\section{Introduction}

Penetration of magnetic flux into hollow superconducting cylinders is a long standing field of interest. The Little-Parks effect and the quantization of trapped flux were intensively studied during the last fifty years~\cite{LITTLE,DOU,VEKHT}. Recent advances in nanotechnology have made it possible for studying experimentally superconducting properties of thin films with different arrays of antidots, see for example,~\cite{Motta1} and references therein. In particular, for the observation of the aforementioned effects, cylinders or antidots of small diameter are required. At the same time, the study of hollow thin-walled cylinders with macroscopic sizes in magnetic fields parallel to its axis has been much less well studied. It was expected that quantum phenomena cannot be observed in such samples
because of the fact that one flux quanta for cylinders with a cross section area of $\approx 1$ cm$^2$ corresponds to a magnetic field about $10^{-7}$ Oe. In this case magnetization will be a smooth function of the magnetic field. However, experimental results obtained recently for thin-walled macroscopic cylinders do not agree with this expectation. Namely, in such Nb cylinders we succeeded in monitoring the magnetic moment of the current circulating in the walls and observed dc magnetic moment jumps even in fields much lower than $H_{c1}$ of the film itself~\cite{Katz1}.
So far it is not clear what mechanism is responsible for such flux jumps. Under an axial magnetic field the cylinder walls screen weak external fields, provided that $L\equiv Rd/\lambda^{2} \gg 1$, where $R$ is the cylinder radius, $d$ is the wall thickness, and $\lambda$ is the London penetration depth ~\cite{DOU,PG,KITTEL}. Therefore, it is expected, that a dc magnetic field, $H_0$, will penetrate into the cylinder as soon as the current in the wall exceeds the critical current and no field penetration should be observed at lower fields. Only above $H_{c1}$, vortices created at the outer cylinder surface can move into the cylinder. For a magnetic field oriented perpendicular to the Nb film surface such vortex motion leads to flux jumps~\cite{NOWAK,STAM}. These flux jumps were interpreted as a thermomagnetic instability of the critical state. It was demonstrated that in a sample with an array of antidots flux jump propagates along the antidots row \cite{MOTTA2}.

Nucleation of the superconducting phase in a thin surface sheath in decreasing magnetic fields parallel to the sample surface was predicted by Saint-James and de Gennes~\cite{DSJ}. They showed that nucleation occurs in a magnetic field $H_0\leq H_{c3}\approx 1.695 H_{c2}$. Experimental confirmations of this prediction were obtained soon after their work appeared. The experimental methods for this confirmation were dc resistivity and ac susceptibility measurements~\cite{ROLL}. It was found that low frequency losses in superconductors in surface superconducting states (SSS) can exceed losses in the normal state~\cite{ BURGER,ROLL}.

A swept dc magnetic field qualitatively changes the character of the ac response. Specifically, the penetration of the ac magnetic field into the sample takes place not only for $H_{c2}<H_0<H_{c3}$ but also for $H_{c1}<H_0<H_{c2}$, in sharp contrast to the case of constant dc fields~\cite{STR2,MAX,GENKIN22}. The effect of a swept dc field can more suitably be investigated by using hollow thin-walled superconducting cylinders, rather than by bulk samples, because one can control the field transmission through their walls. Previously, we have shown~\cite{Genkin1} that in a thin-walled cylinder in the mixed state, the effect of sweeping a dc field on the ac response is due to an enhancement of the vortex motion through the wall. Above $H_{c2}$, however, this picture is no longer appropriate and the experimental data were explained within the framework of a simple relaxation model~\cite{Katz1}.

The goal of this paper is to study how antidots affect the penetration of dc and ac magnetic fields into thin-walled superconducting Nb cylinders of macroscopic sizes, with a rectangular cross section.
 We show that at low enough temperatures for both, a flat and a patterned samples, even in the \emph{vortex-free regime} at $H< H_{c1}$, the dc magnetic field penetrates through the cylinder walls in an ``\emph{avalanche}''-like fashion. Jumps of the dc magnetic moment also become apparent at fields above $H_{c1}$ at low temperatures. For both samples, the field values at which jumps occur vary from one measurement to another, indicating that one deals with transitions between metastable states. At temperatures above $0.66T_c$ and $0.78T_c$ the magnetization curves become smooth for the patterned and the as-prepared sample, respectively.

The ac response of both cylinders was studied in the point-by-point and swept field modes. In these, the signals of the first, second and third harmonics were measured concurrently. The ac response of as-prepared and patterned samples is qualitatively different in a swept field mode.

\section{Experimental}

The cylindrical samples were prepared by dc magnetron sputtering at room temperature on a rotated sapphire substrate. The sizes of the substrate with rounded corners (radius 0.2 mm) are $1.5\times3\times7.5$ mm$^3$. We fabricated, therefore, a thin-walled hollow superconducting cylinder with a rectangular cross section. The nominal film thickness of both samples was $d=100$ nm. A sketch of the sample geometry is presented in Fig.~\ref{f1}.

The reference sample $A$ was kept as-grown, while the second one, sample $B$, was patterned with a row of antidots at the mid of the larger surface over the entire length of the sample.
The row of antidots was milled by focused ion beam (FIB) in a scanning electron microscope (FEI, Nova Nanolab 600). The beam parameters were 30\,kV/0.5\,nA, while the defocus and blur were 560\,$\mu$m and 3\,$\mu$m, respectively. The pitch was equal to the antidot center-to-center distance of 1.8\,$\mu$m and the number of beam passes needed to mill 150\,nm-deep antidots was 2000. The antidots row with a length of $7.5$\,mm was milled by iteratively stitching the processing window with a long size of $400\,\mu$m. SEM images of the patterned surface of sample $B$ are shown in Fig.~\ref{f2}. The antidots have an average diameter of 1.5\,$\mu$m and an average edge-to-edge distance of 300\,nm.

The dc magnetic properties were measured using a commercial superconducting quantum interference device (SQUID), Quantum Desing MPMS5, magnetometer. The ac response was measured by the pick-up coil method. The sample was inserted into one coil of a balanced pair of coils, and the unbalanced signal was measured by means of lock-in amplifier. The ac magnetic susceptibilities were measured in absolute units, see~\cite{LEV2}. A ``home-made'' measurement cell of the experimental setup was adapted to the SQUID magnetometer. A block diagram of the experimental setup can be found elsewhere~\cite{LEV2}.

The ac response as a function of the dc field were carried out by two methods: (i) - point-by-point (PBP) mode, where the dc field was kept constant during the measurement, and (ii) - swept field (SF) mode, where the dc field was ramped with a rate of 20 Oe/s. Both external ac and dc fields were directed parallel to cylinder axis and hence, to the film surface.
\begin{figure
\begin{center}
\leavevmode
\includegraphics[width=0.6\linewidth]{Fig1}
\caption{Sketch of the $B$ sample. Here $\text{L}_s = 7.5$ mm, $\text{W}_s=3$ mm, and $2\text{D}=1.4$ mm are the substrate length, width and thickness, respectively. Both dc and ac fields were parallel to $Z$-axis. Dimensions are not to scale.}
\label{f1}
\end{center}
\end{figure}

\begin{figure
\centering
       \includegraphics[width=0.94\linewidth]{Fig2a.eps}
      \includegraphics[width=0.94\linewidth]{Fig2b.eps}
    \caption{SEM images of the surface of sample $B$. The antidots have an average diameter of 1.5 $\mu$m and an average edge-to-edge distance of 300 nm. An overview SEM image is presented in the bottom panel where the row of FIB-milled antidots is clearly seen. }
    \label{f2}
\end{figure}

\section{Results}

\subsection{dc magnetization}

The upper and lower panels of Fig.~\ref{f3} show the temperature dependences of the magnetic moments, $M_0$, in magnetic field $20\pm 2$ Oe, of samples $A$ and $B$, respectively. The critical temperatures, $T_c$, of both samples are almost the same, 8.3 K, the transition width for sample $A$ is 1.3 K but 2.7 K for sample $B$. Sample $B$ demonstrates a two-stage transition, see the inset to the lower panel of Fig.~\ref{f3}.
At low temperatures, the magnetic moment of sample $A$ is a factor of two larger than that of sample $B$.
Temperature and field dependences of the magnetic moment were measured after cooling the sample down to the desired temperatures in zero field (ZFC).

\begin{figure
    \centering
    \includegraphics[width=0.98\linewidth]{Fig3a.eps}
    \includegraphics[width=0.98\linewidth]{Fig3b.eps}
    \caption{(Color online). Temperature dependences of the magnetic moment of samples $A$ and $B$, upper and lower panels, respectively. Inset to lower panel shows temperature dependence of $M_0$ of $B$ sample near $T_c$. }
    \label{f3}
\end{figure}

\begin{figure
    \centering
    \includegraphics[width=0.98\linewidth]{Fig4a.eps}
    \includegraphics[width=0.98\linewidth]{Fig4b.eps}
    \caption{(Color online). $M_0(H_0)$ of samples $A$ and $B$ after ZFC, upper panel. Expanded view of the magnetization curves in low magnetic fields for samples $A$ and $B$, lower panel. }
    \label{f5}
\end{figure}

The $M_0(H_0)$ dependences for samples $A$ and $B$ at 4.5 K are shown in the upper panel of Fig.~\ref{f5}. The magnetization curves in the ascending branch were measured in the hysteresis mode with 5 Oe step at low fields. Fig.~\ref{f5} shows that the $H_{c2}$ values are different. Determination of $H_{c2}$ for sample $B$ is less accurate than that of sample $A$, due to the magnetic moment relaxation, which at high fields is larger for sample $B$~\cite{MIT}.
 An expanded view of the magnetization curves at low fields is shown in the lower panel of Fig.~\ref{f5}. The fields of the first jumps, $H^*$, are around 20 Oe and 10 Oe, while the number of jumps in magnetic fields up to 100 Oe are 5 and 7 for samples $A$ and $B$, respectively. Jumps of the magnetic moment were observed in a wide range of magnetic fields, including fields below $H_{c1}$ for both samples. This behavior is reminiscent of magnetic flux jumps in Nb thin films for $H_0$ perpendicular to the film surface~\cite{NOWAK,STAM}. The jumps observed in these papers were interpreted as a thermomagnetic instability of the Abrikosov vortex lattice~\cite{NOWAK,STAM}. However, existence of jumps in fields below than $H_{c1}$ and parallel to the surface have been reported in our recent work only \cite{Katz1}. $H_{c1}$ is $\approx 350$ Oe at 4.5 K in our samples. Direct determination of $H_{c1}$ for thin-walled cylindrical samples is impossible due to magnetic moment jumps at low fields. However, the estimation of $H_{c1}$ can be done using magnetization curves of the planar film as it shown in inset to Fig.\ref{f13}.

\subsection{ac response}

The effective ac magnetic susceptibility of the sample in the external field $H(t)=H_0(t)+h_{ac}\sin (\omega t)$
is given by
  \begin{equation}\label{Eq2}
M(t)=Vh_{ac}\sum_n\{\chi_n^{'} \sin (n\omega t)-\chi_n^{''}\cos (n\omega t)\},
\end{equation}
and it exhibits the appearance of the ac field penetration into the sample, i.e. $\chi_1^{'}\neq -1/4\pi$, ac losses $\chi_1^{''}>0$  and harmonics of the fundamental frequency, $\chi_n$. Here, $M(t)$ is the magnetic moment of the sample and $V$ is its volume. In what follows we consider the results of the ac measurements in both PBP and SF modes.

\begin{figure
\begin{center
    \includegraphics[width=0.98\linewidth]{Fig5a.eps}
    \includegraphics[width=0.98\linewidth]{Fig5b.eps}
    \caption{(Color online). Field dependences of $\chi_1(H_0)$ of samples $A$ and $B$ in the PBP mode at 1465 and 293 Hz, upper and lower panels, respectively. Measurements were done at 4.5 K. Arrows on the lower panel show $H_{c3}$ for both samples.}
    \label{f5aa}
\end{center}
\end{figure}

