align*}
where the last equality follows from the induction assumption for $1\leq w\leq \tau-1$, and is trivial for $w=\tau$.
\end{proof}
Using these results, we further have the following.
\begin{corollary}\label{cor:extra}
Using the notation above, we have
\[ I^{(\alpha)}_{m} = c_0 + \sum_{\tau=1}^{\alpha}a_{\tau}^{(\alpha)}R_{\alpha,b^{m-\tau+1}}. \]
\end{corollary}
\begin{proof}
Plugging the expression \eqref{eq:extra_seq} into \eqref{eq:extra_final} and then using Lemma~\ref{lem:extra2}, we have
\begin{align*}
I^{(\alpha)}_{m} & = \sum_{\tau=1}^{\alpha}a_{\tau}^{(\alpha)} \left( c_0 + \sum_{w=1}^{\alpha-1}\frac{c_w}{b^{w(m-\tau+1)}} + R_{\alpha,b^{m-\tau+1}}\right) \\
& = c_0 \sum_{\tau=1}^{\alpha}a_{\tau}^{(\alpha)} + \sum_{w=1}^{\alpha-1}\sum_{\tau=1}^{\alpha}a_{\tau}^{(\alpha)}\frac{c_w}{b^{w(m-\tau+1)}}+ \sum_{\tau=1}^{\alpha}a_{\tau}^{(\alpha)}R_{\alpha,b^{m-\tau+1}} \\
& = c_0 \sum_{\tau=1}^{\alpha}a_{\tau}^{(\alpha)} + \sum_{w=1}^{\alpha-1}\frac{c_w}{b^{wm}}\sum_{\tau=1}^{\alpha}a_{\tau}^{(\alpha)}b^{w(\tau-1)}+ \sum_{\tau=1}^{\alpha}a_{\tau}^{(\alpha)}R_{\alpha,b^{m-\tau+1}}\\
& = c_0 + \sum_{\tau=1}^{\alpha}a_{\tau}^{(\alpha)}R_{\alpha,b^{m-\tau+1}}.
\end{align*}
This completes the proof.
\end{proof}
\section{Extrapolated polynomial lattice rules}\label{sec:explr}
The main idea for coming up with extrapolated polynomial lattice rules is to look at the approximate value of a polynomial lattice rule, as shown in \eqref{eq:poly_lattice_error}, in the following way:
\begin{align*}
I(f; P(p,\boldsymbol{q})) & = I(f)+ \sum_{\substack{\boldsymbol{k}\in P^{\perp}(p,\boldsymbol{q})\setminus \{\boldsymbol{0}\}\\ \exists j\colon b^m\nmid k_j}} \hat{f}(\boldsymbol{k})+\sum_{\substack{\boldsymbol{k}\in P^{\perp}(p,\boldsymbol{q})\setminus \{\boldsymbol{0}\}\\ \forall j\colon b^m\mid k_j}} \hat{f}(\boldsymbol{k}) \\
& = I(f) + \sum_{\substack{\boldsymbol{k}\in P^{\perp}(p,\boldsymbol{q})\setminus \{\boldsymbol{0}\}\\ \exists j\colon b^m\nmid k_j}} \hat{f}(\boldsymbol{k})+\sum_{\boldsymbol{k}\in \mathbb{N}_0^s\setminus \{\boldsymbol{0}\}} \hat{f}(b^m\boldsymbol{k}),
\end{align*}
where the second equality follows from Remark~\ref{rem:poly_lattice}.
By considering the character property of regular grids
\[ P_{\mathrm{grid},b^m} = \left\{ \left( \frac{n_1}{b^m},\ldots,\frac{n_s}{b^m}\right)\in [0,1)^s\colon 0\leq n_1,\ldots,n_s< b^m\right\},\]
we see that the third term in the last expression is nothing but the approximation error of $f$ when using $P_{\mathrm{grid},b^m}$ as quadrature nodes in a QMC integration.
Therefore we have
\[ I(f; P(p,\boldsymbol{q})) = I(f) + \sum_{\substack{\boldsymbol{k}\in P^{\perp}(p,\boldsymbol{q})\setminus \{\boldsymbol{0}\}\\ \exists j\colon b^m\nmid k_j}} \hat{f}(\boldsymbol{k})+ \left(I(f;P_{\mathrm{grid},b^m}) -I(f)\right). \]
Plugging in the Euler-Maclaurin formula for $I(f;P_{\mathrm{grid},b^m})$, which is shown later in Theorem~\ref{thm:euler-maclaurin}, into the right-hand side above, we obtain
\begin{align}\label{eq:decomp}
I(f; P(p,\boldsymbol{q})) = I(f) + \sum_{\substack{\boldsymbol{k}\in P^{\perp}(p,\boldsymbol{q})\setminus \{\boldsymbol{0}\}\\ \exists j\colon b^m\nmid k_j}} \hat{f}(\boldsymbol{k})+ \sum_{\tau=1}^{\alpha -1}\frac{c_{\tau}(f)}{b^{\tau m}} + R_{s,\alpha,b^m},
\end{align}
where $c_{\tau}(f)$ depends only on $f$ and $\tau$, and the remainder term $R_{s,\alpha,b^m}$ is proven to decay with order $b^{-\alpha m}$.
Now suppose that we have polynomial lattice rules with consecutive sizes of nodes, $b^{m-\alpha+1},b^{m-\alpha+2},\ldots,b^m$.
For ease of notation, we denote by $P_{b^n}$ a polynomial lattice point set with the number of nodes equal to $b^n$, and by $P^{\perp}_{b^n}$ the dual polynomial lattice of $P_{b^n}$.
Then we can obtain a chain of $\alpha$ approximate values of the integral, i.e., $I(f;P_{b^{m-\alpha+1}}),\ldots,I(f;P_{b^m})$.
By applying Richardson extrapolation in a recursive way as described in Section~\ref{subsec:extrapolation}, it follows from Lemma~\ref{lem:extra}, Corollary~\ref{cor:extra} and \eqref{eq:decomp} that the final value is given by
\begin{align}\label{eq:extra_approximation}
\sum_{\tau=1}^{\alpha}a_{\tau}^{(\alpha)} I(f;P_{b^{m-\tau+1}}) = I(f) + \sum_{\tau=1}^{\alpha}a_{\tau}^{(\alpha)}\left( \sum_{\substack{\boldsymbol{k}\in P^{\perp}_{b^{m-\tau+1}}\setminus \{\boldsymbol{0}\}\\ \exists j\colon b^{m-\tau+1}\nmid k_j}} \hat{f}(\boldsymbol{k})+ R_{s,\alpha,b^{m-\tau+1}}\right).
\end{align}
If we can construct good polynomial lattice rules such that the inner sum on the right-hand side of \eqref{eq:extra_approximation} decays with order $b^{-(\alpha-\epsilon)m}$ (with arbitrarily small $\epsilon >0$) for any function $f\in W_{s,\alpha,\boldsymbol{\gamma},q,r}$, the integration error
\[\sum_{\tau=1}^{\alpha}a_{\tau}^{(\alpha)} I(f;P_{b^{m-\tau+1}})-I(f)\]
decays with the almost optimal order.
(Note that we use $N=b^{m-\alpha+1}+\cdots + b^m$ quadrature nodes in total, which does not affects the order of convergence.)
This is our key observation for introducing extrapolated polynomial lattice rules.
In what follows, we start with showing the worst-case error bound of extrapolated polynomial lattice rules, and then in Section~\ref{subsec:euler-maclaurin}, we prove the Euler-Maclaurin formula on the regular grid quadrature.
In Section~\ref{subsec:existence}, we prove the existence of such good polynomial lattice rules for $W_{s,\alpha,\boldsymbol{\gamma},q,r}$ with general weights $\boldsymbol{\gamma}=(\gamma_u)_{u\subset \mathbb{N}}$.
In Section~\ref{sec:cbc}, by restricting to product weights, i.e., the case where the weights are given by the form $\gamma_u=\prod_{j\in u}\gamma_j$ for a sequence of reals $(\gamma_j)_{j\in \mathbb{N}}$, we show that good polynomial lattice rules can be constructed by the fast component-by-component (CBC) algorithm.
\subsection{Worst-case error bound}\label{subsec:worst-case}
Using the equality \eqref{eq:extra_approximation}, the absolute integration error of an extrapolated polynomial lattice rule is bounded by
\begin{align}\label{eq:bound_error}
\left| \sum_{\tau=1}^{\alpha}a_{\tau}^{(\alpha)} I(f;P_{b^{m-\tau+1}}) -I(f)\right| \leq \sum_{\tau=1}^{\alpha}|a_{\tau}^{(\alpha)}|\left( \sum_{\substack{\boldsymbol{k}\in P^{\perp}_{b^{m-\tau+1}}\setminus \{\boldsymbol{0}\}\\ \exists j\colon b^{m-\tau+1}\nmid k_j}} |\hat{f}(\boldsymbol{k})|+ |R_{s,\alpha,b^{m-\tau+1}}|\right).
\end{align}
In the following, we write
\[ P_{b^{m-\tau+1},u}^{\perp}= \left\{\boldsymbol{k}_u\in \mathbb{N}^{|u|} \colon (\boldsymbol{k}_u,\boldsymbol{0})\in P_{b^{m-\tau+1}}^{\perp} \right\}, \]
for a subset $\emptyset \neq u\subseteq \{1,\ldots,s\}$. Note that we have
\[ P_{b^{m-\tau+1}}^{\perp}\setminus \{\boldsymbol{0}\} = \bigcup_{\emptyset \neq u\subseteq \{1,\ldots,s\}}P_{b^{m-\tau+1},u}^{\perp}. \]
We now obtain a worst-case error bound as follows.
\begin{theorem}\label{thm:bound_wrst_error}
Let $\alpha,s\in \mathbb{N}$, $\alpha\geq 2$, $1\leq q,r\leq \infty$, and let $\boldsymbol{\gamma}=(\gamma_u)_{u\subset \mathbb{N}}$ be a set of weights.
Let $q'$ and $r'$ be the H\"older conjugates of $q$ and $r$, respectively.
For $m\geq \alpha$, we have
\begin{align*}
& \sup_{\substack{f\in W_{s,\alpha,\boldsymbol{\gamma},q,r}\\ \|f\|_{s,\alpha,\boldsymbol{\gamma},q,r}\leq 1}}\left| \sum_{\tau=1}^{\alpha}a_{\tau}^{(\alpha)} I(f;P_{b^{m-\tau+1}}) -I(f)\right| \\
& \qquad \qquad \leq \sum_{\tau=1}^{\alpha}|a_{\tau}^{(\alpha)}| \left( B_{\boldsymbol{\gamma},r}(P_{b^{m-\tau+1}})+\frac{H_{s,\boldsymbol{\gamma},q,r}}{b^{\alpha(m-\tau+1)}}\right),
\end{align*}
where
\[ B_{\boldsymbol{\gamma},r}(P_{b^{m-\tau+1}}) = \Bigg(\sum_{\emptyset \neq u\subseteq \{1,\ldots,s\}}\Bigg(\gamma_uC_{\alpha}^{|u|}\sum_{\substack{\boldsymbol{k}_u\in P_{b^{m-\tau+1},u}^{\perp}\\ \exists j\in u\colon b^m\nmid k_j}}b^{-\mu_{\alpha}(\boldsymbol{k}_u)}\Bigg)^{r'} \Bigg)^{1/r'}, \]
and
\[ H_{s,\boldsymbol{\gamma},q,r} = \Bigg(\sum_{u\subseteq \{1,\ldots,s\}}\gamma_u^{r'}(\alpha+1)^{|u|r'/q'}D_{\alpha}^{r'|u|}\Bigg)^{1/r'}, \]
with $D_{\alpha}=\max\left\{|b_1|,\ldots,|b_{\alpha-1}|,\sup_{x\in [0,1)}|\tilde{b}_{\alpha}(x)|\right\}.$
\end{theorem}
\begin{proof}
Let us consider the inner sum on the right-hand side of \eqref{eq:bound_error} first.
Using the bound on the Walsh coefficient in Lemma~\ref{lem:walsh_bound} and H\"older inequality, we have
\begin{align*}
\sum_{\substack{\boldsymbol{k}\in P^{\perp}_{b^{m-\tau+1}}\setminus \{\boldsymbol{0}\}\\ \exists j\colon b^{m-\tau+1} \nmid k_j}} |\hat{f}(\boldsymbol{k})| & = \sum_{\emptyset \neq u\subseteq \{1,\ldots,s\}}\sum_{\substack{\boldsymbol{k}_u\in P^{\perp}_{b^{m-\tau+1},u}\setminus \{\boldsymbol{0}\}\\ \exists j\in u\colon b^{m-\tau+1}\nmid k_j}} |\hat{f}(\boldsymbol{k}_u,\boldsymbol{0})| \\
& \leq \sum_{\emptyset \neq u\subseteq \{1,\ldots,s\}}\|f_u\|_{s,\alpha,\boldsymbol{\gamma},q,r}\gamma_uC_{\alpha}^{|u|} \sum_{\substack{\boldsymbol{k}_u\in P^{\perp}_{b^{m-\tau+1},u}\setminus \{\boldsymbol{0}\}\\ \exists j\in u\colon b^{m-\tau+1}\nmid k_j}} b^{-\mu_{\alpha}(\boldsymbol{k}_u)} \\
& \leq \Bigg( \sum_{\emptyset \neq u\subseteq \{1,\ldots,s\}}\|f_u\|_{s,\alpha,\boldsymbol{\gamma},q,r}^r\Bigg)^{1/r} \\
& \qquad \times \Bigg( \sum_{\emptyset \neq u\subseteq \{1,\ldots,s\}}\Bigg(\gamma_uC_{\alpha}^{|u|} \sum_{\substack{\boldsymbol{k}_u\in P^{\perp}_{b^{m-\tau+1},u}\setminus \{\boldsymbol{0}\}\\ \exists j\in u\colon b^{m-\tau+1}\nmid k_j}} b^{-\mu_{\alpha}(\boldsymbol{k}_u)}\Bigg)^{r'}\Bigg)^{1/r'} \\
& \leq \|f\|_{s,\alpha,\boldsymbol{\gamma},q,r}B_{\boldsymbol{\gamma},r}(P_{b^{m-\tau+1}}).
\end{align*}
Regarding the bound on $R_{\alpha,b^{m-\tau+1}}$, it follows from Theorem~\ref{thm:euler-maclaurin} below that
\[ |R_{s,\alpha,b^{m-\tau+1}}| \leq \frac{\|f\|_{s,\alpha,\boldsymbol{\gamma},q,r} H_{s,\boldsymbol{\gamma},q,r}}{b^{\alpha(m-\tau+1)}}.\]
Plugging these bounds into the right-hand side of \eqref{eq:bound_error} and then taking the supremum among $f\in W_{s,\alpha,\boldsymbol{\gamma},q,r}$ such that $\|f\|_{s,\alpha,\boldsymbol{\gamma},q,r}\leq 1$, the result follows.
\end{proof}
\begin{remark}
As already pointed out in \cite{DKLNS14}, since we have $B_{\boldsymbol{\gamma},r}(P_{b^{m-\tau+1}})\leq B_{\boldsymbol{\gamma},\infty}(P_{b^{m-\tau+1}})$ and $H_{\boldsymbol{\gamma},q,r}\leq H_{\boldsymbol{\gamma},q,\infty}$ for any $r$, it is convenient to work with an upper bound which can be obtained by setting $r=\infty$ and thus $r'=1$. In the rest of this paper, we always consider the case $r=\infty$. The bound $B_{\boldsymbol{\gamma}, r}$ is used below to construct good generating vectors for polynomial lattice rules. The choice $r'=1$ simplifies the computation of $B_{\boldsymbol{\gamma}, r}$.
\end{remark}
\subsection{Euler-Maclaurin formula for regular grid quadrature}\label{subsec:euler-maclaurin}
Here we show the Euler-Maclaurin formula on $I(f;P_{\mathrm{grid},N})$, where
\[ P_{\mathrm{grid},N} = \left\{ \left( \frac{n_1}{N},\ldots,\frac{n_s}{N}\right)\in [0,1)^s\colon 0\leq n_1,\ldots,n_s< N\right\}. \]
As preparation, we prove the following lemma.
\begin{lemma}\label{lem:bernoulli_sum}
For $\tau, N\in \mathbb{N}$ and $x\in [0,1)$, we have
\[ \frac{1}{N}\sum_{n=0}^{N-1}b_{\tau}\left( \frac{n}{N}\right) = \frac{b_{\tau}}{N^{\tau}} \quad \text{and} \quad \frac{1}{N}\sum_{n=0}^{N-1}\tilde{b}_{\tau}\left( x-\frac{n}{N}\right) = \frac{\tilde{b}_{\tau}(Nx)}{N^{\tau}}.\]
\end{lemma}
\begin{proof}
For $\tau=1$, we obtain the results by direct calculation, which is omitted here.
We assume $\tau\ge 2$.
By using the Fourier series of $b_{\tau}$, we have
\begin{align*}
\frac{1}{N}\sum_{n=0}^{N-1}b_{\tau}\left( \frac{n}{N}\right) & = \frac{1}{N}\sum_{n=0}^{N-1}\frac{-1}{(2\pi i)^{\tau}}\sum_{h\in \mathbb{Z}\setminus \{0\}}\frac{e^{2\pi i hn/N}}{h^{\tau}} \\
& = \frac{-1}{(2\pi i)^{\tau}}\sum_{h\in \mathbb{Z}\setminus \{0\}}\frac{1}{h^{\tau}}\left(\frac{1}{N}\sum_{n=0}^{N-1}e^{2\pi i hn/N}\right) \\
& = \frac{-1}{(2\pi i)^{\tau}}\sum_{\substack{h\in \mathbb{Z}\setminus \{0\}\\ N\mid h}}\frac{1}{h^{\tau}} = \frac{-1}{(2\pi i)^{\tau}}\sum_{h\in \mathbb{Z}\setminus \{0\}}\frac{1}{(hN)^{\tau}}= \frac{b_{\tau}}{N^{\tau}},
\end{align*}
which completes the proof of the first equality.
Since the second equality can be proven in exactly the same way by using the Fourier series of $\tilde{b}_{\tau}$, we omit the proof.