The real and imaginary components of the ac susceptibility at 4.5 K for both samples measured in the PBP mode as a function of $H_0$ at two frequencies are shown in Fig.~\ref{f5aa}. Almost complete screening up to 12.5 kOe of the ac field by the superconducting walls is observed for both samples. This value is higher than $H_{c2} =11\pm 0.5$ kOe of sample $A$ (Fig.~\ref{f5}, upper panel). Complete screening of ac fields by a type II superconductor at low frequencies ($\omega \ll \omega_p$, here $\omega_p$ is a depinning frequency) and amplitudes of excitation (ac current much lower than depinning current) in dc fields lower than $H_{c2}$ was observed years ago \cite{STR2}.
The frequency dispersion of $\chi_1$ is weak for both samples. Third critical magnetic field was determined using ac data as follows. At low amplitude of excitation a loss peak located between $H_{c2}$ and $H_{c3}$. Losses disappear at $H_0>H_{c3}$ because in a normal state $\delta>>d$. Here $\delta$ is a skin depth in a normal state. Such determination of $H_{c3}$ was proposed years ago by Rollins and Silcox~\cite{ROLL}.
The lower panel of Fig.\ref{f5aa} shows an example of determination of the third critical magnetic field. It was found that $H_{c3}\approx 17.5\pm 0.5$ and $16\pm 0.5$ kOe at 4.5 K for $A$ and $B$ samples, respectively. $H_{c3}/H_{c2} \approx 1.6$ for sample $A$. An accurate determination of $H_{c2}$ for sample $B$ is difficult, due to magnetic relaxation, as discussed above. The absorption line, $\chi_1^{''}(H_0)$, near $H_{c3}$ is different for samples $A$ and $B$. Thus, this line is nonuniform for sample $A$ and it is uniform but broadened for sample $B$.
The ac response of superconductors even at very low amplitude of excitation, e.g., less than 1 Oe, is strongly nonlinear in the SSS \cite{ROLL,Genkin1}. The second harmonic signal is absent in the PBP mode in the entire range of magnetic fields. At the same time, the third-harmonic signal exists in the vicinity of $H_{c3}$ only. The absence of the second harmonic in PBP mode is a common feature for the bulk samples as well~\cite{CAMP}. Fig. \ref{f6a} shows the field dependences of $\chi_3$, $\chi_{2,3}\equiv \sqrt{(\chi_{2,3}^{'})^{2}+(\chi_{2,3}^{''})^{2}}$, in PBP mode
for samples $A$ and $B$, in the upper and lower panels, respectively. Perturbation theory with respect to the amplitude of excitation is not applicable for interpreting these experimental data. For example, according to perturbation theory, $\chi_3$ should be proportional $h_{ac}^{2}$ and this is not the case in our findings, Fig. \ref{f6a}. It is known that perturbation theory cannot explain experimental data for bulk samples too \cite{ROLL,GENKIN22}. We also note that there is a difference for the third harmonic signal between samples $A$ and $B$ in the PBP mode.
\begin{figure
\begin{center}
\leavevmode
\includegraphics[width=0.9\linewidth]{Fig6a.eps}
\includegraphics[width=0.9\linewidth]{Fig6b.eps}

\caption{(Color online). Field dependences of $\chi_3$ of samples $A$ and $B$  (upper and lower panels, respectively) in the PBP mode at 4.5 K.}
\label{f6a}
\end{center}
\end{figure}

A swept field affects the ac response more strongly at low frequencies or/and low excitation amplitudes for a given sweep rate. This was confirmed in experiments with bulk and thin-walled cylinders samples \cite{MAX,GENKIN22} and \cite{Genkin1,Katz1}, respectively. Fig. \ref{f7c} shows the field dependences $\chi_1$ for both samples $A$ and $B$ in the PBP and SF modes at 293 Hz and amplitude 0.04 Oe. The difference between the PBP and SF modes can easily be seen for both samples. The ac response at low magnetic fields in the SF mode are fluctuating due to magnetic flux jumps, Fig.~\ref{f5}. Near $H_{c3}$ the curves of $\chi_1$ coincide well in PBP and SF modes for both samples, Fig. \ref{f7c}. The difference between the two samples in the SF mode is very pronounced in fields above 5 kOe. In particular, $\chi_1^{''}$ is a smooth function of the dc field for sample $A$, but for sample $B$ it shows step-like features in fields near 7 and 10 kOe.

\begin{figure
 \begin{center
    \includegraphics[width=0.98\linewidth]{Fig7a.eps}
    \includegraphics[width=0.98\linewidth]{Fig7b.eps}
    \caption{(Color online). Field dependences of $\chi_1$ for samples $A$ and $B$ (upper and lower panels, respectively) in the PBP and SF modes at 293 Hz and an excitation amplitude of 0.04 Oe. Measurements were done at 4.5 K.}
    \label{f7c}
\end{center}
\end{figure}

A nonlinearity can clearly be seen not only in the second and third harmonics, but in the first harmonic too. Fig. \ref{f11a} shows the field dependences of $\chi_1$ of samples $A$ and $B$ at $h_{ac}=0.04$ and 0.2 Oe and T = 4.5 and 5.5 K in the SF mode. Panels $a$ and $b$ demonstrate: (i) that at low magnetic fields, losses in sample $A$ are significantly larger than losses in sample $B$ and: (ii) an increase of the excitation amplitude leads to a decrease of $\chi_1{''}$ for both samples. At $h_{ac}= 0.2$ Oe for $H_0> 5$ kOe there is a plateau and $\chi_1^{''}$ for both samples coincides with high precision. The plateau in the SF mode at high excitation amplitudes was observed at T = 4.5 K, Fig. \ref{f11a}$c$ and also at 5.5 K for, Fig. \ref{f11a}$b$. It appears that in this range of magnetic fields and at high enough amplitude, the first harmonic signal of the two samples is almost identical. However, a qualitative difference remains for the signals of the second and third harmonics, see Figs.~\ref{f9a} and \ref{f10a}.
\begin{figure
\begin{center
\leavevmode
\includegraphics[width=0.9\linewidth]{Fig8a.eps}
\includegraphics[width=0.9\linewidth]{Fig8b.eps}
\includegraphics[width=0.9\linewidth]{Fig8c.eps}
\caption{(Color online). Field dependences of $\chi_1$ for samples $A$ and $B$  in the SF mode at 5.5 K (panels $a$ and $b$, respectively) and 4.5 K (panel $c$).}
\label{f11a}
\end{center}
\end{figure}

As for the second harmonic signal it is absent for both samples in the whole range of magnetic fields in the PBP mode, but becomes visible in the SF mode. Fig. \ref{f10a} shows the field dependences of $\chi_2$ in the SF mode. Perturbation theory cannot explain the data for $\chi_2$ in the SF mode and $\chi_3$ in both modes. In accordance to this theory one could expect that $\chi_3\propto h_{ac}^2$ and  $\chi_2\propto h_{ac}$. However, this is not the case in our experiment at any magnetic field. In our experiment, an increase of the excitation amplitude leads to a suppression of $\chi_2$. In the SF mode $\chi_2$ is larger than $\chi_3$ under the conditions of the experiment, see Figs. \ref{f9a} and \ref{f10a}. We note that the data for $\chi_1$, $\chi_2$ and $\chi_3$ fluctuate strongly at fields lower than 4 kOe at 4.5 K for sample $A$ due to magnetic flux jumps.
\begin{figure
\begin{center}
\leavevmode
\includegraphics[width=0.9\linewidth]{Fig9a.eps}
\includegraphics[width=0.9\linewidth]{Fig9b.eps}

\caption{(Color online). Field dependences of $\chi_3$ of $A$ and $B$ samples (upper and lower panels, respectively) in the SF mode at 4.5 K.}
\label{f9a}
\end{center}
\end{figure}
\begin{figure
\begin{center}
\leavevmode
\includegraphics[width=0.9\linewidth]{Fig10a.eps}
\includegraphics[width=0.9\linewidth]{Fig10b.eps}

\caption{(Color online). Field dependences of $\chi_2$ of $A$ and $B$ samples, upper and lower panels, respectively, in the SF mode at 4.5 K.}
\label{f10a}
\end{center}
\end{figure}

It is interesting to note the following concerning the relation between field dependences of $\chi_1^{''}$ and $\chi_3$. Figs.~\ref{f8c} and \ref{f8b} show field dependences of normalized $\chi_1^{''}$ and $\chi_3$ for samples $A$ and $B$. Upper panels in both figures correspond to the PBP mode and lower panels to SF mode. At low magnetic fields  $\chi_1^{''}$ and $\chi_3$ are very small in the PBP mode for both samples. Both signals become measurable near $H_{c3}$ and the shape of these signals is identical with high precision.  In the SF mode the shapes of  $\chi_1^{''}$ and $\chi_3$ are again the same in the vicinity of $H_{c3}$. However, at low magnetic fields this similarity vanishes in the SF mode. Such similarity in the PBP mode can be proved in the frame of perturbation theory~\cite{PAV}, but it has not yet proven in the general case which we face in our experiment.

\begin{figure
\begin{center}
\leavevmode
\includegraphics[width=0.9\linewidth]{Fig11a.eps}
\includegraphics[width=0.9\linewidth]{Fig11b.eps}

\caption{(Color online). Field dependences of normalized $\chi_1^{''}$ and $\chi_3$ of sample $A$ in point-by-point and swept field modes (upper and lower panels, respectively). The shapes of $\chi_1^{"}$ and $\chi_3$ are with high accuracy identical in PBP and SF modes near $H_{c3}$. This similarity breaks in a SF mode at low magnetic fields.}
\label{f8c}
\end{center}
\end{figure}

\begin{figure
\begin{center}
\leavevmode
\includegraphics[width=0.9\linewidth]{Fig12a.eps}
\includegraphics[width=0.9\linewidth]{Fig12b.eps}

\caption{(Color online). Field dependences of normalized of $\chi_1^{"}$ and $\chi_3$ of sample $B$ in point-by-point and swept field modes (upper and lower panels, respectively). The shapes of $\chi_1^{"}$ and $\chi_3$ are with high accuracy identical in PBP mode and in large fields in SF mode.}
\label{f8b}
\end{center}
\end{figure}

\section{Discussion}

\subsection{dc magnetization curves}
The physical reasons for the observed flux jumps at small magnetic fields are not clear. One can suggest that the alignment of the magnetic field with respect to the sample surface is not perfect. Indeed, the latter cannot be ruled out completely, and a small field component perpendicular to the surface, $H_{\bot}$, should create vortices which might be responsible for the flux jumps at small magnetic fields. Hence, one may expect that flux jumps could be present at small magnetic fields in a reference planar film as well. This assumption has been examined in an additional control experiment with a reference planar film. Figure \ref{f13} displays ascending branches of the magnetization curves of the planar Nb film of 240\,nm thickness sputtered onto a silicon substrate, for the magnetic field inclination angles $\varphi =0^{\circ}$, $10^{\circ}$, and $45^{\circ}$. For $\varphi = 10^{\circ}$ and $45^{\circ}$ the component $H_{\bot}\approx0.17H_0$ and $H_{\bot}\approx0.71H_0$, respectively. Vortices created by this field component exist at small magnetic fields. This experiment demonstrates that in small fields the magnetic moment is a linear function of the magnetic field value and vortices created by $H_{\bot}$ \emph{do not induce any flux jumps} at small fields. The magnetic moment at small fields remains a linear function of the magnetic field for planar films of different thicknesses. Magnetic moment jumps first appear in the magnetization curve at inclination angles larger than $10^\circ$. Such a field inclination angle is at least a factor of 3 larger than the orientational misalignment of the sample orientation with respect to the field direction in our experiment. Therefore, the results obtained for planar films suggest that the vortices created by the small field component perpendicular to the surface are not the cause for magnetic moment jumps at small magnetic fields in the cylindrical samples.
\begin{figure
\begin{center}
\leavevmode
\includegraphics[width=0.9\linewidth]{Fig13.eps}
\caption{Ascending branches of magnetization curves of planar film in parallel and tilted magnetic fields. Inset shows determination of $H_{c1}$ of the planar film.}
\label{f13}
\end{center}
\end{figure}

The experimental data demonstrate the existence of magnetic instabilities in fields lower than $H_{c1}$. At 4.5 K, the flux starts to penetrate into the cylinders $A$ and $B$ at $H_0 = 20$ and 10 Oe, respectively, Fig.~\ref{f5} (lower panel). The field of the first jump, $H^*$, is defined by the some critical current (not to be confused with a depairing current). If we assume that the critical current density in the isthmus between two antidots is the same as in the film, then the ratio  $H_B^*/ H_A^*$ should be $\approx 0.16$. However, the experiment shows that this ratio is about 0.5, see Fig.~\ref{f5}. This means that the critical current density in the isthmuses is higher than in the as-prepared film. We note that the ratio of the magnetic moments in ZFC in field 20 Oe  for samples $B$ and $A$ is 0.5, see Fig. \ref{f3}. In accordance to the thermodynamic criterion \cite{Katz1} $H^* \propto\sqrt{d}$. Comparison $H^*$ for $A$ sample and samples from \cite{Katz1} shows that the thermodynamics cannot describe these magnetization jumps in samples without antidots.