\end{proof}
As shown in Lemma~\ref{lem:func_represent}, we have the following pointwise representation for a function $f\in W_{s,\alpha,\boldsymbol{\gamma},q,r}$:
\begin{align}\label{eq:func_represent}
f(\boldsymbol{y}) & = \sum_{u\subseteq \{1,\ldots,s\}}\sum_{v\subseteq u}\sum_{\boldsymbol{\tau}_{u\setminus v}\in \{1,\ldots,\alpha\}^{|u\setminus v|}}\prod_{j\in u\setminus v}b_{\tau_j}(y_j) \nonumber \\
& \qquad \times (-1)^{(\alpha+1)|v|}\int_{[0,1)^s} f^{(\boldsymbol{\tau}_{u\setminus v},\boldsymbol{\alpha}_v,\boldsymbol{0})}(\boldsymbol{x}) \prod_{j\in v}\tilde{b}_{\alpha}(x_j-y_j) \, \mathrm{d} \boldsymbol{x}.
\end{align}
By using Lemma~\ref{lem:bernoulli_sum}, we obtain the Euler-Maclaurin formula on $I(f;P_{\mathrm{grid},N})$.
\begin{theorem}\label{thm:euler-maclaurin}
For $f\in W_{s,\alpha,\boldsymbol{\gamma},q,r}$, we have
\[ I(f;P_{\mathrm{grid},N}) = I(f)+\sum_{\tau=1}^{\alpha -1}\frac{c_{\tau}(f)}{N^{\tau}} + R_{s,\alpha,N}, \]
where $c_{\tau}(f)$ depends only on $f$ and $\tau$, and is given by
\[ c_{\tau}(f) = \sum_{\substack{\boldsymbol{\tau} \in \{0,1,\ldots,\alpha-1\}^{s}\\ |\boldsymbol{\tau}|_1=\tau}}\prod_{j=1}^{s}b_{\tau_j} \int_{[0,1)^{s}}f^{(\boldsymbol{\tau})}(\boldsymbol{x})\, \mathrm{d} \boldsymbol{x} \]
with $ |\boldsymbol{\tau}|_1 = \sum_{j=1}^{s}|\tau_j|$. Further we have
\[ |R_{s,\alpha,N}| \leq \frac{\|f\|_{s,\alpha,\boldsymbol{\gamma},q,r} H_{s,\boldsymbol{\gamma},q,r}}{N^{\alpha}},\]
where $H_{s,\boldsymbol{\gamma},q,r}$ is given as in Theorem~\ref{thm:bound_wrst_error}.
\end{theorem}
\begin{proof}
Plugging the representation \eqref{eq:func_represent} into $I(f;P_{\mathrm{grid},N})$ and using Lemma~\ref{lem:bernoulli_sum}, we have
\begin{align*}
I(f;P_{\mathrm{grid},N}) & = \frac{1}{N^s}\sum_{n_1=0}^{N-1}\cdots\sum_{n_s=0}^{N-1}f\left( \frac{n_1}{N},\ldots,\frac{n_s}{N}\right) \\
& = \sum_{u\subseteq \{1,\ldots,s\}}\sum_{v\subseteq u}\sum_{\boldsymbol{\tau}_{u\setminus v}\in \{1,\ldots,\alpha\}^{|u\setminus v|}}\prod_{j\in u\setminus v}\frac{1}{N}\sum_{n_j=0}^{N-1}b_{\tau_j}\left(\frac{n_j}{N}\right) \\
& \qquad \times (-1)^{(\alpha+1)|v|}\int_{[0,1)^s}f^{(\boldsymbol{\tau}_{u\setminus v},\boldsymbol{\alpha}_{v},\boldsymbol{0})}(\boldsymbol{x}) \prod_{j\in v}\frac{1}{N}\sum_{n_j=0}^{N-1}\tilde{b}_{\alpha}\left(x_j-\frac{n_j}{N}\right)\, \mathrm{d} \boldsymbol{x} \\
& = \sum_{u\subseteq \{1,\ldots,s\}}\sum_{v\subseteq u}\sum_{\boldsymbol{\tau}_{u\setminus v}\in \{1,\ldots,\alpha\}^{|u\setminus v|}}\frac{1}{N^{|\boldsymbol{\tau}_{u\setminus v}|_1+\alpha|v|}}\prod_{j\in u\setminus v}b_{\tau_j} \\
& \qquad \times (-1)^{(\alpha+1)|v|}\int_{[0,1)^{s}}f^{(\boldsymbol{\tau}_{u\setminus v},\boldsymbol{\alpha}_{v},\boldsymbol{0})}(\boldsymbol{x})\prod_{j\in v}\tilde{b}_{\alpha}(Nx_j)\, \mathrm{d} \boldsymbol{x}.
\end{align*}
Let us reorder the summands with respect to the value of $|\boldsymbol{\tau}_{u\setminus v}|_1+\alpha|v|$, which appears in the exponent of $N$. If $|\boldsymbol{\tau}_{u\setminus v}|_1+\alpha|v|=0$, we must have $u=v=\emptyset$ and the corresponding summand is nothing but $I(f)$. If $|\boldsymbol{\tau}_{u\setminus v}|_1+\alpha|v|=\tau$ with $1\leq \tau<\alpha$, we must have $v=\emptyset$ and thus
\begin{align*}
c_{\tau}(f) & = \sum_{\emptyset \neq u\subseteq \{1,\ldots,s\}}\sum_{\substack{\boldsymbol{\tau}_u\in \{1,\ldots,\alpha-1\}^{|u|}\\ |\boldsymbol{\tau}_u|_1=\tau}}\prod_{j\in u}b_{\tau_j} \int_{[0,1)^s}f^{(\boldsymbol{\tau}_u,\boldsymbol{0})}(\boldsymbol{x})\, \mathrm{d} \boldsymbol{x} \\
& = \sum_{\substack{\boldsymbol{\tau} \in \{0,1,\ldots,\alpha-1\}^{s}\\ |\boldsymbol{\tau}|_1=\tau}}\prod_{j=1}^{s}b_{\tau_j} \int_{[0,1)^{s}}f^{(\boldsymbol{\tau})}(\boldsymbol{x})\, \mathrm{d} \boldsymbol{x}.
\end{align*}
The other summands have the exponents $|\boldsymbol{\tau}_{u\setminus v}|_1+\alpha|v| \geq \alpha$ and belong to $R_{s,\alpha,N}$.
Next we prove the bound on $R_{s,\alpha,N}$.
From the above argument, it is obvious that $R_{s,\alpha,N}$ is bounded by
\begin{align*}
|R_{s,\alpha,N}| & \leq \frac{1}{N^{\alpha}}\Bigg|\sum_{u\subseteq \{1,\ldots,s\}}\sum_{v\subseteq u}\sum_{\boldsymbol{\tau}_{u\setminus v}\in \{1,\ldots,\alpha\}^{|u\setminus v|}}\prod_{j\in u\setminus v}b_{\tau_j} \\
& \qquad \times (-1)^{(\alpha+1)|v|}\int_{[0,1)^{s}}f^{(\boldsymbol{\tau}_{u\setminus v},\boldsymbol{\alpha}_{v},\boldsymbol{0})}(\boldsymbol{x})\prod_{j\in v}\tilde{b}_{\alpha}(Nx_j)\, \mathrm{d} \boldsymbol{x} \Bigg| .
\end{align*}
By applying H\"older's inequality, we have
\begin{align*}
& \Bigg|\int_{[0,1)^{s}}f^{(\boldsymbol{\tau}_{u\setminus v},\boldsymbol{\alpha}_{v},\boldsymbol{0})}(\boldsymbol{x})\prod_{j\in v}\tilde{b}_{\alpha}(Nx_j)\, \mathrm{d} \boldsymbol{x} \Bigg| \\
& \quad \leq \int_{[0,1)^{|v|}}\Bigg|\int_{[0,1)^{s-|v|}} f^{(\boldsymbol{\tau}_{u\setminus v},\boldsymbol{\alpha}_{v},\boldsymbol{0})}(\boldsymbol{x})\, \mathrm{d} \boldsymbol{x}_{-v}\Bigg| \cdot \Bigg|\prod_{j\in v}\tilde{b}_{\alpha}(Nx_j)\Bigg|\, \mathrm{d} \boldsymbol{x}_v \\
& \quad \leq D_{\alpha}^{|v|}\Bigg(\int_{[0,1)^{|v|}}\Bigg|\int_{[0,1)^{s-|v|}} f^{(\boldsymbol{\tau}_{u\setminus v},\boldsymbol{\alpha}_{v},\boldsymbol{0})}(\boldsymbol{x})\, \mathrm{d} \boldsymbol{x}_{-v}\Bigg| ^q\, \mathrm{d} \boldsymbol{x}_v\Bigg)^{1/q},
\end{align*}
for $1\leq q\leq \infty$. Using the above inequality and H\"older's inequality twice, we obtain
\begin{align*}
& |R_{s,\alpha,N}| \\
& \leq \frac{1}{N^{\alpha}}\sum_{u\subseteq \{1,\ldots,s\}}\sum_{v\subseteq u}\sum_{\boldsymbol{\tau}_{u\setminus v}\in \{1,\ldots,\alpha\}^{|u\setminus v|}}D_{\alpha}^{|u|} \\
& \qquad \times \Bigg(\int_{[0,1)^{|v|}}\Bigg|\int_{[0,1)^{s-|v|}} f^{(\boldsymbol{\tau}_{u\setminus v},\boldsymbol{\alpha}_{v},\boldsymbol{0})}(\boldsymbol{x})\, \mathrm{d} \boldsymbol{x}_{-v}\Bigg| ^q\, \mathrm{d} \boldsymbol{x}_v\Bigg)^{1/q} \\
& \leq \frac{1}{N^{\alpha}}\sum_{u\subseteq \{1,\ldots,s\}}\Bigg(\sum_{v\subseteq u}\sum_{\boldsymbol{\tau}_{u\setminus v}\in \{1,\ldots,\alpha\}^{|u\setminus v|}}\gamma_u^{q'}D_{\alpha}^{q'|u|}\Bigg)^{1/q'} \\
& \qquad \times \Bigg(\sum_{v\subseteq u}\sum_{\boldsymbol{\tau}_{u\setminus v}\in \{1,\ldots,\alpha\}^{|u\setminus v|}}\gamma_u^{-q}\int_{[0,1)^{|v|}}\Bigg|\int_{[0,1)^{s-|v|}} f^{(\boldsymbol{\tau}_{u\setminus v},\boldsymbol{\alpha}_{v},\boldsymbol{0})}(\boldsymbol{x})\, \mathrm{d} \boldsymbol{x}_{-v}\Bigg| ^q\, \mathrm{d} \boldsymbol{x}_v\Bigg)^{1/q} \\
& \leq \frac{1}{N^{\alpha}}\Bigg(\sum_{u\subseteq \{1,\ldots,s\}}\gamma_u^{r'}(\alpha+1)^{|u|r'/q'}D_{\alpha}^{r'|u|}\Bigg)^{1/r'} \\
& \qquad \times \Bigg(\sum_{u\subseteq \{1,\ldots,s\}} \Bigg(\gamma_u^{-q}\sum_{v\subseteq u}\sum_{\boldsymbol{\tau}_{u\setminus v}\in \{1,\ldots,\alpha\}^{|u\setminus v|}} \\
& \qquad \qquad \qquad \int_{[0,1)^{|v|}}\Bigg|\int_{[0,1)^{s-|v|}} f^{(\boldsymbol{\tau}_{u\setminus v},\boldsymbol{\alpha}_{v},\boldsymbol{0})}(\boldsymbol{x})\, \mathrm{d} \boldsymbol{x}_{-v}\Bigg| ^q\, \mathrm{d} \boldsymbol{x}_v\Bigg)^{r/q}\Bigg)^{1/r} \\
& = \frac{\|f\|_{s,\alpha,\boldsymbol{\gamma},q,r}H_{s,\boldsymbol{\gamma},q,r}}{N^{\alpha}}.
\end{align*}
This completes the proof of this theorem.
\end{proof}
\subsection{Existence results}\label{subsec:existence}
Here we prove the existence of good extrapolated polynomial lattice rules which achieve the almost optimal order of convergence.
Since each point set $P_{b^{m-\tau+1}}$ can be constructed independently, it suffices to prove the existence of a good polynomial lattice rule of size $b^m$ which achieves the almost optimal order of the term $B_{\boldsymbol{\gamma},\infty}(P_{b^m})$ for any $m\in \mathbb{N}$.
In order to emphasize the role of the modulus $p$ and generating vector $\boldsymbol{q}$, instead of $B_{\boldsymbol{\gamma},\infty}(P_{b^m})$ we write
\[ B_{\boldsymbol{\gamma}}(p,\boldsymbol{q}) = \sum_{\emptyset \neq u\subseteq \{1,\ldots,s\}}\gamma_uC_{\alpha}^{|u|}\sum_{\substack{\boldsymbol{k}_u\in P_u^{\perp}(p,\boldsymbol{q})\\ \exists j\in u\colon b^m\nmid k_j}}b^{-\mu_{\alpha}(\boldsymbol{k}_u)}, \]
where $m=\deg(p)$.
First we recall the following auxiliary result. See \cite[Lemma~7]{G16} for the proof.
\begin{lemma}\label{lem:sum_mu_alpha}
For $\alpha\geq 2$ and $1/\alpha<\lambda\leq 1$, we have
\[ \sum_{k=1}^{\infty}b^{-\lambda\mu_{\alpha}(k)} = \sum_{w=1}^{\alpha-1}\prod_{i=1}^{w}\left( \frac{b-1}{b^{\lambda i}-1}\right)+ \left( \frac{b^{\lambda \alpha}-1}{b^{\lambda \alpha}-b}\right)\prod_{i=1}^{\alpha}\left( \frac{b-1}{b^{\lambda i}-1}\right) =: E_{\alpha,\lambda}. \]
\end{lemma}
Now we prove the existence result.
\begin{theorem}\label{thm:existence}
Let $p\in \mathbb{F}_b[x]$ with $\deg(p)=m$ be irreducible.
For a set of weights $\boldsymbol{\gamma}=(\gamma_u)_{u\subset \mathbb{N}}$, there exists at least one $\boldsymbol{q}^*=(q_1^*,\ldots,q_s^*)\in (G^*_{b,m})^s$ such that
\begin{align*}
B_{\boldsymbol{\gamma}}(p,\boldsymbol{q}^*) \leq \frac{1}{(b^m-1)^{1/\lambda}}\left[ \sum_{\emptyset \neq u\subseteq \{1,\ldots,s\}}\gamma_u^{\lambda}C_{\alpha}^{\lambda|u|}E_{\alpha,\lambda}^{|u|}\right]^{1/\lambda}
\end{align*}
holds for any $1/\alpha<\lambda\leq 1$.
\end{theorem}
\begin{proof}
Let $\boldsymbol{q}^*$ be given by
\[ \boldsymbol{q}^* = \arg\min_{\boldsymbol{q} \in (G^*_{b,m})^s}B_{\boldsymbol{\gamma}}(p,\boldsymbol{q}). \]
Using Jensen's inequality, for any $1/\alpha < \lambda\leq 1$ we have
\begin{align*}
(B_{\boldsymbol{\gamma}}(p,\boldsymbol{q}^*))^{\lambda} & \leq \frac{1}{(b^m-1)^s}\sum_{\boldsymbol{q} \in (G^*_{b,m})^s}(B_{\boldsymbol{\gamma}}(p,\boldsymbol{q}))^{\lambda} \\
& \leq \frac{1}{(b^m-1)^s}\sum_{\boldsymbol{q} \in (G^*_{b,m})^s}\sum_{\emptyset \neq u\subseteq \{1,\ldots,s\}}\gamma_u^{\lambda}C_{\alpha}^{\lambda |u|}\sum_{\substack{\boldsymbol{k}_u\in P_u^{\perp}(p,\boldsymbol{q})\\ \exists j\in u\colon b^m\nmid k_j}}b^{-\lambda \mu_{\alpha}(\boldsymbol{k}_u)} \\
& = \sum_{\emptyset \neq u\subseteq \{1,\ldots,s\}}\gamma_u^{\lambda}C_{\alpha}^{\lambda |u|}\sum_{\substack{\boldsymbol{k}_u\in \mathbb{N}^{|u|}\\ \exists j\in u\colon b^m\nmid k_j}}b^{-\lambda \mu_{\alpha}(\boldsymbol{k}_u)}\\
& \qquad \times \frac{1}{(b^m-1)^{|u|}}\sum_{\substack{\boldsymbol{q}_u \in (G^*_{b,m})^{|u|}\\ \mathrm{tr}_m(\boldsymbol{k}_u)\cdot \boldsymbol{q}_u=0 \pmod p}}1.
\end{align*}
If there exists at least one component $k_j$ with $j\in u$ such that $b^m\nmid k_j$, the number of polynomials $\boldsymbol{q}_u\in (G^*_{b,m})^{|u|}$ which satisfies $\mathrm{tr}_m(\boldsymbol{k}_u)\cdot \boldsymbol{q}_u=0\pmod p$ is $(b^m-1)^{|u|-1}$. Thus we obtain
\begin{align*}
(B_{\boldsymbol{\gamma}}(p,\boldsymbol{q}^*))^{\lambda} & \leq \frac{1}{b^m-1}\sum_{\emptyset \neq u\subseteq \{1,\ldots,s\}}\gamma_u^{\lambda}C_{\alpha}^{\lambda |u|}\sum_{\substack{\boldsymbol{k}_u\in \mathbb{N}^{|u|}\\ \exists j\in u\colon b^m\nmid k_j}}b^{-\lambda \mu_{\alpha}(\boldsymbol{k}_u)} \\
& \leq \frac{1}{b^m-1}\sum_{\emptyset \neq u\subseteq \{1,\ldots,s\}}\gamma_u^{\lambda}C_{\alpha}^{\lambda |u|}E_{\alpha,\lambda}^{|u|}.
\end{align*}
This completes the proof.
\end{proof}
\subsection{Dependence of the upper bound on the dimension}\label{subsec:dependence}
Here we study the dependence of the worst-case error bound on the dimension.