It was demonstrated that at low temperature and at magnetic fields higher than some critical value, $H_{th}$, the magnetization curve becomes smooth and $H_{th}$ is sufficiently larger in the sample with an array of antidots~\cite{Motta1}. The latter experiments were carried out with the field perpendicular to the film surface. In our case we deal with the row of antidots and the magnetic field parallel to the surface. We believe that this is the main reason why $H_{th}$ is lower for the sample with antidots, see upper panel of Fig. \ref{f5}. We have to mention that the difference between perpendicular and parallel geometries is crucial. For example vortex velocity in the perpendicular geometry is a few orders magnitude larger than for the parallel one, see Ref. \cite{Genkin1}.

\subsection{ac response}
The field dependences of $\chi_1(H_0)$ in the PBP mode are different for $A$ and $B$ samples, Fig. \ref{f5aa}. Losses appear and screening decreases in magnetic fields above $H_{c2}$. Near $H_{c3}$ there is a loss peak and the shape of this peak is different for samples $A$ and $B$. The shape of the loss peak for sample $A$ is nonuniform and for sample $B$ it is broadened. The third critical field of sample $A$ is larger than for sample $B$, Fig. \ref{f5aa} lower panel. However, the determination of $H_{c3}$ for sample $B$ is questionable.

The difference in the ac response of samples $A$ and $B$ becomes qualitative in the SF mode, Figs.~\ref{f7c}, \ref{f9a} and \ref{f10a}. Whereas the field dependences of $\chi_1$, $\chi_2$ and $\chi_3$ are smooth for sample $A$, they have peculiarities in 7 and 10 kOe for sample $B$. As we have mentioned above, the data for $\chi_1$, $\chi_2$ and $\chi_3$ are noisy and fluctuating at fields lower than 4 kOe at 4.5 K and 2 kOe at 5.5 K due to magnetic flux jumps. The behavior of the ac response in the SF mode has some similar features for both samples. Thus, an increase of the excitation amplitude and frequency leads to a decrease of $\chi_1^{''}$ in fields down to $H_{c2}$ and $\chi_2$ in the whole field range. The reason for this behavior is the following. The main physical parameter defining the difference between the PBP and SF modes is $Q=\frac{\dot{H}_0}{\omega h_{ac}}$ \cite{MAX,FINK2}. The PBP mode corresponds to $Q=0$. Parameter $Q$ decreases with the excitation amplitude and/or frequency tending to zero. This is why $\chi_1^{''}$ and $\chi_2$ decrease with $h_0$ and $\omega$ and
in consequence of this in the SF mode perturbation theory is not applicable. In the limiting case of high frequencies, for example, in the GHz range, a swept field with sweep rate of few tens or hundred oersted per second does not affect the ac response \cite{VAT}.

The ac response of sample $A$ in the SF mode is similar to that reported in our previous papers \cite{Genkin1,Katz1}. In this sample we observe a smooth field dependence of $\chi_1^{''}$, $\chi_2$ and $\chi_3$. The models proposed in \cite{Genkin1,Katz1} can explain the experimental data for sample $A$ in magnetic fields lower and higher than $H_{c2}$. The case with the sample $B$ is more complicated. It turned out that $\chi_1^{''}$ at magnetic fields of 4 kOe ($H_0< H_{c2}$) is lower for sample $B$ than for sample $A$, see Fig. \ref{f11a}$c$. The following may be the reason for this. Vortex pinning and the current induced by ac and swept fields play an important role in ac response in a swept magnetic field \cite{Genkin1}. The area under the row of antidots is much smaller than the total film area. This is why vortex pinning by this row antidots cannot explain loss reduction. At the same time the total induced current is lower in sample $B$ than in sample $A$, Fig. \ref{f3}. This reduces the forces dragging vortices into the substrate and leads to loss reduction \cite{Genkin1}. The jump at $H_0\approx 5$ kOe takes place only for sample $B$, see Figs. \ref{f9a} and \ref{f10a}. At fields higher than the jump field the losses for both samples at $h_{ac} =0.2$ Oe are equal, panels $b$ and $c$ of Fig. \ref{f11a}. The weakening of pinning in high magnetic fields could be a cause for such behavior.

The nature of the jump of $\chi_{3,2}$ in magnetic fields of 10 kOe (see panels \textit{b} of Figs.\ref{f9a} and \ref{f10a}) for sample $B$ in SF mode is not clear. ac amplitude is not smeared this jump completely in contrast with $\chi_1^{''}$, see panels \textit{a} and \textit{c} of Fig.\ref{f11a}. This jump takes place in magnetic fields near $H_{c2}$ of sample $B$.  Decreasing of ac losses and harmonics jump near $H_{c2}$ in a swept field was observed in single crystal Nb \cite{GENKIN22,MT}. However, single crystal Nb has a well defined vortex structure and $H_{c2}$ but it is not the case with our sample.

\section{conclusion}

We have studied the dc and ac magnetic properties of thin-walled cylinders of superconducting Nb with and without a row of antidots.  Experiment showed that the critical current density is higher in the isthmus between antidots than in the film itself. The dc magnetization curves demonstrate an "avalanche''-like penetration of the magnetic flux into the cylinder for both samples. The effect was observed at a temperature of 4.5 K and completely disappeared at 7 and 5.5 K for samples $A$ and $B$, respectively.
Such a behavior resembles a thermomagnetic instability of vortices but it was observed in fields below $H_{c1}$ of the films, i.e. in a vortex-free state. The effect of end faces, consisting in that the magnetic force lines is bending near the sample ends, could be another reason for flux jumps. The influence of the sample end faces on the flux jumps in such samples has to be studied using a local probe technique.

The ac response of thin-walled cylinders with and without antidots is strongly nonlinear and perturbation theory cannot explain the experimental data. The ac response of $A$ and $B$ samples is similar in the point-by-point mode. However, in the swept field mode there is a qualitative difference between losses for samples $A$ and $B$. Thus, at low magnetic fields, losses in sample $B$ are lower than in sample $A$. There are jumps in $\chi_1$, $\chi_2$ and $\chi_3$ in high magnetic fields for sample $B$, but these quantities are smooth functions of the magnetic field in sample $A$.

We demonstrate that field dependences of $\chi_1^{''}$ and $\chi_3$ have the same shapes in the point-by-point mode with high accuracy. In the swept field mode the shapes of  $\chi_1^{''}$ and $\chi_3$ are the same in the vicinity of $H_{c3}$.  This similarity has yet not been proved in the case of strong nonlinear response that we encounter in our experiment.

The models developed in \cite{Genkin1,Katz1} could describe the ac response of the as-prepared sample. However, these models are not applicable to the sample with a row of antidots. New models for samples with antidots have to be elaborated. As well as further experimental studies of samples with different lengths, wall thicknesses, sizes and geometry of antidots row or array have to be carried out.

\section{acknowledgments}
We thank J. Kolacek, P. Lipavsky and V.A. Tulin for fruitful discussions.
This work was done within the framework of the NanoSC-COST Action MP1201.
Financial support of the grant agency VEGA in projects nos. 2/0173/13 and 2/0120/14 are kindly appreciated.


\section{Introduction}
\label{sec:intro}
One of the most important results obtained by RHIC (Relativistic Heavy Ion Collider) experiment is an elliptic flow of QCD particles~\cite{Adams:2005dq,Adcox:2004mh}.
The elliptic flow data using a Boltzmann-type equation for gluon
scattering indicates a large cross section above the phase transition
temperature ($T_c$)~\cite{Molnar:2001ux}.
Furthermore, this elliptic flow is well explained by the hydrodynamics of the quark-gluon-plasma (QGP) state~\cite{Teaney:2000cw,Kolb:2000fha,Huovinen:2001cy,Teaney:2001av,Hirano:2001eu,Hirano:2002ds,Teaney:2003kp,Teaney:2009qa}.
Thus, the QGP state is not described by a free gas system of perturbed gluons.
It might be natural since around $T_c$ the interaction force between the quarks and gluons becomes strong.
On the other hand, the numerical simulation of the relativistic liquid system gives that the upper limit of the ratio between the shear viscosity ($\eta$) to the thermal entropy ($s$)  is very small, $\eta/s  \le 0.4$~\cite{Teaney:2009qa}.
It turns out that the QGP has the perfect liquid property.
The result gives us a non-sticky image of QGP states even though the QCD particles are strongly interacting.
To make these curious properties of QGP clear,  a considerable number of studies have been conducted on the determination of the shear viscosity from the first principle calculation in the pure SU($3$) gauge theory~\cite{ Aarts:2002cc, Nakamura:2004sy,Meyer:2007ic,Moore:2008ws, Meyer:2008dq, Meyer:2009jp, Meyer:2011gj, Mages:2015rea,Astrakhantsev:2017nrs, Astrakhantsev:2018oue,Pasztor:2018yae}.

The shear viscosity is given by the first-differential coefficient of the spectral function at zero frequency ($\eta (T) = \pi d \rho (\omega)/ d \omega $ at $\omega=0$).
The spectral function is defined by the Euclidean correlation function of the renormalized spacial energy-momentum tensor (EMT) in the static state as follows:
\beq
C(\tau) &=& \frac{1}{T^5} \int d \vec{x} \langle T^R_{12} (0, \vec{0}) T^R_{12}(\tau, \vec{x}) \rangle= \int^{+\infty}_{-\infty} d \omega K(\tau, \omega) \rho(\omega).\label{eq:fisrt-eq}
\eeq
Here, $K(\tau,\omega)$ denotes the kernel of an integral transform.
In the lattice simulations, $C(\tau)$ is measured by generated configurations using the first equality sign, and then $\rho(\omega)$ is estimated through the second one.

During these calculations, there are at least three difficulties to obtain $\rho(\omega)$:
(i) How to define the renormalized EMT on the lattice (ii) how to improve a bad signal-to-noise ratio of the correlation function of EMT (iii) how to estimate $\rho(\omega)$ from the limited number of the data $C(\tau)$ on the lattice.

As for the first and second problems, a promising method, called the gradient flow method, has been provided in Ref.~\cite{Suzuki:2013gza,Asakawa:2013laa}.
By using the UV finiteness of the flowed operators in the gradient flow~\cite{Luscher:2010iy,Luscher:2011bx},
we can obtain the renormalized EMT without hard numerical calculation of $Z$-factor for quenched QCD.
Furthermore, thank  the smearing property of the gradient flow, the statistical uncertainty is suppressed in this method.
On the other hand, it is notable that we have to analyze the data only in the fiducial window of the flow-time to obtain the thermodynamic quantities using the gradient flow method in Ref.~\cite{Asakawa:2013laa}.
The range is given by $2a \ll \sqrt{8t} \ll T^{-1} = N_\tau a$, where both a strong lattice discretization and  over-smearing corrections are suppressed.

The third problem is well known as an ill-posed inverse problem.
Especially in the finite temperature system, the number of sites in the temporal direction is very limited, then it makes harder to solve the inverse problem.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.45]{./plot-corr-T12T12-rT-deps.pdf}
\caption{Flow-time dependence of the $T_{12}$ correlation function. Here, the data of $\beta=6.72, N_s^3\times N_\tau=48^3 \times12$ ($T=1.65T_c$) are plotted. Magenta curves denotes the best-fit function with 1-$\sigma$ error of the fitting parameters which is reconstructed by the estimated spectral function by using the Breit-Wigner ansatz given in Ref.~\cite{Meyer:2007ic} . It indicates a non-smeared correlation function. $r_{\mbox{smear}}T=\sqrt{8t}/aN_\tau$, where $t$ denotes the flow-time. This plot is firstly shown in Ref.~\cite{Itou:2017ypy}.
}
\label{fig:T12T12-corr-flowtime-deps}
\end{center}
\end{figure}
If we use the gradient flow method to obtain the correlation function, taking the data only in the fiducial window makes the third problem worse, since the available number of the data is further limited.
To see the origin of the demerit, we show the smeared correlation function $C(t,\tau)$ in Fig.~\ref{fig:T12T12-corr-flowtime-deps}.
We find that
the slope of $C(t,\tau)$ in the short (and long because of the periodicity) $\tau$-regime shows the discrepancy from the non-smeared data presented in Ref.~\cite{Meyer:2007ic}.
The discrepancy becomes large if the distance of two operators ($\tau$) gets shorter than the smeared length ($\sqrt{8t}$) by the gradient flow since the smeared regime of one operator reaches at the position of the other operator in the correlation function.