For $1/\alpha < \lambda<1$, we write
\[ J_{s,\lambda,\boldsymbol{\gamma}}=\left[ \sum_{\emptyset \neq u\subseteq \{1,\ldots,s\}}\gamma_u^{\lambda}C_{\alpha}^{\lambda|u|}E_{\alpha,\lambda}^{|u|}\right]^{1/\lambda}. \]
From Theorem~\ref{thm:bound_wrst_error} together with Theorem~\ref{thm:existence}, we have
\begin{align*}
& \sup_{\substack{f\in W_{s,\alpha,\boldsymbol{\gamma},q,\infty}\\ \|f\|_{s,\alpha,\boldsymbol{\gamma},q,\infty}\leq 1}}\left| \sum_{\tau=1}^{\alpha}a_{\tau}^{(\alpha)} I(f;P_{b^{m-\tau+1}}) -I(f)\right| \\
& \qquad \leq \sum_{\tau=1}^{\alpha}|a_{\tau}^{(\alpha)}| \left( \frac{J_{s,\lambda,\boldsymbol{\gamma}}}{(b^{m-\tau+1}-1)^{1/\lambda}}+\frac{H_{s,\boldsymbol{\gamma},q,\infty}}{b^{\alpha(m-\tau+1)}}\right) \\
& \qquad \leq \sum_{\tau=1}^{\alpha}|a_{\tau}^{(\alpha)}| \frac{b^{\tau/\lambda}J_{s,\lambda,\boldsymbol{\gamma}}+b^{\alpha(\tau-1)}H_{s,\boldsymbol{\gamma},q,\infty}}{(b^m-1)^{1/\lambda}} \leq \alpha |a_{1}^{(\alpha)}|\frac{b^{\alpha/\lambda}J_{s,\lambda,\boldsymbol{\gamma}}+b^{\alpha(\alpha-1)}H_{s,\boldsymbol{\gamma},q,\infty}}{(b^m-1)^{1/\lambda}},
\end{align*}
for any $1/\alpha < \lambda<1$. Here we recall
\[ H_{s,\boldsymbol{\gamma},q,\infty} = \sum_{u\subseteq \{1,\ldots,s\}}\gamma_u(\alpha+1)^{|u|/q'}D_{\alpha}^{|u|}. \]
The dependence of the upper bound on the dimension can be stated as follows.
\begin{corollary}\label{cor:dependence}
Let $\alpha > 1$ be an integer and $N = b^m + b^{m-1} + \cdots + b^{m-\alpha+1}$ be the number of function evaluations used in the extrapolated polynomial lattice rule.
\begin{enumerate}
\item For general weights, assume that
\[ \lim_{s\to \infty}J_{s,\lambda,\boldsymbol{\gamma}}<\infty\quad \text{and}\quad \lim_{s\to \infty}H_{s,\boldsymbol{\gamma},q,\infty} <\infty, \]
for some $1/\alpha<\lambda\leq 1$. Then the worst-case error for extrapolated polynomial lattice rules converges with order $\mathcal{O}(N^{-1/\lambda})$ with the constant bounded independently of the dimension.
\item For general weights, assume that there exists a positive real $q$ such that
\[ \limsup_{s\to \infty}\frac{J_{s,\lambda,\boldsymbol{\gamma}}}{s^q}<\infty\quad \text{and}\quad \limsup_{s\to \infty}\frac{H_{s,\boldsymbol{\gamma},q,\infty}}{s^q} <\infty, \]
holds for some $1/\alpha<\lambda\leq 1$. Then the worst-case error bound for extrapolated polynomial lattice rules converges with order $\mathcal{O}(N^{-1/\lambda})$ with the constant depending polynomially on the dimension.
\item For product weights $\gamma_u=\prod_{j\in u}\gamma_j$, assume that
\[ \sum_{j=1}^{\infty}\gamma_j^{\lambda}<\infty, \]
for some $1/\alpha<\lambda\leq 1$. Then the worst-case error for extrapolated polynomial lattice rules converges with order $\mathcal{O}(N^{-1/\lambda})$ with the constant bounded independently of the dimension.
\item For product weights $\gamma_u=\prod_{j\in u}\gamma_j$, assume that
\[ \limsup_{s\to \infty}\frac{\sum_{j=1}^{s}\gamma_j^{\lambda}}{\log (s+1)}<\infty, \]
for some $1/\alpha<\lambda\leq 1$. Then the worst-case error bound for extrapolated polynomial lattice rules converges with order $\mathcal{O}(N^{-1/\lambda})$ with the constant depending polynomially on the dimension.
\end{enumerate}
\end{corollary}
\begin{proof}
The results for general weights follows immediately.
The proof of the results for product weights can be also completed by following essentially the same argument as in \cite[Proof of Theorem~5.3]{DP07}.
\end{proof}
\begin{remark}
For product weights, good extrapolated polynomial lattice rules can be constructed as discussed in the next section. As can be seen from the error bound obtained in Theorem~\ref{thm:cbc}, if the same condition as Item~3 or 4 of Corollary~\ref{cor:dependence} holds, we also have exactly the same result for the dependence of the worst-case error bound on the dimension.
\end{remark}
\section{Component-by-component construction}\label{sec:cbc}
\subsection{Convergence analysis}
Here we only consider the case of product weights and prove that the CBC construction algorithm can find a good polynomial lattice rule which achieves the almost optimal order bound on the criterion $B_{\boldsymbol{\gamma}}(p,\boldsymbol{q})$. Remark~\ref{rem_prod} below points out the challenge in generalizing the result to general weights.
The CBC construction algorithm proceeds as follows:
\begin{algorithm}
\label{alg:cbc}
For $m,s\in \mathbb{N}$, $\alpha\geq 2$ and $\boldsymbol{\gamma}=(\gamma_j)_{j\in \mathbb{N}}$.
\begin{enumerate}
\item Choose an irreducible polynomial $p\in \mathbb{F}_b[x]$ with $\deg(p)=m$.
\item Set $q_1^*=1$.
\item For $2\leq d \leq s$, find $q_d^*\in G^*_{b,m}$ which minimizes $$B_{\boldsymbol{\gamma}}(p,(q^*_1,\ldots,q^*_{d-1},q_d))=\sum_{\emptyset \neq u\subseteq \{1,\ldots,d \}}\gamma_uC_{\alpha}^{|u|}\sum_{\substack{\boldsymbol{k}_u\in P_u^{\perp}(p,(q^*_1,\ldots,q^*_{d-1},q_d))\\ \exists j\in u\colon b^m\nmid k_j}}b^{-\mu_{\alpha}(\boldsymbol{k}_u)}$$ as a function of $q_d$.
\end{enumerate}
\end{algorithm}
\noindent In Section~\ref{sec_fast} we simplify the formula for $B_{\boldsymbol{\gamma}}(p,(q^*_1,\ldots,q^*_{d-1},q_d))$ to obtain a criterion which can be computed efficiently.
\begin{theorem}\label{thm:cbc}
Let $p\in \mathbb{F}_b[x]$ with $\deg(p)=m$ and $\boldsymbol{q}_s^*=(q_1^*,\ldots,q_s^*)\in (G^*_{b,m})^s$ be found by Algorithm~\ref{alg:cbc}. Then for $1\leq d \leq s$ we have
\begin{align*}
B_{\boldsymbol{\gamma}}(p,\boldsymbol{q}_d^*)\leq \frac{1}{(b^m-1)^{1/\lambda}}\prod_{j=1}^{d}\left[ 1+ \gamma_j^{\lambda} C_{\alpha}^{\lambda}E_{\alpha,\lambda}\right]^{1/\lambda}
\end{align*}
holds for any $1/\alpha<\lambda\leq 1$.
\end{theorem}
\begin{proof}
Without loss of generality, we assume that the modulus $p$ is monic.
We prove the theorem by induction on $d$.
First let $d=1$.
Since we assume $q_1^*=1$, the dual polynomial lattice is given by
\[ P^{\perp}(p,1) = \{k\in \mathbb{N}_0\colon \mathrm{tr}_m(k)=0\pmod p\} = \{k\in \mathbb{N}_0\colon b^m\mid k\}. \]
Thus we have
\[ B_{\boldsymbol{\gamma}}(p,1)= C_{\alpha}\gamma_1 \sum_{\substack{k\in P^{\perp}(p,1)\setminus \{0\}\\ b^m\nmid k}}b^{-\mu_{\alpha}(k)}=0 \leq \frac{1}{(b^m-1)^{1/\lambda}}\left(1+\gamma^{\lambda}_1C^{\lambda}_{\alpha}E_{\alpha,\lambda}\right)^{1/\lambda}, \]
for any $1/\alpha<\lambda\leq 1$.
Next suppose that we have already found the first $d-1$ components of the generating vector $\boldsymbol{q}_{d-1}^*=(q^*_1,\ldots,q^*_{d-1})\in (G^*_{b,m})^{d-1}$ such that
\begin{align*}
B_{\boldsymbol{\gamma}}(p,\boldsymbol{q}_{d-1}^*)\leq \frac{1}{(b^m-1)^{1/\lambda}}\prod_{j=1}^{d-1}\left[ 1+ \gamma_j^{\lambda} C_{\alpha}^{\lambda}E_{\alpha,\lambda}\right]^{1/\lambda}
\end{align*}
holds for any $1/\alpha<\lambda\leq 1$.
Putting $\boldsymbol{q}_d=(\boldsymbol{q}_{d-1}^*,q_d)$ with $q_d \in G^*_{b,m}$ we have
\begin{align}
B_{\boldsymbol{\gamma}}(p,\boldsymbol{q}_d) & = \sum_{\emptyset \neq u\subseteq \{1,\ldots,d-1\}}\gamma_u C_{\alpha}^{|u|}\sum_{\substack{\boldsymbol{k}_u\in P_u^{\perp}(p,\boldsymbol{q}_d)\\ \exists j\in u\colon b^m\nmid k_j}}b^{-\mu_{\alpha}(\boldsymbol{k}_u)} \nonumber \\
& \qquad + \sum_{\emptyset \neq u\subseteq \{1,\ldots,d-1\}}\gamma_{u\cup\{d\}} C_{\alpha}^{|u|+1}\sum_{\substack{\boldsymbol{k}_{u\cup\{d\}}\in P_{u\cup\{d\}}^{\perp}(p,\boldsymbol{q}_d)\\ \exists j\in u\colon b^m\nmid k_j\\ b^m\mid k_d}}b^{-\mu_{\alpha}(\boldsymbol{k}_{u\cup\{d\}})} \nonumber \\
& \qquad + \sum_{u\subseteq \{1,\ldots,d-1\}}\gamma_{u\cup\{d\}}C_{\alpha}^{|u|+1}\sum_{\substack{\boldsymbol{k}_{u\cup\{d\}}\in P_{u\cup\{d\}}^{\perp}(p,\boldsymbol{q}_d)\\ b^m\nmid k_d}}b^{-\mu_{\alpha}(\boldsymbol{k}_{u\cup\{d\}})} \nonumber \\
& = B_{\boldsymbol{\gamma}}(p,\boldsymbol{q}^*_{d-1}) + \sum_{\emptyset \neq u\subseteq \{1,\ldots,d-1\}}\gamma_{u\cup\{d\}}C_{\alpha}^{|u|+1}\sum_{\substack{\boldsymbol{k}_u\in P_u^{\perp}(p,\boldsymbol{q}_{d-1}^*)\\ \exists j\in u\colon b^m\nmid k_j}}\sum_{\substack{k_d\in \mathbb{N}\\ b^m\mid k_d}}b^{-\mu_{\alpha}(\boldsymbol{k}_u,k_d)} \nonumber \\
& \qquad + \sum_{u\subseteq \{1,\ldots,d-1\}}\gamma_{u\cup\{d\}}C_{\alpha}^{|u|+1}\sum_{\substack{\boldsymbol{k}_{u\cup\{d\}}\in P_{u\cup\{d\}}^{\perp}(p,\boldsymbol{q}_d)\\ b^m\nmid k_d}}b^{-\mu_{\alpha}(\boldsymbol{k}_{u\cup\{d\}})} \nonumber \\
& = B_{\boldsymbol{\gamma}}(p,\boldsymbol{q}^*_{d-1}) \left(1+ \gamma_d C_{\alpha}\sum_{\substack{k_d\in \mathbb{N}\\ b^m\mid k_d}}b^{-\mu_{\alpha}(k_d)}\right) \nonumber \\
& \qquad + \sum_{u\subseteq \{1,\ldots,d-1\}}\gamma_{u\cup\{d\}}C_{\alpha}^{|u|+1}\sum_{\substack{\boldsymbol{k}_{u\cup\{d\}}\in P_{u\cup\{d\}}^{\perp}(p,\boldsymbol{q}_d)\\ b^m\nmid k_d}}b^{-\mu_{\alpha}(\boldsymbol{k}_{u\cup\{d\}})} , \label{eq:decomp_error}
\end{align}
where the second equality stems from the fact that since $b^m\mid k_d$, we have $\mathrm{tr}_m(k_d)=0$ and thus $\mathrm{tr}_m(\boldsymbol{k}_{u\cup \{d\}})\cdot (\boldsymbol{q}_u^*, q_d)=\mathrm{tr}_m(\boldsymbol{k}_u)\cdot \boldsymbol{q}_u^*$, which yields
\[ \{\boldsymbol{k}_{u\cup \{d\}}\in P_{u\cup \{d\}}^{\perp}(p,\boldsymbol{q}_d)\colon b^m\mid k_d\}= \{(\boldsymbol{k}_u,k_d)\in \mathbb{N}^{|u|+1}\colon \boldsymbol{k}_u\in P_u^{\perp}(p,\boldsymbol{q}_{d-1}^*), b^m \mid k_d\}. \]
It is clear that the first term of \eqref{eq:decomp_error} does not depend on the choice of $q_d$.
Thus denoting the second term of \eqref{eq:decomp_error} by
\[ \psi_{p,\boldsymbol{q}_{d-1}^*}(q_d):= \sum_{u\subseteq \{1,\ldots,d-1\}}\gamma_{u\cup\{d\}}C_{\alpha}^{|u|+1}\sum_{\substack{\boldsymbol{k}_{u\cup\{d\}}\in P_{u\cup\{d\}}^{\perp}(p,\boldsymbol{q}_d)\\ b^m\nmid k_d}}b^{-\mu_{\alpha}(\boldsymbol{k}_{u\cup\{d\}})}, \]
we have
\[ q_d^*=\arg\min_{q_d\in G^*_{b,m}} B_{\boldsymbol{\gamma}}(p,\boldsymbol{q}_d) = \arg\min_{q_d\in G^*_{b,m}} \psi_{p,\boldsymbol{q}_{d-1}^*}(q_d) . \]
Using Jensen's inequality, as long as $1/\alpha<\lambda\leq 1$, we have
\begin{align*}
& (\psi_{p,\boldsymbol{q}_{d-1}^*}(q_d^*))^{\lambda} \\
& \leq \frac{1}{b^m-1}\sum_{q_d\in G^*_{b,m}}(\psi_{p,\boldsymbol{q}_{d-1}^*}(q_d))^{\lambda} \\
& \leq \frac{1}{b^m-1}\sum_{q_d\in G^*_{b,m}}\sum_{u\subseteq \{1,\ldots,d-1\}}\gamma^{\lambda}_{u\cup\{d\}}C_{\alpha}^{\lambda(|u|+1)}\sum_{\substack{\boldsymbol{k}_{u\cup\{d\}}\in P_{u\cup\{d\}}^{\perp}(p,\boldsymbol{q}_d)\\ b^m\nmid k_d}}b^{-\lambda\mu_{\alpha}(\boldsymbol{k}_{u\cup\{d\}})} \\
& = \frac{1}{b^m-1}\sum_{u\subseteq \{1,\ldots,d-1\}}\gamma^{\lambda}_{u\cup\{d\}}C_{\alpha}^{\lambda(|u|+1)}\sum_{\substack{\boldsymbol{k}_{u\cup\{d\}}\in \mathbb{N}^{|u|+1}\\ b^m\nmid k_d}}b^{-\lambda\mu_{\alpha}(\boldsymbol{k}_{u\cup\{d\}})} \\
& \qquad \times \sum_{\substack{q_d\in G^*_{b,m}\\ \mathrm{tr}_m(\boldsymbol{k}_u)\cdot \boldsymbol{q}_u^{*}+\mathrm{tr}_m(k_d)q_d = 0\pmod p}}1.
\end{align*}
Since $b^m\nmid k_d$, we have $\mathrm{tr}_m(k_d)\neq 0$.
For $\boldsymbol{k}_u\in P_u^{\perp}(p,\boldsymbol{q}_{d-1}^*)$, it follows from the definition of the dual polynomial lattice that $\mathrm{tr}_m(\boldsymbol{k}_u)\cdot \boldsymbol{q}_u^{*}=0\pmod p$, and thus there is no polynomial $q_d\in G^*_{b,m}$ such that the condition $\mathrm{tr}_m(k_d)q_d = 0\pmod p$ is satisfied.
For $\boldsymbol{k}_u\notin P_u^{\perp}(p,\boldsymbol{q}_{d-1}^*)$, there exists exactly one $q_d\in G^*_{b,m}$ such that $\mathrm{tr}_m(k_d)q_d = -\mathrm{tr}_m(\boldsymbol{k}_u)\cdot \boldsymbol{q}_u^{*} \pmod p$. From these facts and Lemma~\ref{lem:sum_mu_alpha}, we obtain
\begin{align*}
(\psi_{p,\boldsymbol{q}_{d-1}^*}(q_d^*))^{\lambda} & \leq \frac{1}{b^m-1}\sum_{u\subseteq \{1,\ldots,d-1\}}\gamma^{\lambda}_{u\cup\{d\}}C_{\alpha}^{\lambda(|u|+1)}\sum_{\substack{\boldsymbol{k}_u\in \mathbb{N}^{|u|}\\ \boldsymbol{k}_u\notin P_u^{\perp}(p,\boldsymbol{q}_{d-1}^*)}}\sum_{\substack{k_d\in \mathbb{N}\\ b^m\nmid k_d}}b^{-\lambda\mu_{\alpha}(\boldsymbol{k}_u,k_d)} \\
& \leq \frac{1}{b^m-1}\sum_{u\subseteq \{1,\ldots,d-1\}}\gamma^{\lambda}_{u\cup\{d\}}C_{\alpha}^{\lambda(|u|+1)}\sum_{\boldsymbol{k}_u\in \mathbb{N}^{|u|}}b^{-\lambda\mu_{\alpha}(\boldsymbol{k}_u)}\sum_{\substack{k_d\in \mathbb{N}\\ b^m\nmid k_d}}b^{-\lambda\mu_{\alpha}(k_d)} \\
& = \frac{1}{b^m-1}\prod_{j=1}^{d-1}\left[ 1+ \gamma_j^{\lambda} C_{\alpha}^{\lambda}E_{\alpha,\lambda}\right]\cdot \gamma_d^{\lambda} C_{\alpha}^{\lambda}\sum_{\substack{k_d\in \mathbb{N}\\ b^m\nmid k_d}}b^{-\lambda\mu_{\alpha}(k_d)}.