Now, a question arises: whether we can straightforwardly apply ordinary estimation methods of $\rho(\omega)$ to such a smeared correlation functions.
Until now, the fitting with some ansatz ({\it e.g.}, the Breit-Wigner ansatz) and the Bayesian methods ({\it e.g.}, the maximal entropy method~\cite{Nakahara:1999vy,Asakawa:2000tr}) have been utilized for the  estimation of the spectral function in the case of the non-smeared correlation function, $C(\tau)$.
 For the smeared EMT correlation function obtained by the gradient flow method in $N_f=2+1$ QCD, the fitting analyses with the Breit-Wigner ansatz and the hard thermal loop ansatz for $\rho(\omega)$ have been applied~\cite{Taniguchi:2019eid}.
The authors have tried to fit the data of $C(t,\tau)$ only in the limited $\tau$-regime to eliminate the over-smeared data.
However, the analysis for the eliminated $C(t, \tau)$ using the same functional form of $\rho(\omega)$ is questionable since originally the non-smeared $C(\tau)$ and $\rho(\omega)$ are related via the integration equation Eq.(\ref{eq:fisrt-eq}) and $C(\tau)$ at each $\tau$ gets the corrections from all $\omega$-regime.
Thus, the lack of $C(t,\tau)$ at several $\tau$ may change the functional form of the spectral function.
As the other directions, several methods to reconstruct the spectral function with smearing have been also proposed in Refs.~\cite{Burnier:2013nla,Hansen:2017mnd,Tripolt:2018xeo,Hansen:2019idp,Kades:2019wtd, Bailas:2020qmv}.

In this work, we propose the sparse modeling method to estimate the spectral function at finite flow-time, which is based on Bayesian statistics.
Here, we need not introduce any functional form of the spectral function.
The sparse modeling method can be applied to both non-smeared and smeared correlation functions.
It has been utilized in board topics, {\it e.g.} MRI~\cite{MRI} and taking a picture of a large blackhole~\cite{Blackhole} (see also a recent review~Ref.~\cite{review-sparse}).
As for the estimation of spectral function, it has been succeeded for several analyses in the many-body system~\cite{Shinaoka2017a, Shinaoka2017b}.
Utilizing the intermediate-representation basis (IR basis)~\cite{Shinaoka2017a,Nagai2018,Chikano2018a,Chikano2018b,Li2019} and a few reasonable constraints, the sparse modeling makes it possible to estimate the real-frequency representation of $\rho(\omega)$ from a sparse imaginary-time correlation function.

We formulate the sparse modeling analysis for $C(t,\tau)$ only in the fiducial window.
Here, we carry out  the singular value decomposition (SVD) of the kernel $K(\tau,\omega)$ in the IR basis, which is independent of the Monte Carlo data.
The singular values of the kernel decay exponentially or even faster, then we drop the degree of freedom for the correlation function and the spectral function in the IR basis associated with the sufficiently small singular values.
Then, we find the spectral function to minimize the square error given by the difference between the input correlation function and the output one reconstructed by $\rho(\omega)$.
During this process, the $L_1$ regularization term is added to be consistent with the truncation of the degree of freedom in the IR basis.
Technically, we utilize the ADMM algorithm to solve the optimization problem with such a regularization term, which is recently proposed by  S.~Boyd {\it et al.}~\cite{ADMM1, ADMM2}.
In this work, we demonstrate that the sparse modeling analyses using very limited data  for the quenched QCD at  finite temperature.

This paper is organized as follows:
In \S.~\ref{sec:sparse-modeling}, we give a review of the sparse modeling method.
In \S.~\ref{sec:EMT}, we explain how to calculate the correlation function of the renormalized EMT for the quenched QCD in the gradient flow method.
 In \S.~\ref{sec:sparse-Ctau}, we give a standard description to obtain the shear viscosity, where we assume that the correlation function in the whole $\tau$-regime is available for the sparse modeling analysis.
 Then, we modify the standard description to analyze the smeared correlation function in which the over-smeared data are eliminated in \S.~\ref{sec:tech-step}.
Section~\ref{sec:setup} presents the simulation setup of this work.
We show the result of the sparse modeling analysis using the central value of the smeared correlation function in \S.~\ref{sec:SpM}.
We find that it works well since the reconstructed correlation function from the obtained spectral function is almost consistent with the input data.
\S.~\ref{sec:error} contains the error estimations.
We investigate several systematic errors; $\omega_{cut}$, $\tau$-regime, and the flow-time dependence in our analysis, and also estimate the statistical uncertainty..
Although the statistical error of our result is sizable, we show that the bootstrap analysis is promising for estimating the statistical error of this analysis, since an expected relationship between the correlation function and the spectral function in the IR basis is satisfied.
In this work, we have only $2,000$ configurations for the measurement of the correlation function, and it is very few in comparison with $800,000$ and $6$ million configurations in Refs.~\cite{Nakamura:2004sy,Pasztor:2018yae}.
Judging from such a poor statistic, we will give up the precise determination of the shear viscosity and focus on the feasibility of the sparse modeling method to obtain the spectral function combing with the gradient flow method.
The last section is devoted to the summary.

\section{Sparse modeling method}\label{sec:sparse-modeling}
Here, we give a brief review of the sparse modeling method in the IR basis following Refs.~\cite{Shinaoka2017a, Shinaoka2017b} (see also a review paper~\cite{review-sparse}).

The input is the imaginary-time Green's function $C(\tau)$.
Its Fourier transform $C(i \omega_n)$, where $\omega_n$ denotes a Matsubara frequency, is related with
the spectral function $\rho(\omega)$ as $\rho(\omega)= \frac{1}{\pi} \mbox{Im} C(\omega + i 0)$ by replacing $i \omega_n$ with $\omega + i0$.
Then, the input $C(\tau)$ is written by the integral form of the spectral function,
\beq
C(\tau) = \int^{+\infty}_{-\infty} d \omega K(\tau, \omega) \rho(\omega), \label{eq:def-C}
\eeq
where $0 \le \tau \le 1/T$.
The kernel $K$ in this work is given by
\beq
K(\tau, \omega) = \frac{\cosh \left( \omega (\frac{1}{2T}-\tau) \right)}{\sinh (\frac{\omega}{2T})},
\eeq
but in this section the discussion does not depend on the details of the kernel.
Here, we assume that the kernel is described by some exponential function as in many systems.

For simplicity,  we denote Eq.(\ref{eq:def-C}) using the dimensionless vectors as
\beq
\vec{C} \equiv K \vec{\rho}. \label{eq:ckrho}
\eeq
The component of the vector $\vec{C}$ is $C_i \equiv C(\tau_i)$, where $\tau_i$ labels the temporal site on the lattice with $0 \le \tau_i \le N_\tau-1$.
The kernel $K$ becomes $N_\tau \times N_\omega$ matrix, while $\vec{\rho}$ denotes a vector whose component is $\rho_j \equiv \rho(\omega_j)$ with $j = 1, \cdots ,N_\omega$.

Our goal is to find $\vec{\rho}$ to minimize the square error
\beq
\chi^2 (\vec{\rho}) = \frac{1}{2} \| \vec{C} - K \vec{\rho}  \|^2_2.\label{eq:chi-sq}
\eeq
Here, $\| \cdot \|_2$ stands for the $L_2$ norm defined by $\| \vec{\rho} \|_2 \equiv (\sum_j \rho^2_j)^{1/2}$.
Note that the vector $\vec{C}$ is sparse, so that the amount of the information in $\vec{C}$ is much smaller than that in $\vec{\rho}$.
The fact leads to the instability of $\vec{\rho}$.

We have only the Monte Carlo data of $C(\tau)$, which is not the same with the true value of $\vec{C}$ and the data have the statistical uncertainty.
Here, we call the discrepancy between the Monte Carlo data and the true value of $\vec{C}$ a ``noise".

Equation~(\ref{eq:ckrho}) gives simultaneous linear equations, where the number of equations is $N_\tau$ while one of the unknown variables is $N_\omega$.
We can solve the equation if $N_\tau = N_\omega$ and the kernel is given by a regular matrix, while the unique solution does not exist and several possible solutions are allowed in the case of $N_\tau < N_\omega$.

Here, we would like to choose a stable solution against the noise among these possible solutions.
For this purpose, we take an efficient basis called an IR basis.
By introducing the singular value decomposition (SVD) of the matrix $K$ defined as
\beq
K = U S V^\dag.
\eeq
We rewrite Eq.(\ref{eq:ckrho}) as
\beq
\vec{C} \equiv  U S V^\dag \vec{\rho},
\eeq
where $S$ is an $N_\tau \times N_\omega$ diagonal matrix, and $U$ and $V$ are unitary matrices of size $N_\tau \times N_\tau$ and $N_\omega \times N_\omega$, respectively.
The new vectors in the IR basis
\beq
\vec{\rho}' \equiv V^t \vec{\rho}, ~~~~~~~~ \vec{C}' \equiv U^t \vec{C}.\label{eq:rho-C-IR-basis}
\eeq
imply the square error Eq.(\ref{eq:chi-sq}) as
\beq
\chi^2(\vec{\rho}) = \frac{1}{2} \| \vec{C}' - S \vec{\rho}' \|^2_2 = \frac{1}{2} \sum_l (C'_l - s_l \rho'_l)^2.\label{eq:chi-sq-SVD}
\eeq
Thus, at the minimum point of the square error the $l$-th element  almost satisfies
\beq
 [\vec{C}']_l =  s_{l} [\vec{\rho}']_l.\label{eq:Cl-sl-rhol}
\eeq
It turns out that the contribution of $\rho'_l$ to $\chi^2 (\vec{\rho}')$ is weighted by the corresponding $s_l$.
The singular values $s_l$ with $l = 1,2,\cdots $ of this type of kernels decay exponentially or even faster.
Then, for example, if double precision numbers are used, $C'_l$ can not include the information about $\rho'_l$, where $s_{l}/s_{1}$ is less than $10^{-16}$ on a computer.
It {\it does not } depend on simulation details since a kernel $K$ is fixed.
This fact naturally  explains the reason why a tiny noise of $C(\tau)$ leads to a large difference of $\rho(\omega)$.
Therefore, a tiny fluctuation in the left-hand side can affect the ``best" inference of $\rho(\omega)$, and several ``likely" solutions satisfy Eq.~(\ref{eq:def-C}).

In actual calculations, the component  $s_l\rho'_l$ with sufficiently small singular values  gives a negligible contribution to $\chi^2 (\vec{\rho})$.
Then, by introducing some threshold $s_{cut}$, we can drop such a component $l > l_{cut}$, where $s_{l} < s_{cut}$.
This truncation leads us to obtain a stable solution which is robust against the noise of $C(\tau)$.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.4]{./L1-reg.pdf}
\caption{Image of $L_1$ and $L_2$ regularizations in 2-dimensional $\rho'_l$ plane. The $L_1$ ($L_2$) regularization can be described by a line (a circle).
The intersection point (open symbol) appears on the horizontal axis, where $\rho'_2$ is zero.
}
\label{fig:L1-reg}
\end{center}
\end{figure}

Now, the vectors in the IR basis are reduced to the $l_{cut}$-component vectors.
Therefore,  we would like to find the solution $\rho'_l$  in Eq.~(\ref{eq:chi-sq-SVD}), where  many components of $\vec{\rho}'$ takes zero to be consistent with the truncation.
To search for such a solution $\rho'_l$, we consider the cost function with an $L_1$ regularization term,
\beq
F(\vec{\rho}') \equiv \frac{1}{2} \| \vec{C}' - S \vec{\rho}'  \|_2^2 + \lambda \| \vec{\rho}' \|_1,\label{eq:cost-fn-L1}
\eeq
where $\lambda$ is a positive constant and $\| \cdot \|_1$ denotes the $L_1$ norm defined by
\beq
\| \vec{\rho}' \|_1 \equiv \sum_l | \rho_l'|.
\eeq
This $\lambda$ plays the role of the Lagrange multiplier.
If the value of $\lambda$ is smaller than the contribution of the noise for $\vec{C}$, then the obtained $\vec{\rho}$ must be consistent with the true spectral function within the uncertainty coming from the noise.

An intuitive picture of the role of the $L_1$ regularization is following.
The $L_1$ regularization ($\| \vec{\rho}' \|_1 = $const.) gives a linear constraint for the components of $\vec{\rho}'$, and is graphically drawn by dotted and solid lines in Fig.~\ref{fig:L1-reg}.
The minimize problem of the cost function turns  to tune $\lambda$ to minimize the sum of two constants: $\|  \vec{C}' - S \vec{\rho}'    \|_2^2 $ is a constant and $\| \vec{\rho}' \|_1  $ is the other constant.
In Fig.~\ref{fig:L1-reg}, we see that the open-circle point becomes the most favorable.
Note that at the open-circle point, $\rho_2$ vanishes and the number of the effective components is reduced.

Actually, according to the example analyses in Ref.~\cite{Shinaoka2017b, review-sparse}, the spectral function becomes featureless for $\lambda > \lambda^{opt}$, while artificial spikes appear for $\lambda < \lambda^{opt}$.
In other words, the former case corresponds to the under-fitting, where the $L_1$ regularization term is too strong and the number of components  $\rho'_l$ is too reduced.
On the other hand, the latter case does to the over-fitting, where the $L_1$ term is too weak and the vector $\vec{\rho}'$ has redundancy.

The optimization problem with these $L_1$ and/or $L_2$ regularization is called  the LASSO (Least Absolute Shrinkage and Selection Operators) problem~\cite{LASSO}.
It allows obtaining the global minimum regardless of initial conditions~\cite{ADMM1, ADMM2}.
As an additional constraint, we will add the non-negativity condition of the spectral function,
\beq
\rho(\omega) \ge 0,
\eeq
in this work.
Furthermore, we can also use the sum rule,
\beq
\sum_j \rho_j=1,
\eeq
as an additional condition, while this constraint is not  used in this work.
The numerical algorithm to solve the optimization problems with these constraints has been developed~\cite{ADMM1, ADMM2}, and it is called the alternating direction method of multipliers (ADMM) algorithm.
See Appendix~\ref{sec:ADMM} for the detail  of the algorithm.

\section{Shear viscosity in quenched QCD}\label{sec:shear-viscosity}
In this section, we explain how to obtain the shear viscosity for the quenched QCD in the lattice simulation.
Firstly, we explain the calculation strategy of the correlation function of EMT using the gradient flow.
Secondly, we give a standard formula for the sparse modeling method.
Here, we assume that the correlation function in the whole $\tau$-regime is available for the sparse modeling analysis.
Then, we modify the standard formula for the smeared correlation function at  finite flow-time, where we eliminate the over-smeared data.

\subsection{Measurement of the correlation function of EMT in the gradient flow method}\label{sec:EMT}
The renormalized EMT for the quenched QCD in the gradient flow method~\cite{Suzuki:2013gza} is given by
\begin{align}
   T_{\mu\nu}^R(x)
   =\lim_{t\to0}\left\{\frac{1}{\alpha_U(t)}U_{\mu\nu}(t,x)
   +\frac{\delta_{\mu\nu}}{4\alpha_E(t)}
   \left[E(t,x)-\left\langle E(t,x)\right\rangle_0 \right]\right\},
\label{eq:(4)}
\end{align}
at a flow-time $t$.
Here, we utilize the small flow-time expansion and drop ${\cal O} (t)$ corrections.
The coefficients $\alpha_U,\alpha_E$ are calculated perturbatively in Ref.~\cite{Suzuki:2013gza}.
An advantage of the usage of the gradient flow method is the absence of $Z$-factor to define the renormalized EMT.
Once we use the renormalized coupling constant in the coefficient $\alpha_U, \alpha_E$, all composite operators constructed by the flowed gauge field are the UV finite at positive flow-time ($t>0$)~\cite{Luscher:2011bx}.
The last term in right-hand-side, $\langle\cdot\rangle_0$, describes the vacuum expectation value (v.e.v.), which relates to the zero point energy.
$U_{\mu \nu}(t,x)$ and $E(t,x)$ denote gauge-invariant local products of dimension~$4$.

In the continuum theory,
$U_{\mu\nu}(t,x)\equiv G_{\mu\rho}(t,x)G_{\nu\rho}(t,x)
-\delta_{\mu\nu} E(t,x)$
and~$E(t,x)\equiv\frac{1}{4}G_{\mu\nu}(t,x)G_{\mu\nu}(t,x)$~\footnote{Here, we drop the color indices.}.
Here, $G_{\mu \nu}$ represents the field strength constructed by the flowed gauge field ($B_{\mu}(t,x)$).
This flowed gauge field is a solution to the gradient flow equation,
\beq
\partial_t B_{\mu} = D_{\nu} G_{\nu \mu}, ~~~~ B_{\mu} (t=0,x) = A_{\mu}(x),\label{eq:gflow}
\eeq
where $A_{\mu}(x)$ denotes the original quantum gauge field variable.
The Fourier components $B_\mu (p)$ have a suppression factor $e^{-p^2t}$.
 Therefore, the flow-time ($t$) plays a role of the UV cutoff in momentum space.
In the coordinate space, the flowed gauge field can be interpreted by the smeared field in the range of $|x| < \sqrt{8t}$.

On the lattice, $G_{\mu \nu}$ operator can be calculated by the clover-leaf operator, whose size is $2a$.
The relationship Eq.~(\ref{eq:(4)}) is useful in the range of $2a < \sqrt{8t}$, where the corrections of the lattice discretization  would be mild.

The numerical calculation using this formula has firstly done in Ref.~\cite{Asakawa:2013laa}, where the expectation value of the EMT has been measured for the quenched QCD at finite temperature.
The expectation value of EMT is directly related to the bulk quantities, {\it e.g.} integration measure and thermal entropy.
The results are consistent with the ones obtained by the integration method within the statistical error.
Thanks to the smearing effect of the flowed fields, the statistical uncertainty is smaller than the other method.
On the other hand, we need to take a fiducial window of the flow-time, $2a < \sqrt{8t} < aN_\tau/2$, to avoid a strong lattice discretization and  over-smearing corrections.

Now, we calculate the two-point function of EMT.
It is related to the shear and bulk viscosities.
In this work, we focus on the former one, which is given by the correlation function of $T^R_{12}$ component.
The Euclidean correlation function is defined as
\beq
C(\tau) = \frac{1}{T^5} \int d \vec{x} \langle T^R_{12} (0) T^R_{12}(x) \rangle,\label{eq:def-Ctau-latt}
\eeq
where $x=(\tau, \vec{x})$.

On the lattice at a finite flow-time $t$, we measure the two-point function of $U_{12} (t,x)$,
\beq
C (t,\tau/a)= N_\tau^5 \sum_{\vec{x}} \frac{1}{\alpha_U(t)^2} \langle U_{12} (t, 0) U_{12}(t,x) \rangle,\label{eq:def-smear-C}
\eeq
where the second argument of $U_{12} (t,x)$ denotes a four-dimensional vector $x=(\tau/a, \vec{x}/a)$.
Although in an ideal double limits, $a \rightarrow 0$ and then $t\rightarrow 0$ limits, this smeared correlation function goes to the Euclidean correlation function (Eq.~(\ref{eq:def-Ctau-latt})), practically it is better to calculate physical observables from the smeared correlation function and take the double limits for the observables directly.
In this work, we would like to obtain the shear viscosity from $C(t,\tau/a)$ at finite flow-time.
How to treat $C(t,\tau/a)$ in the sparse modeling analysis will be discussed in \S.~\ref{sec:tech-step}.

\subsection{Estimation of the spectral function using the sparse modeling method}\label{sec:sparse-Ctau}
Here, we give a standard formula for the sparse modeling method.
We assume that $C(\tau)$ is obtained after the double limits or $C(t,\tau)$ in the whole $\tau$-regime is available for the analysis.
In this section, we simply denote these correlation functions as $C(\tau)$.

The correlation function can be expressed in terms of the corresponding spectral functions ($\rho(\omega)$) as
\beq
C(\tau) = \frac{1}{T^5} \int_{0}^{\infty} d \omega \rho(\omega) \frac{\cosh \omega \left( \frac{1}{2T} - \tau \right)}{\sinh(\frac{\omega}{2T})}.\label{eq:C-rho-cont}
\eeq
The spectral function has the following properties,
\beq
\frac{\rho(\omega)}{\omega} \ge 0, \quad \rho(-\omega) = -\rho(\omega).
\eeq
The shear viscosity is given by
\beq
\eta (T) = \pi \frac{d \rho}{d \omega} |_{\omega=0},\label{eq:def-shear}
\eeq
from the spectral function.

On the lattice, Eq.~(\ref{eq:C-rho-cont}) turns to
\beq
C(\hat{\tau})&=& N_\tau^5 \int^{\infty}_0 d \hat{\omega}  \hat{\rho}( \hat{\omega}) \frac{\cosh \hat{\omega} \left( \frac{N_\tau}{2} -\hat{\tau} \right)}{\sinh (\frac{\hat{\omega} N_\tau }{2})},\label{eq:EMT-corr-org}
\eeq
where $\hat{\tau}=\tau/a$, $\hat{\omega}=a \omega$ and $\hat{\rho} = a^4 \rho $, respectively.

Equation (\ref{eq:def-shear}) shows that the shear viscosity is the first-differential coefficient at $\omega=0$, then it is convenient to define
\beq
\tilde{\rho}(\hat{\omega}) \equiv \frac{\hat{\rho} (\hat{\omega})}{2  \tanh (\hat{\omega} N_\tau/2)} \label{eq:rho-tilde}.
\eeq
Note that $\tilde{\rho}(\hat{\omega})$ turns to an even-function in terms of $\hat{\omega}$.
Then, the relationship with the correlation function (\ref{eq:EMT-corr-org}) is expressed by
\beq
C(\hat{\tau}) = N_\tau ^5 \int_{-\infty}^\infty d \hat{\omega} \tilde{\rho}(\hat{\omega}) \frac{\cosh(\hat{\omega} \left( \frac{N_\tau}{2} -\hat{\tau} \right))}{ \cosh ( \hat{\omega} N_\tau/2 )}.\label{eq:C-tilde-rho}
\eeq

To obtain the spectral function in Eq.(\ref{eq:C-tilde-rho}), we introduce the cutoff of $\omega$ and the rescaled $\tau$ as follows:
\beq
\omega' \equiv \frac{\omega}{\omega_{cut}} = \frac{\hat{\omega}}{\hat{\omega}_{cut}}, ~~~ \tau' \equiv \frac{\tau}{aN_\tau}=\frac{\hat{\tau}}{N_\tau}.
\eeq
Here, the regimes of the primed variables are $-1 \le \omega' \le 1$ and $ 0 \le \tau' < 1$.
Using $\Lambda = N_\tau \hat{\omega}_{cut}$, the relationship between the correlation function and the spectral function is rewritten by
\beq
C(\tau')&=&N_\tau^4 \Lambda \int_{-1}^1 d \omega ' \tilde{\rho}(\omega ') \frac{\cosh [\frac{\omega ' \Lambda }{2} (2 \tau' -1)]}{\cosh (\omega ' \Lambda /2)}.
\eeq

Discretizing the integral into a finite sum and taking a replacement $ (d\omega ' ) \rightarrow \Delta \omega'$ with $\Delta \omega ' = 2/(N_{\omega}-1)$,
the finite sum is given by~\footnote{In the quadrature by parts, the upper value of the sum is ($N_\omega-1)$ in Eq.~(\ref{eq:Nomega-sum}), here we take $N_\omega$ in our analysis.
As a result, the integrand $\rho(\omega)$ is approximately zero at $\omega = \pm \omega_{cut}$, so that practically the contribution of $N_\omega$-th data to the sum is negligible.}
\beq
\frac{C(\tau_i')}{N_\tau ^4 \Lambda} = \sum_{j}^{N_\omega} (\Delta \omega ' ) \tilde{\rho}(\omega_j')  \frac{\cosh [\frac{\omega_j ' \Lambda }{2} (2 \tau_i' -1)]}{\cosh (\omega_j ' \Lambda /2)}.\label{eq:Nomega-sum}
\eeq
To make the $N_\omega$ dependence mild in the sparse modeling analysis, we include $\sqrt{\Delta \omega'}$ in the kernel as follows:
\beq
K(\tau'_i,\omega'_j)  \equiv \frac{\cosh [\frac{\omega_j ' \Lambda }{2} (2 \tau'_i -1)]}{\cosh (\omega_j ' \Lambda /2)} \sqrt{\Delta \omega'}.
\eeq
Note that the kernel at $\omega'=0$ is independent of $\tau$-coordinate.

Now, the equation we have to solve is
\beq
\frac{C(\tau'_i)}{N_\tau^4 \Lambda} &=& \sum_j^{N_\omega} K(\tau'_i,\omega'_j) \tilde{\rho}_{new} (\omega_j ')~~~ \mbox{with}~~~
\tilde{\rho}_{new} (\omega_j ') = \tilde{\rho} (\omega_j ') \sqrt{\Delta \omega'}. \label{eq:C-rho-new}
\eeq
The spectral function $\tilde{\rho}_{new}$ satisfies the sum rule
\beq
\sum_j^{N_\omega} \tilde{\rho}_{new} (\omega_j ') =\frac{C(\tau_0')}{N_\tau^4 \Lambda \sqrt{\Delta \omega '}},
\eeq
which is given by Eq.~(\ref{eq:C-rho-new}) at $i=0$.
Therefore, a normalized spectral function,
\beq
\tilde{\rho}_{calc}  \equiv \tilde{\rho}_{new}  \left[ \frac{C(\tau'_0)}{N_\tau^4 \Lambda \sqrt{\Delta \omega '}} \right]^{-1},
\eeq
implies a simple form of the sum rule,
\beq
\sum_j \tilde{\rho}_{calc} (\omega_j ') = 1.\label{eq:sum-rule-Sp}
\eeq
We will obtain this spectral function $\tilde{\rho}_{calc}$ as an output of the sparse modeling analysis.