\end{align*}
Finally by applying Jensen's inequality to \eqref{eq:decomp_error} and using Lemma~\ref{lem:sum_mu_alpha}, we have
\begin{align*}
(B_{\boldsymbol{\gamma}}(p,\boldsymbol{q}^*_d))^{\lambda} & \leq (B_{\boldsymbol{\gamma}}(p,\boldsymbol{q}^*_{d-1}))^{\lambda}\left(1+ \gamma_d^{\lambda}C^{\lambda}_{\alpha}\sum_{\substack{k_d\in \mathbb{N}\\ b^m\mid k_d}}b^{-\lambda\mu_{\alpha}(k_d)}\right) \\
& \qquad + \frac{1}{b^m-1}\prod_{j=1}^{d-1}\left[ 1+ \gamma_j^{\lambda} C_{\alpha}^{\lambda}E_{\alpha,\lambda}\right]\cdot \gamma_d^{\lambda} C_{\alpha}^{\lambda}\sum_{\substack{k_d\in \mathbb{N}\\ b^m\nmid k_d}}b^{-\lambda\mu_{\alpha}(k_d)} \\
& \leq \frac{1}{b^m-1}\prod_{j=1}^{d-1}\left[ 1+ \gamma_j^{\lambda} C_{\alpha}^{\lambda}E_{\alpha,\lambda}\right] \cdot \left[ 1+\gamma_d^{\lambda}C^{\lambda}_{\alpha}\sum_{k_d\in \mathbb{N}}b^{-\lambda\mu_{\alpha}(k_d)}\right] \\
& = \frac{1}{b^m-1}\prod_{j=1}^{d}\left[ 1+ \gamma_j^{\lambda} C_{\alpha}^{\lambda}E_{\alpha,\lambda}\right] .
\end{align*}
This completes the proof.
\end{proof}
\begin{remark}\label{rem_prod}
In the above proof, we use the property of product weights to obtain the equality \eqref{eq:decomp_error}.
In fact, this is a crucial step to get the almost optimal order upper bound on $B_{\boldsymbol{\gamma}}(p,\boldsymbol{q})$.
Thus it is an open question whether a similar proof goes through for general weights.
\end{remark}
\subsection{Fast construction algorithm}\label{sec_fast}
In the convergence analysis above, we used the criterion $B_{\boldsymbol{\gamma}}(p,\boldsymbol{q}_d)$.
However, since the quantity
\[ \sum_{\emptyset \neq u\subseteq \{1,\ldots,d\}}\gamma_uC_{\alpha}^{|u|}\sum_{\substack{\boldsymbol{k}_u\in P_u^{\perp}(p,\boldsymbol{q}_d)\\ \forall j\in u\colon b^m\mid k_j}}b^{-\mu_{\alpha}(\boldsymbol{k}_u)} = \sum_{\emptyset \neq u\subseteq \{1,\ldots,d\}}\gamma_uC_{\alpha}^{|u|}\sum_{\boldsymbol{k}_u\in \mathbb{N}^{|u|}}b^{-\mu_{\alpha}(b^m\boldsymbol{k}_u)} \]
does not depend on the choice of generating vector $\boldsymbol{q}_d$, we can add this quantity to the criterion $B_{\boldsymbol{\gamma}}(p,\boldsymbol{q}_d)$ to get another criterion
\begin{align*}
\tilde{B}_{\boldsymbol{\gamma}}(p,\boldsymbol{q}_d) & = B_{\boldsymbol{\gamma}}(p,\boldsymbol{q}_d) + \sum_{\emptyset \neq u\subseteq \{1,\ldots,d\}}\gamma_u C_{\alpha}^{|u|}\sum_{\substack{\boldsymbol{k}_u\in P_u^{\perp}(p,\boldsymbol{q}_d)\\ \forall j\in u\colon b^m\mid k_j}}b^{-\mu_{\alpha}(\boldsymbol{k}_u)} \\
& = \sum_{\emptyset \neq u\subseteq \{1,\ldots,d\}}\gamma_u C_{\alpha}^{|u|}\sum_{\boldsymbol{k}_u\in P_u^{\perp}(p,\boldsymbol{q}_d)}b^{-\mu_{\alpha}(\boldsymbol{k}_u)} \\
& = -1+\frac{1}{b^m}\sum_{n=0}^{b^m-1}\sum_{u\subseteq \{1,\ldots,d\}}\gamma_u C_{\alpha}^{|u|}\sum_{\boldsymbol{k}_u\in \mathbb{N}^{|u|}}b^{-\mu_{\alpha}(\boldsymbol{k}_u)}\mathrm{wal}_{(\boldsymbol{k}_u,\boldsymbol{0})}(\boldsymbol{x}_n) \\
& = -1+\frac{1}{b^m}\sum_{n=0}^{b^m-1}\prod_{j=1}^{d}\left[ 1+ \gamma_j C_{\alpha} w_{\alpha}\left(v_m\left( \frac{nq_j}{p}\right) \right)\right],
\end{align*}
where we used Lemma~\ref{lem:character} in the third equality, and the function $w_{\alpha}\colon [0,1)\to \mathbb{R}$ is defined by
\[ w_{\alpha}(x) = \sum_{k=1}^{\infty}b^{-\mu_{\alpha}(k)}\mathrm{wal}_k(x). \]
As shown in \cite[Theorem~2]{BDLNP12}, one can compute $w_{\alpha}$ efficiently when $x$ is a $b$-adic rational.
More precisely, if $x$ is of the form $a/b^m$ for $m\in \mathbb{N}$ and $0\leq a< b^m$, $w_{\alpha}(x)$ can be computed in at most $O(\alpha m)$ operations.
Furthermore, in case of $b=2$, we have explicit formulas for $w_2$ and $w_3$, see \cite[Corollary~1]{BDLNP12}.
In what follows, we show how one can use the fast CBC construction algorithm to find suitable polynomials $q_1^*,\ldots,q_s^*\in G^*_{b,m}$ by employing $\tilde{B}_{\boldsymbol{\gamma}}(p,\boldsymbol{q})$ as a quality measure.
Assume that $q^*_1=1,q^*_2,\ldots,q^*_{d-1}$ are already found.
Let
\[ P_{n,d-1}=\prod_{j=1}^{d-1}\left[ 1+ \gamma_j C_{\alpha} w_{\alpha}\left(v_m\left( \frac{nq^*_j}{p}\right) \right)\right], \]
for $0\leq n<b^m$. Note that we have
\[ P_{0,d-1}=\prod_{j=1}^{d-1}\left[ 1+ \gamma_j C_{\alpha} w_{\alpha}\left(0 \right)\right] , \]
regardless of the choice $q^*_1,q^*_2,\ldots,q^*_{d-1}$.
Now the criterion $\tilde{B}_{\boldsymbol{\gamma}}(p,\boldsymbol{q}_d)$ is given by
\begin{align*}
\tilde{B}_{\boldsymbol{\gamma}}(p,\boldsymbol{q}_d) & = -1+\frac{1}{b^m}\sum_{n=0}^{b^m-1}P_{n,d-1}\left[ 1+ \gamma_d C_{\alpha} w_{\alpha}\left(v_m\left( \frac{nq_d}{p}\right) \right)\right] \\
& = -1+\frac{P_{0,d}}{b^m} + \frac{1}{b^m}\sum_{n=1}^{b^m-1}P_{n,d-1}\left[ 1+ \gamma_d C_{\alpha} w_{\alpha}\left(v_m\left( \frac{nq_d}{p}\right) \right)\right] \\
& = -1+\frac{P_{0,d}}{b^m} + \frac{1}{b^m}\sum_{n=1}^{b^m-1}P_{n,d-1}+ \frac{\gamma_d C_{\alpha}}{b^m}\sum_{n=1}^{b^m-1}P_{n,d-1} w_{\alpha}\left(v_m\left( \frac{nq_d}{p}\right) \right).
\end{align*}
Thus it is obvious that the CBC algorithm finds a component $q_d^*$ which minimizes the last sum.
Since the modulus $p$ is assumed to be irreducible, there exists a primitive polynomial $g\in \mathbb{F}_b[x]/p$ for which we have $\{g^0=g^{b^m-1}=1, g^{1},\ldots,g^{b^m-2}\}=(\mathbb{F}_b[x]/p)\setminus \{0\}$, and then the last sum for a polynomial $q_d=g^{z}$ with $1\leq z\leq b^m-1$ is equivalent to
\begin{align*}
\sum_{n=1}^{b^m-1}P_{n,d-1} w_{\alpha}\left(v_m\left( \frac{nq_d}{p}\right) \right) = \sum_{n=1}^{b^m-1} P_{g^{-n},d-1} w_{\alpha}\left(v_m\left( \frac{g^{z-n}}{p}\right) \right) =: \eta_z,
\end{align*}
where we note that the subscript $g^{-n}$ appearing in $P_{g^{-n},d-1}$ is identified with the integer in $\{1,\ldots,b^m-1\}$.
We define the circulant matrix
\[ A = \omega_{\alpha}\left( v_m\left( \frac{g^{z-n}}{p}\right)\right)_{1\leq z,n\leq b^m-1},\]
and compute
\[ (\eta_1,\ldots,\eta_{b^m-1})^{\top} = A \cdot (P_{g^{-1},\tau-1},P_{g^{-2},\tau-1},\ldots,P_{g^{-b^m+1},d-1})^{\top}.\]
Let $z_0$ be an integer such that $\eta_{z_0}\leq \eta_{z}$ holds for any $1\leq z\leq b^m-1$. Then we set $q_d^*=g^{z_0}$.
Since the matrix $A$ is circulant, the matrix-vector multiplication above can be done by using the fast Fourier transform in $O(mb^m)$ arithmetic operations with $O(b^m)$ memory space for $P_{n,d-1}$, see \cite{NC06a,NC06b}.
Therefore, we can compute the vector $(\eta_1,\ldots,\eta_{b^m-1})$ in a fast way.
After finding $q_d^*=g^{z_0}$, each $P_{n,d-1}$ is updated simply by
\[ P_{g^{-n},d} = P_{g^{-n},d-1} \left( 1+ \gamma_d C_{\alpha} w_{\alpha}\left(v_m\left( \frac{g^{z_0-n}}{p}\right) \right)\right). \]
Since each element of the circulant matrix $A$ can be calculated in at most $O(\alpha m)$ arithmetic operations, calculating one row (or one column) of $A$ requires $O(\alpha mb^m)$ arithmetic operations as the first step of the CBC algorithm.
Then the CBC algorithm proceeds in an inductive way as described above, yielding $O((s+\alpha)mb^m)$ arithmetic operations with $O(b^m)$ memory space for finding the generating vector $\boldsymbol{q}^*\in (G^*_{b,m})^s$.
Further, for an extrapolated polynomial lattice rule, we need to construct polynomial lattice rules with $\alpha$ consecutive sizes of nodes, $b^{m-\alpha+1},\ldots,b^{m}$, implying that the total number of points is $N=b^{m-\alpha+1}+\cdots+b^{m}$.
The obvious inequality
\[ \sum_{\tau=1}^{\alpha}(s+\alpha)(m-\tau+1)b^{m-\tau+1} \leq (s+\alpha)m N \leq (s+\alpha)N\log_b N\]
shows that the total construction cost is of $O((s+\alpha)N \log N)$ together with $O(N)$ memory space, which improves the currently known result for an interlaced polynomial lattice rule that requires $O(s\alpha N\log N)$ arithmetic operations with $O(N)$ memory space \cite{G15,GD15}.
\section{Numerical experiments}\label{sec:experiment}
As a low-dimensional problem, let us consider a simple bi-variate test function
\[ f(x,y)=\frac{ye^{xy}}{e-2}, \]
whose exact value of $I(f)$ equals 1. This function has been often used in the literature, see for instance \cite[Chapter~8]{SJbook}. We approximate $I(f)$ by using extrapolated polynomial lattice rules over $\mathbb{F}_2$ and also by using interlaced polynomial lattice rules over $\mathbb{F}_2$ for comparison. Here extrapolated polynomial lattice rules are constructed by the fast CBC algorithm as described in Section~\ref{sec_fast} with the constant $C_{\alpha}=1$, which is justified as mentioned in Remark~\ref{rem:walsh_bound}, whereas interlaced polynomial lattice rules are constructed by the fast CBC algorithm based on a computable quality criterion given in \cite[Corollary~3]{G15}. For both the rules, we set $\gamma_1=\gamma_2=1$ within the CBC algorithm.
\begin{figure}[!b]
\centering
\includegraphics[width=0.49\textwidth]{testfunc0_order2.eps}
\includegraphics[width=0.49\textwidth]{testfunc0_order3.eps}
\caption{The results for $f(x,y)=ye^{xy}/(e-2)$ by using extrapolated polynomial lattice rules (solid) and interlaced polynomial lattice rules (dashed) with $\alpha=2$ (left) and $\alpha=3$ (right).}
\label{fig:test0}
\end{figure}
Figure~\ref{fig:test0} shows the results for the cases $\alpha=2$ (left) and $\alpha=3$ (right). The absolute integration errors as functions of $\log_2 N$ are shown in each graph. The solid lines denote the results for extrapolated polynomial lattice rules and the dashed lines for interlaced polynomial lattice rules. For reference, the dotted lines correspond to $O(N^{-1})$ and $O(N^{-2})$ convergences for $\alpha=2$, and to $O(N^{-2})$ and $O(N^{-3})$ convergences for $\alpha=3$. For the case $\alpha=2$, both the rules perform comparably and achieve approximately the desired rate of the error convergence $O(N^{-2})$. For the case $\alpha=3$, although interlaced polynomial lattice rules outperform extrapolated polynomial lattice rules, we see that the rate of the error convergence for extrapolated polynomial lattice rules asymptotically improves towards the expected $O(N^{-3})$, which supports our theoretical funding.
Next let us consider the following high-dimensional test integrands
\begin{align*}
f_1(\boldsymbol{x}) & = \prod_{j=1}^{s}\left[ 1+\gamma_j\left(x_j^{c_1}-\frac{1}{1+c_1}\right)\right] ,\\
f_2(\boldsymbol{x}) & = \prod_{j=1}^{s}\left[ 1+\frac{\gamma_j}{1+\gamma_j x_j^{c_2}} \right] ,
\end{align*}
for positive constants $c_1,c_2>0$.
Note that the exact values of the integrals for $f_1$ and for $f_2$ with the special cases $c_2=1$ and $c_2=2$ are known. We put $s=100$ and $\gamma_j=j^{-2}$. We construct both extrapolated polynomial lattice rules and interlaced polynomial lattice rules by using the fast CBC algorithm with the same choice of the weights $\gamma_j=j^{-2}$. Note that, in our experiments, we do not observe the phenomenon that the same elements of the generating vector repeat as pointed out in \cite{GS16}.
\begin{figure}[!p]
\centering
\includegraphics[width=0.49\textwidth]{testfunc1_order2.eps}
\includegraphics[width=0.49\textwidth]{testfunc1_order3.eps}\\
\includegraphics[width=0.49\textwidth]{testfunc2_1_order2.eps}
\includegraphics[width=0.49\textwidth]{testfunc2_1_order3.eps}\\
\includegraphics[width=0.49\textwidth]{testfunc2_2_order2.eps}
\includegraphics[width=0.49\textwidth]{testfunc2_2_order3.eps}
\caption{The results for $f_1$ with $c_1=1.3$ (top), $f_2$ with $c_2=1$ (middle), and $f_2$ with $c_2=2$ (bottom) by using extrapolated polynomial lattice rules (solid) and interlaced polynomial lattice rules (dashed) with $\alpha=2$ (left) and $\alpha=3$ (right).}
\label{fig:test1}
\end{figure}
Figure~\ref{fig:test1} shows the results for the cases $\alpha=2$ (left column) and $\alpha=3$ (right column). Each row corresponds to the results for $f_1$ with $c_1=1.3$, $f_2$ with $c_2=1$, and $f_2$ with $c_2=2$, respectively. Again, for reference, the dotted lines correspond to $O(N^{-1})$ and $O(N^{-2})$ convergences for $\alpha=2$, and to $O(N^{-2})$ and $O(N^{-3})$ convergences for $\alpha=3$. For the case $\alpha=2$, extrapolated polynomial lattice rules perform competitively with interlaced polynomial lattice rules and achieve approximately the desired rate of the error convergence $O(N^{-2})$. For the case $\alpha=3$, similarly to the result for the bi-variate test function, interlaced polynomial lattice rules outperform extrapolated polynomial lattice rules, but the rate of the error convergence for extrapolated polynomial lattice rules improves as the number of points increases.
These numerical results indicate that extrapolated polynomial lattices rule can be quite useful in fast QMC matrix-vector multiplication with higher order convergence, which shall be undertaken in the near future.
\section*{Acknowledgments}
The second author would like to thank Professor Josef Dick for his hospitality while visiting the University of New South Wales where most of this research was carried out.
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in Data Studio
StarCoder2 Extras
This is the dataset of extra sources (besides Stack v2 code data) used to train the StarCoder2 family of models. It contains the following subsets:
- Kaggle (
kaggle): Kaggle notebooks from Meta-Kaggle-Code dataset, converted to scripts and prefixed with information on the Kaggle datasets used in the notebook. The file headers have a similar format to Jupyter Structured but the code content is only one single script. - StackOverflow (
stackoverflow): stackoverflow conversations from this StackExchange dump. - Issues (
issues): processed GitHub issues, same as the Stack v1 issues. - OWM (
owm): the Open-Web-Math dataset. - LHQ (
lhq): Leandro's High quality dataset, it is a compilation of high quality code files from: APPS-train, CodeContests, GSM8K-train, GSM8K-SciRel, DeepMind-Mathematics, Rosetta-Code, MultiPL-T, ProofSteps, ProofSteps-lean. - Wiki (
wikipedia): the English subset of the Wikipedia dump in RedPajama. - ArXiv (
arxiv): the ArXiv subset of RedPajama dataset, further processed the dataset only to retain latex source files and remove preambles, comments, macros, and bibliographies from these files. - IR_language (
ir_cpp,ir_low_resource,ir_python,ir_rust): these are intermediate representations of Python, Rust, C++ and other low resource languages. - Documentation (
documentation): documentation of popular libraries.
For more details on the processing of each subset, check the StarCoder2 paper or The Stack v2 GitHub repository.
Usage
from datasets import load_dataset
# replace `kaggle` with one of the config names listed above
ds = load_dataset("bigcode/starcoder2data-extras", "kaggle", split="train")
Citation
@article{lozhkov2024starcoder,
title={Starcoder 2 and the stack v2: The next generation},
author={Lozhkov, Anton and Li, Raymond and Allal, Loubna Ben and Cassano, Federico and Lamy-Poirier, Joel and Tazi, Nouamane and Tang, Ao and Pykhtar, Dmytro and Liu, Jiawei and Wei, Yuxiang and others},
journal={arXiv preprint arXiv:2402.19173},
year={2024}
}
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