The original spectral function $\tilde{\rho}(\omega'_j)$ in Eq.~(\ref{eq:Nomega-sum}) is transformed from the output of the sparse modeling analysis ($\tilde{\rho}_{calc}$),
\beq
\tilde{\rho} (\omega_j ') &= &\frac{\tilde{\rho}_{new}}{\sqrt{\Delta \omega '}}
= \frac{C(\tau'_0)}{N_\tau^4 \Lambda \Delta \omega '} \tilde{\rho}_{calc} (\omega_j ').\label{eq:tilde-rho-calc}
\eeq
Finally, the shear viscosity divided by $T^3$ to be a dimensionless quantity is given by
\beq
\frac{\eta}{T^3} &\equiv & \pi \frac{1}{T^3} \frac{d \rho (\omega)}{d \omega} |_{\omega =0} \nonumber\\
&=& \pi {N_\tau^4} \tilde{\rho} (0). \label{eq:eta-rho-calc}
\eeq

Let us summarize the technical steps of our strategy to obtain the shear viscosity using the sparse modeling method as follows:

{\bf Step 1:} Measure the correlation function $C(\tau_i)$ on the lattice, and then normalize it using the value of $C(\tau_i=0)$.
\beq
C_{calc} (\tau_i') &=& \frac{C(\tau'_i)}{N_\tau^4 \Lambda} \left[  \frac{C(\tau'_0)}{N_\tau^4 \Lambda  \sqrt{\Delta \omega'}}   \right]^{-1}= \frac{C(\tau_i')}{C(\tau'_0)} \sqrt{\Delta \omega '}.\label{eq:norm-C}
\eeq

{\bf Step 2:} Construct the discretized kernel
\beq
K(\tau'_i,\omega'_j)  \equiv \frac{\cosh [\frac{\omega_j ' \Lambda }{2} (2 \tau_i' -1)]}{\cosh (\omega_j ' \Lambda /2)} \sqrt{\Delta \omega'}\label{eq:kernel-SVD}
\eeq
and carry out SVD decomposition. Here, we utilize DGESDD routines of LAPACK.

{\bf Step 3:} Estimate an optimal value of $\lambda$ as follows:
We first fix a searching range of $\lambda$ as $[\lambda_{min},\lambda_{max}]$.
Next, we define a line segment function in log scale, $f(\lambda)=a \lambda^b$, which connects $f(\lambda_{mix})$ with $f(\lambda_{max})$.
Then, we search $\lambda^{opt}$, where the ratio $f(\lambda)/\chi^2(\vec{\rho})$ has a peak since it corresponds to the position of kink in $\chi^2 (\vec{\rho})$.
Furthermore, a simple check of whether an obtained $\lambda$ is a ``correctly" optimized value or not can be done by seeing the scaling property of $\lambda^{opt}$.
The ADMM algorithm has free parameters ($\mu, \mu'$).
The correct $\lambda^{opt}$ is inversely scaled by the values of ($\mu$, $\mu'$) in the ADMM algorithm (see Eq.~(\ref{eq:ADMM-nu}) in Appendix~\ref{sec:ADMM}).

{\bf Step 4:} Carry out the sparse modeling analysis using the ADMM algorithm to find the most likely spectral function $\tilde{\rho}_{calc}$ using $\lambda^{opt}$.
We can  reconstruct the correlation function using the obtained $\tilde{\rho}_{calc}$ as
\beq
C_{output}(\tau'_i) = \frac{C(\tau'_0)}{\sqrt{\Delta \omega}} \sum_j K(\tau'_i, \omega'_i) \tilde{\rho}_{calc} (\omega'_i) .
\eeq
We check the feasibility of the analysis whether  $C_{output} (\tau_i)$ reproduces the input correlation function $C(\tau_i)$.

{\bf Step 5:} Calculate the shear viscosity ($\eta(T)/T^3$) on the lattice from the obtained spectral function using Eqs.~(\ref{eq:tilde-rho-calc}) and (\ref{eq:eta-rho-calc}).
Carry out the same calculations from {\bf Step 1} to {\bf Step 4} using the other lattice spacings with keeping a temperature, and then obtain $\eta(T)/T^3$ in the continuum extrapolation.

We show an example sparse-modeling analysis using this standard formula in Appendix~\ref{sec:All-tau-analysis}, where
 the correlation function is the smeared correlation function at a finite flow-time~\footnote{We put the numerical code and the data on the arXiv page of this paper.}.
Because of the over-smearing problem in the smeared correlation function, we find that the standard formula does not work for the smeared correlation function.

\subsection{Shear viscosity at a finite flow-time}\label{sec:tech-step}
We actually have a smeared correlation function $C(t,\tau)$ instead of $C(\tau)$.
The short $\tau$-regime of $C(t,\tau)$ suffers from the over-smearing correction, and then we have to eliminate these data whose slope is different from the non-smeared ones.
On the other hand, as we will see later (the right panel in Fig.~\ref{fig:Ctau-ft-deps}), the slope of $C(t,\tau)$ in the fiducial $\tau$-regime is universal and does not depend on the flow-time.
Then, we consider the kernel for $C(t,\tau)$ in the fiducial $\tau$-regime can be the same functional form with the one for the non-smeared correlation function.
Thus, the spectral function bears the flow-time dependence of $C(t,\tau)$ while the (reduced) kernel is universal.

Therefore, the integration equation for $C(t,\tau)$ turns to
\beq
C(t, \tau') = N_\tau ^4 \Lambda \int_{-1}^1 d \hat{\omega}' \tilde{\rho}(t, \hat{\omega}') \frac{\cosh [\frac{\omega ' \Lambda }{2} (2 \tau' -1)]}{\cosh (\omega ' \Lambda /2)}.
\eeq
where $\tau'$ runs to the site in the fiducial regime.
From the spectral function in the integral equation above, we will obtain the shear viscosity
\beq
\eta (t,T) = \pi {N_\tau} \tilde{\rho} (t,\omega'=0),\label{eq:eta-t}
\eeq
which depends on the flow-time.
Furthermore, it depends on the lattice spacing, so that we will take the double limits, $a\rightarrow 0$ and then $t \rightarrow 0$ limits, for the obtained $\eta(t,T)$ at a fixed $T$.
After these processes, we will find $\eta(T)/T^3$ at the temperature.

Applying the  sparse modeling method to the smeared correlation function needs three modifications to the standard formula:
Firstly, the input data $C(t,\tau)$ should be restricted only in the fiducial $\tau$-regime as same as the calculation of the one-point function of EMT in Ref.~\cite{Asakawa:2013laa}.
Simultaneously, the kernel matrix is reduced to $K^{red}(\tau'_i,\omega'_j)$, where $\tau'_i$ runs the site only in the fiducial $\tau$-regime.
Secondly, we remove the sum rule given in Eq.~(\ref{eq:sum-rule-Sp}) in the cost function, since the sum rule is related to the correlation function at $\tau=0$ but $C(t,\tau=0)$ is always contaminated by the over-smearing at  finite flow-time.
The third one is the modification of the normalization factor $C(\tau'_0)$ in Eq.~(\ref{eq:norm-C}) to a different value. This modification is not so essential but useful.
Here, we utilize $C(t,\tau'_{ini})$ instead of $C(\tau'_0)$, where $\tau_{ini}$ is the smallest number of site in the fiducial $\tau$-regime.

The modified strategy to obtain the shear viscosity using the gradient flow and the sparse modeling methods can be summarized as follows:

{\bf Step 1':} Measure the correlation function $C(t,\tau/a)$ at  flow-time $t$ using the gradient flow method, and then normalize it using the value of $C(t,\tau/a=\tau_{ini})$.
\beq
C_{calc} (t,\tau'_i) &=& \frac{C(t,\tau'_i)}{N_\tau^4 \Lambda} \left[  \frac{C(t,\tau'_{ini})}{N_\tau^4 \Lambda  \sqrt{\Delta \omega'}}   \right]^{-1}= \frac{C(t,\tau')}{C(t,\tau'_{ini})} \sqrt{\Delta \omega '}.
\eeq

{\bf Step 2':} Construct the reduced kernel
\beq
K^{red}(\tau'_i,\omega'_j)  \equiv \frac{\cosh [\frac{\omega_j ' \Lambda }{2} (2 \tau'_i -1)]}{\cosh (\omega_j ' \Lambda /2)} \sqrt{\Delta \omega'}\label{eq:kernel-SVD}
\eeq
and carry out SVD decomposition.
Here, $\tau'_i$ runs only in the fiducial range for each flow-time, namely $\sqrt{8t}/a \le \tau_i$ and $\sqrt{8t} \le N_\tau- \tau_i$. Here, the second inequality comes from the periodic boundary condition.
For simplicity, we will drop the second inequality in the present draft and only show the first one.
Note that we assume that the functional form of the kernel does not change for the reduced data of the smeared correlation function.
We will see that the numerical data of  $C(t,\tau)$ in the fiducial $\tau$-regime support this assumption, and actually this reduced kernel works well in our analyses.

{\bf Step 3':} Estimate an optimal value of $\lambda$. This is the same process as the standard formula.

{\bf Step 4':} Carry out the sparse modeling {\it without the sum rule} (set $\nu=0$ in Eq.~(\ref{eq:App-cost})).
We obtain the spectral function at a finite flow-time from the output of the sparse modeling analysis ($\tilde{\rho}_{calc}$) as
\beq
\tilde{\rho} (t,\omega')= \frac{C(\tau_{ini})}{N_\tau^4 \Lambda \Delta \omega '} \tilde{\rho}_{calc} (t,\omega_j ').\label{eq:tilde-rho-calc-t}
\eeq
We reconstruct the correlation function using the obtained $\tilde{\rho}_{calc}$,
\beq
C_{output}(t,\tau') = \frac{C(t,\tau'_{ini})}{\sqrt{\Delta \omega}} \sum_j K^{red}(\tau'_i, \omega'_j) \tilde{\rho}_{calc} (t,\omega'_j) .
\eeq
Then, we can check the feasibility of the analysis whether  $C_{output} (t,\tau/a)$ reproduces the input correlation function $C(t,\tau/a)$.

{\bf Step 5':}
Calculate the shear viscosity ($\eta(t, T)/T^3$) on the lattice from the obtained spectral function using Eqs.~(\ref{eq:eta-t}) and (\ref{eq:tilde-rho-calc-t}).
Carry out the same calculations from {\bf Step 1'} to {\bf Step 4'} using the other lattice spacings with keeping a temperature, and then obtain $\eta(t,T)/T^3$ after taking the continuum extrapolation.
Finally, we have $\eta(T)/T^3$ after taking $t \rightarrow 0$ limit.
The demonstration of these processes is out of the present work, and here our goal is to find a stable $\tilde{\rho}(t,\omega')$ from the smeared correlation function.

\section{Simulation setup}\label{sec:setup}
We consider the Wilson plaquette gauge action under the periodic boundary condition at $\beta=6.93$ on $N_s^3 \times N_\tau = 64^3 \times 16$ lattices.
The lattice parameter realizes the system at $T=1.65T_c$.
We use the relation between $a/r_0$, where $r_0$ denotes the Sommer scale, and $\beta$ in Ref.~\cite{Guagnelli:1998ud}.
Then,  $T/T_c$ is fixed by the resultant values of $Tr_0 = (N_\tau (a/r_0))^{-1}$ using the result at $\beta= 6.20$ in Ref.~\cite{Boyd:1996bx}.
 The gradient flow method in Ref.~\cite{Asakawa:2013laa} gives the thermal entropy, $s/T^3=4.98(24)$, after $a \rightarrow 0$ and then $t\rightarrow 0$ extrapolations.

Gauge configurations are generated by the pseudo-heatbath algorithm with the over-relaxation.
We call one pseudo-heatbath update sweep plus several over-relaxation sweeps
as a ``Sweep".
To eliminate the autocorrelation, we take $200$ Sweeps between measurements.
The number of gauge configurations for the measurements is $2,000$.
Note that it is quite a small number of configurations in comparison with $800,000$ in Ref.~\cite{Nakamura:2004sy} and $6$ million in Ref.~\cite{Pasztor:2018yae}.

Judging from such a poor statistic, we give up a serious estimation of the statistical error and focus on the feasibility of the sparse modeling method to obtain the shear viscosity combing with the gradient flow method.
In \S.~\ref{sec:SpM} we explain how to estimate the spectral function using the central value of the correlation function.
Then, in \S.~\ref{sec:error}, we discuss the systematic and statistical errors in the analysis.
We demonstrate the bootstrap analyses to estimate  statistical uncertainty.

The flowed gauge field is obtained by solving the ordinary first-order differential equation (Eq.~(\ref{eq:gflow})).
Numerically, we utilize the third-order Runge-Kutta method in which the error per step ($t \rightarrow t+ \varepsilon$) is ${\cal O}(\varepsilon^5)$.
We take $\varepsilon= 0.01$, and confirm that the accumulation errors are sufficiently smaller than the statistical errors.
The gauge action of the flow is the Wilson plaquette gauge action.

\section{Results of the sparse modeling method for the central value of $C(t,\tau/a)$}\label{sec:SpM}
Now, we demonstrate the sparse modeling analysis using the central value of the measured correlation functions.
Table~\ref{table:parameter} shows the technical parameters in the analysis in this section.
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
 $s_{cut}$ & $a \omega_{cut}$  & $N_\omega$  & $[\lambda_{min},\lambda_{max}]$ & $N_\lambda$  &  $(\mu,\mu')$  \\
 \hline
 $10^{-10}$ & $ 4.0 $  & $3001$  &  $ [10^{-15} , 10^{2}]$  & $100$  &  $(1.0,1.0)$  \\
 \hline
\end{tabular}
\caption{ Parameters for the present sparse-modeling analysis } \label{table:parameter}
\end{center}
\end{table}
The reason why we take these values of $s_{cut}$ and $a \omega_{cut}$ will be explained below and \S.~\ref{sec:omega-deps}.
Fixing $[\lambda_{min},\lambda_{max}]$, $N_\lambda$, and $(\mu,\mu')$ are correlated with each other.
This set is a possible choice to find an optimal value of $\lambda$.

In {\bf Step 1'} listed the end of \S.~\ref{sec:tech-step}, we measure the correlation function using the gradient flow method in the range of the flow-times $0.50 \le t/a^2 \le 2.50$ with the interval $\Delta t/a^2=0.10$.
Take $t/a^2 = 0.50,1.00,1.50,2.00$, and $2.50$ for example in this section.
The data of $C(t,\tau/a)$ appears in Fig.~\ref{fig:Ctau-ft-deps}.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.5]{./Center-Ctau-ft-deps-2.pdf}
\caption{The correlation function of $T_{12}^R$ operator at several flow-time.  }
\label{fig:Ctau-ft-deps}
\end{center}
\end{figure}
We see from Fig.~\ref{fig:Ctau-ft-deps} that the longer flow-time data have a smaller statistical error.
Actually, in these poor statistics, namely $2,000$ configurations, the correlation function of the non-smeared $U_{12}$ operator is quite noisy and its central values at some $\tau/a$ often take negative values because of the large fluctuations.
However, all flowed data shown in the left panel in Fig.~\ref{fig:Ctau-ft-deps} (except for only one data-point at $\tau/a=8$ at $t/a^2=1.00$) indicate correctly positive values.
It is a great advantage of the usage of the gradient flow method to obtain the correlation function.

On the other hand, there is a demerit of the usage of the gradient flow for $C(t,\tau/a)$.
Thus, the smearing changes the slope in the short $\tau$-regime because of the over-smearing.
To see the merit and demerit clearly, we rescale the correlation function by multiplying the flow-time, $(t/a^2)C(t,\tau/a)$ (right panel in  Fig.~\ref{fig:Ctau-ft-deps}).
In $\tau/a \lesssim 3$, the slope of the correlation function strongly depends on the flow-time.
It indicates that the kernel term, which is proportional to hyperbolic cosine function, is not available for the whole $\tau$-regime in the flowed correlation function (see also Appendix~\ref{sec:All-tau-analysis}).
On the other hand, around $\tau/a = N_\tau/2 $ the curve of the correlation function does not change, while the statistical errors are reduced thanks to the smearing.
A similar property of the smearing has been reported in the case of the APE smearing for the Wilson loop~(see Fig.~$5$ in Ref.\cite{Bilgici:2009kh}).
To take only the merits of the gradient flow, it is better to take the fiducial $\tau$-regime at  finite flow-time, $\sqrt{8t} < \tau$, for the estimation of $\rho(\omega)$.
Here, we  fix the $\tau$-regime as $3 \le \tau/a \le 13 $ and investigate the flow dependence of the results.

The next step, {\bf Step 2'},  is the SVD decomposition of the kernel matrix.
We introduce the cutoff $\hat{\omega}_{cut}=4.0$ and discretize $a \omega$ in the range of $-4.0 \le a\omega \le 4.0$ into $N_{\omega}=3001$ data.
The singular values of the SVD decomposition appear in the left panel of Fig.~\ref{fig:log-sl-tau-deps}.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.5]{./log_sl_l_ft-deps.pdf}
\caption{$\tau$-regime (left) and $N_\omega$ (right) dependence of the singular of the kernel matrix. Here, we utilize $a \omega_{cut}=4.0$.  }
\label{fig:log-sl-tau-deps}
\end{center}
\end{figure}
Note that the vertical axis takes a log scale.
The figure tells us that a large hierarchy more than $10^{10}$ exists between $6$-th and $7$-th singular values if we take $3 \le \tau/a \le 13$.
Furthermore, we find that the number of $s_l$ larger than $10^{-16}$ is the same with the number of independent $\hat{\tau}_i$ under the periodic boundary condition.
Refer to several condensed matter studies~\cite{Chikano2018a,Chikano2018b,Li2019}, it will saturate to a specific value even though the number of the data points for $C(\tau)$ increases.
It indicates that the number of $\tau_i$, here at most $7$ in taking $2 \le \tau/a \le 12$ under the periodic boundary condition, to analyze this type of kernels is very sparse~\footnote{
We can also estimate a lattice size in the temporal direction to fully analyze the kernel (see Appendix~\ref{sec:App-sl}).}.
To minimize the square error in Eq.~(\ref{eq:chi-sq-SVD}), the contributions of $s_l \rho'_l$ with a sufficiently small $s_l$ are negligible.
Therefore, we introduce the threshold of  the singular value $s_{cut}$ as $s_{cut}=10^{-10}$ in the present analysis.

The most important point is that this property of the singular values is determined only by the kernel matrix Eq.~(\ref{eq:kernel-SVD}) and is independent of the simulation details.
Here, we just assume that the correlation function can be described by a hyperbolic cosine function.

The right panel in Fig.~\ref{fig:log-sl-tau-deps} depicts the $N_\omega$ dependence of the singular values using $3 \le \tau/a \le 13$.
We find that $s_l$ is almost independent of $N_\omega$ if we include $\sqrt{\Delta \omega'}$ in the kernel matrix as Eq.~(\ref{eq:kernel-SVD}).

It is worth to see the correlation function in the IR basis,
\beq
C'_l = \sum_{\hat{\tau}=3}^{13} U^{t}(l,\hat{\tau}) C(\hat{\tau}) .
\eeq
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.5]{./Center-Cl-ft-tau-deps.pdf}
\caption{The input correlation function in the IR basis ($C_l$) as a function of the label $l$.   }
\label{fig:Cl-ft-deps}
\end{center}
\end{figure}
Figures~\ref{fig:Cl-ft-deps} depicts $C_l$ as a function of $l$.
In comparison with the right panel where $4 \le \tau/a \le 12$, the values of $C_l$ is large.
We take  $3 \le \tau/a \le 13$ and will discuss the results using  $4 \le \tau/a \le 12$ in \S.~\ref{sec:tau-deps}.
We see that $C_l$ with $l \gtrsim 5$ is also sufficiently small, so that it is consistent with the truncation of $s_l$ for $l \ge 7$.

{\bf Step 3'} is the $\lambda$ optimization.
The $\lambda$ dependence of the square error (Eq.~(\ref{eq:chi-sq})) is a monotonically increasing function of $\lambda$~\cite{Shinaoka2017b}.
The optimized $\lambda$ is given by $\lambda^{opt}=1.1 \times 10^{-7}$ -- $1.9 \times 10^{-6}$ in our analysis.
To find this, we scan the value of $\lambda$ in the range of $ 10^{-15} \le \lambda \le 10^{2}$ by changing its exponent with $1 /N_\lambda$ interval.
Once we obtain $\lambda^{opt}$, then we check  whether it shows roughly an inverse-rescaling with $(\mu,\mu')$.

The results of the spectral function are shown in Fig.~\ref{fig:rho-center}.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.5]{./rho-center-ft-deps.pdf}
\caption{The obtained spectral function $\tilde{\rho}(t, a\omega)$ as a function of $a \omega$.  The right panel is an enlarged plot in $\omega \approx 0$ regime.   }
\label{fig:rho-center}
\end{center}
\end{figure}
First of all, we find the tails of spectral function for all flow-times approach to zero.
It tells us that  $a \omega_{cut}=4.0$ is a reasonable choice in this analysis.
Secondly, we see the integration of $\tilde{\rho}(t,a\omega)$ in terms of $a \omega$,
\beq
{\mathcal N}=\int_{-a\omega_{cut}}^{a\omega_{cut}} \tilde{\rho}(t,a\omega) d (a\omega),\label{eq:N}
\eeq
monotonically decreases as a function of flow-time.
In more detail, the spectral functions in large $|\omega|$ regime are highly suppressed in the longer flow-time.
The gradient flow gradually reduces the degree of freedom with high-frequency and can be interpreted as a renormalization group flow.
We can see that the results of the sparse modeling analysis give a good account of such an intuitive picture.

The left panel of Fig.~\ref{fig:rho-center} is an enlarged plot, which focuses on $\omega \approx 0$.
The curvatures of $\tilde{\rho}(t,a \omega)$ at $\omega=0$ has an opposite sign between  $t/a^2 =0.50,1.00$ and $t/a^2=1.50,2.00,2.50$.
It remind us that the gradient flow smears the data in $\tau/a < \sqrt{8t}/a$, then the data at $\tau/a =3$ (and $13$) are over-smeared in $t/a^2 \le 1.12$.
Thus, the results in $t/a^2=0.50,1.00$ may suffer from the over-smearing corrections.
To conclude this, we have to investigate the statistical uncertainty of the input $C(t,\tau/a)$.
We will discuss this point in \S.~\ref{sec:error}.

Finally, in {\bf Step 4'}, we check whether the obtained spectral function correctly reproduces the input correlation function.
Figure~\ref{fig:comp-Ctau-center-ana} depicts the comparison plot between the input $C(t,\tau/a)$ and the $C_{output}(t,\tau/a)$ constructed by the obtained $\tilde{\rho}(t, a\omega)$.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.35]{./Comp-Ctau-in-out-center-ana-shadow.pdf}
\caption{Comparison between the input $C(t,\tau/a)$ with the jackknife error and the output $C_{output}(t,\tau/a)$ constructed by the obtained $\tilde{\rho}(t,a\omega)$.  The input $C(t,\tau/a)$ in shadowing regime are not utilized in the analyses. }
\label{fig:comp-Ctau-center-ana}
\end{center}
\end{figure}
Note that the input data of the analysis is the central value of $C(t,\tau/a)$, while its jackknife errors are also shown.
We see that most data, especially in the short $\tau$-regime, are consistent between the input and output, while the discrepancies exist around $\tau/a= N_\tau/2$.
We consider that it comes from the large fluctuation of the input data, where some of them are consistent with zero or take a negative value.
On the other hand, the output data is constructed to be positive, since we utilize the non-negativity condition to find the spectral function.
Actually, the discrepancy around $\tau/aN_\tau=1/2$ in the longer flow-time with less statistical uncertainty becomes smaller.
Therefore, we believe that the discrepancy will be reduced if we have the input data precisely.

\section{Error estimations}\label{sec:error}
\subsection{ $\omega_{cut}$ dependence}\label{sec:omega-deps}
Now, we study the systematic uncertainty coming from $\omega_{cut}$, which is artificially introduced.
Figure~\ref{fig:rho-omegamax-deps} shows the comparison of the obtained $\tilde{\rho}(t,a\omega)$ between taking $a\omega_{cut}=3.0$ and $a\omega_{cut}=4.0$ for several flow-times data.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.45]{./rho-center-omegamax-deps.pdf}
\caption{The $\omega_{cut}$ dependence of the spectral function for several flow-time data.   }
\label{fig:rho-omegamax-deps}
\end{center}
\end{figure}
The tails of the spectral functions using $a\omega_{cut}=4.0$ for all flow-times approach to zero, while the ones using $a \omega_{cut}=3.0$ for $t/a^2=0.50, 1.00$ are the middle of the decreasing.
Nevertheless, the shape of spectral function for $t/a^2=0.50, 1.00$ does not depend on the value of $\omega_{cut}$ so much.
It implies the stability of this sparse modeling analysis when the artificial parameter $\omega_{cut}$ is changed.

On the other hand, the tails of $\tilde{\rho}(t,a\omega)$ for $t/a^2=1.50, 2.00$ are closed to zero even though we utilize $a\omega_{cut}=3.0$, since the higher frequency modes are suppressed by the long flow processes.
From this point of view, although the sparse-modeling method can apply to the non-smeared data, the smearing may practically stabilize the analysis with the truncation of $\omega$~{\footnote{Applying the sparse modeling analysis  to mock data whose spectral function has a constant mode has been considered~\cite{Tomiya}. }}.

\subsection{$\tau$-regime dependence and the fiducial window of the gradient flow}\label{sec:tau-deps}
Next, we investigate the relationship between a choice of $\tau$-regime and the gradient flow-time.
The fiducial $\tau$-regime is theoretically given as $\tau > \sqrt{8t}$, and we expect that  the data does not suffer from an over-smearing.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.45]{./rho-center-Ntaudata-deps.pdf}
\caption{The $\tau$-regime dependence of the spectral function for several flow-time data.   }
\label{fig:rho-Ntaudata-deps}
\end{center}
\end{figure}
In other words, if we take $3 \le \tau/a \le 13$, then $C(t,\tau/a)$ in $t/a^2 < 1.12$ are in the fiducial regime, while $\tau/a=3$ (and $13$) must have a strong over-smearing correction in $t/a^2 > 1.12$.
On the other hand, if we take $4 \le \tau/a \le 12$, then $C(t,\tau/a)$ in $t/a^2 < 2.00$ stay in the fiducial regime.

Figure~\ref{fig:rho-Ntaudata-deps} shows the comparison of the spectral functions between the $3 \le \tau/a \le 13$ and $4 \le \tau/a \le 12$ analyses for $t/a^2 =0.50, 1.00, 1.50, 2.00$.
First of all, as we explained, the gradient flow with each fixed $\tau$-regime (same color symbols) reduces $\mathcal{N}$ in Eq.~(\ref{eq:N}).
On the other hand, we naively expected that $\mathcal{N}$ must decrease if we take the narrow regime of $\tau$ since the effective degrees of freedom are truncated.
However, in $t/a^2=1.50, 2.00$, $\mathcal{N}$ of the red data ($4\le \tau/a \le 12$) is larger than the one of the black data ($3 \le \tau/a \le 13$).
The discrepancy is slightly beyond the statistical error, which we will study in \S.~\ref{sec:stat-error}.
It implies that the corrections to the spectral function from over-smearing in the black data are strong rather than the influence on $\tilde{\rho}(t,a\omega)$ from the truncation of $C(t,\tau/a)$.

\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.35]{./rho0-center-ft-deps-with-shadow.pdf}
\caption{The flow-time and $\tau$-regime dependence of $\tilde{\rho}(t,\omega=0)$. The fiducial window ($2a < \sqrt{8t} < \tau$)  is indicated
by the dashed lines. $0.50 < t/a^2 < 1.12$ (non-shadowing) is the fiducial window for the  $3 \le \tau/a \le 13$ analysis, while $0.50 < t/a^2 < 2.00$ (non-shadowing and red-shadowing) depicts the one for the $4 \le \tau/a \le 12$ analysis.   }
\label{fig:rho0-ft-deps}
\end{center}
\end{figure}
Figure~\ref{fig:rho0-ft-deps} shows $\tilde{\rho}(t,0)$ as a function of the flow-time.
Here, we summarize that the fiducial regime of the flow-time for each $\tau$-regime analysis.
The orders of all $\tilde{\rho} (t,0)$ in the fiducial window are the same.
In more detail, the value itself seems to have a discrepancy between two $\tau$-regime analyses.
We will discuss the difference in the next section after including the statistical uncertainty of $C(t,\tau/a)$ since a tiny noise of the correlation function often leads to a large difference in the spectral function in this kind of the ill-posed inverse problems.

\subsection{Statistical errors }\label{sec:stat-error}
Now, we try to include the statistical uncertainty of the correlation function in our analysis.
The statistical uncertainty of the correlation function ($C(t,\tau/a)$) is not directly related to the error of the spectral function ($\tilde{\rho}(t,a\omega)$) since these two quantities are related to each other through the integration equation.
On the other hand, the correlation function in the IR basis is linearly related to the spectral function in the same basis (Eq.~(\ref{eq:Cl-sl-rhol})).
Thus, we expect that the statistical uncertainties of these quantities satisfy
\beq
\Delta C_l \sim s_l \Delta \rho_l.\label{eq:error-Cl-rhol}
\eeq

We propose the bootstrap method to estimate the statistical error of the spectral function as follows:
We resample $N_{boot}$ sets of $2,000$ data of $C(t,\tau/a)$ for each configuration, where the overlapping selection of the configurations for one bootstrap sample is allowed.
For each set of the bootstrap sample, we take the average of $2,000$ data of $C(t, \tau/a)$ and carry out the sparse modeling analysis using its mean value.
The statistical errors of the spectral function are calculated by the variance over $N_{boot}$ samples.
It allows finding the asymmetric errors, so that it is a good tool to calculate the errors of $\tilde{\rho}$ which should be non-negative as a constraint.
Here, we take $N_{boot}=1,000$.

\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.45]{./rho-with-booterror.pdf}
\caption{The statistical error of the spectral function using the bootstrap analysis. }
\label{fig:rho-errors}
\end{center}
\end{figure}
Figure~\ref{fig:rho-errors} depicts the spectral function with the bootstrap errors.
We also plot the result of the central-value analysis shown in Fig.~\ref{fig:rho-center}, which is inside the error bound.

We check whether the bootstrap errors satisfy an expected relationship Eq.~(\ref{eq:error-Cl-rhol}).
Here, using Eq.~(\ref{eq:rho-C-IR-basis}), $\Delta C_l$ is estimated as the variation of $C_l$  transformed by the bootstrap sample of $C(t,\tau/a)$,
while $\Delta \rho_l$ is done as the variation of the obtained $\tilde{\rho}(t,\hat{\omega})$ for each bootstrap sample.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.5]{./Cl-rhol-with-error.pdf}
\caption{The correlation function and the spectral function in the IR basis as a function of $l$.  The flow-time of the data is $t/a^2=1.50$. }
\label{fig:Cl-rhol}
\end{center}
\end{figure}
Figure~\ref{fig:Cl-rhol} shows the $l$-dependence of $C_l$ and $\rho_l$.
We see that the statistical errors of them roughly satisfies $\Delta \rho_l \sim \Delta C_l /s_l $.
Note that $\rho_l$ with $l \ge 7$ are consistent with zero since these modes are eliminated by the cut of the singular values in the analysis.

We also show the comparison of the statistical error bars between the input and output $C(t,\tau/a)$ in Fig.~\ref{fig:input-output-Ctau}.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.35]{./input-output-Ctau-boot-shadow.pdf}
\caption{The comparison between the input correlation function with the Jackknife error and the output one with the bootstrap error.  The data $C(t,\tau/a)$ in shadowing regime are not utilized in the analyses.}
\label{fig:input-output-Ctau}
\end{center}
\end{figure}
The error bars between them in the short $\tau$-regime are consistent with each other.
The error bars of the output around $\tau/a=N_\tau/2$ are smaller than the ones of the input.
We consider that it comes from the non-negativity condition of the spectral function during the sparse modeling analysis.
It makes $C(t,\tau/a)$ positive correctly, and then the condition reduces the error bars of $C_{output}$.
We can conclude that the bootstrap method is a reasonable estimation method of the statistical errors for the sparse modeling analysis.

\subsection{Flow-time dependence of the shear viscosity}
Finally, we discuss the flow-time dependence of the shear viscosity with the statistical uncertainty.
Since the number of configurations is quite poor, then we cannot give a conclusive value of the shear viscosity.
Furthermore, we first take the continuum limit of $\eta(t,T)$ obtained by the lattice simulation with several different lattice spacings at the fixed temperature. After that, we could discuss the flow-time dependence~\footnote{
Otherwise, physical observables diverge in the inverse-ordered extrapolations, $t\rightarrow 0$ and then $a \rightarrow 0$.}.
Thus, in the present analysis, we do not include the systematic uncertainties coming from the continuum extrapolation.
Therefore, we show the results to see naive flow-time dependence at a fixed lattice spacing.

\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.25]{./rho0-eta-with-shadow.pdf}
\caption{ (Left) The spectral function at $\omega=0$. (Right) The ratio between the shear viscosity and the thermal entropy ($\eta/s$).
Horizontal axes denote the flow-time. Magenta data at $t/a^2=0$ denotes the result by H.~Meyer at the same temperature~\cite{Meyer:2007ic}.
The fiducial window ($2a < \sqrt{8t} < \tau$)  is indicated by the dashed lines, where the meaning of each line and shadowing region is the same with the one in Fig.~\ref{fig:rho0-ft-deps}.   }
\label{fig:rho0-eta}
\end{center}
\end{figure}
The left panel in Fig.~\ref{fig:rho0-eta} depicts the value of $\tilde{\rho}(t,0)$ in the $3 \le \tau/a \le 13$ and $4 \le \tau/a \le 12$ analyses.
We find that both analyses are consistent with each other
in non-shadowing regime ($0.50 < t/a^2 < 1.12$), where all data in both analyses are in the fiducial $\tau$-regime.
However, the discrepancy appears in red-shadowing regime ($1.12 < t/a^2 < 2.00$), where the data at $\tau/a=3$ is over-smeared.
Furthermore, the slope of the flow-time dependence of the $4 \le \tau/a \le 12$ analysis is milder than the one of the $3 \le \tau/a \le13$ analysis.
It tells us the importance of taking the fiducial window in the sparse modeling analysis.

The ratio between the shear viscosity and the thermal entropy is shown in the right panel in Fig.~\ref{fig:rho0-eta}.
Here, we utilize $s/T^3=4.98$ at this temperature, which is given in Ref.~\cite{Asakawa:2013laa}.
We also plot the result in Ref.~\cite{Meyer:2007ic} at $t/a^2=0$ as a reference.
A theoretical large-$N_c$ analysis based on AdS/CFT correspondence for ${\mathcal N}=4$ super Yang-Miils theory gives the lower bound for $\eta/s=1/4\pi$~\cite{Son:2007vk}~\footnote{The $1/N_c$ correction terms to $\eta/s$ has not yet been determined even for its sign in the finite $N_c$~\cite{Kats:2007mq}.}.
Although our result seems to be smaller than the previous works, we can conclude that our result also indicates the small $\eta/s$, which describes the perfect liquid property in the QGP phase.

\section{Summary}\label{sec:summary}
We have proposed the sparse modeling analysis to estimate the spectral function from the smeared correlation functions.
We have described how to obtain the shear viscosity from the correlation function of the renormalized EMT measured by using the gradient flow method for the quenched QCD at  finite temperature.
The gradient flow method reduces the statistical uncertainty of the correlation functions thanks to its smearing property,
while the smearing breaks the sum rule of the spectral function.
 Therefore, we have eliminated the over-smeared data  when we analyze the spectral function.

We have first given the standard formula of the sparse modeling analysis where we assume  $C(t,\tau)$ in the whole $\tau$-regime is available for the analysis.
Then, we have modified the formulation to investigate the smeared correlation function, where the over-smeared data of $C(t,\tau/a)$ are eliminated.
We have shown that the sparse modeling analyses in the IR basis looks stable even using very limited data of the correlation function and the obtained spectral function reproduces the input correlation function.
Several systematic uncertainties of the analysis are well under control.
We have also demonstrated the bootstrap analysis for estimating the statistical errors.
Although the statistical error of our result is sizable because of poor statistics of our data, we have shown that the bootstrap analysis seems to be promising since the expected relationship of the errors between $C_l$ and $\rho_l$ have been satisfied.

If we will collect $6$ million configurations as the same with the work~\cite{Pasztor:2018yae}, a naive estimation following $1/\sqrt{N_{conf.}}$ would give a few $\%$ relative error for $\eta/s$.
It looks very promising toward the precise determination of the shear viscosity.

\acknowledgments
We are grateful to  T.~Hirano and A.~Tomiya for useful comments.
The numerical simulations were carried out on SX-ACE
 at Cybermedia Center (CMC) and Research Center for Nuclear Physics (RCNP),
Osaka University.
We also acknowledge the help of CMC in tuning the gradient-flow code.
This work partially used computational resources of
 HPCI-JHPCN System Research Project (Project ID: jh200031) in Japan.
 This work was supported in part by Grants-in-Aid for Scientific Research
through Grant Nos.
 15H05855
and 19K0387
, which were
provided by the Japan Society for the Promotion of Science (JSPS), and in part
by the Program for the Strategic Research Foundation at Private Universities
``Topological Science'' through Grant No.\ S1511006, which was supported by the
Ministry of Education, Culture, Sports, Science and Technology (MEXT) of
Japan.
The calculations were partially performed by the supercomputing system SGI ICE X at the Japan Atomic Energy Agency.
This work was partially supported by JSPS-KAKENHI Grant Numbers 18K11345.