\section{Introduction}\label{sec1}
\setcounter{equation}{0}

Transport problems with highly forward-peaked scattering are prevalent in a variety of areas, including astrophysics, medical physics, and plasma physics \cite{HGK,aristova,multiphysics}.
For these problems, solutions of the transport equation converge slowly when using conventional methods such as source iteration (SI) \cite{adamslarsen} and the generalized minimal residual method (GMRES) \cite{gmres}.
Moreover, diffusion-based acceleration techniques like diffusion synthetic acceleration (DSA) \cite{alcouffe} and nonlinear diffusion acceleration (NDA) \cite{smithetall} are generally inefficient when tackling these problems, as they only accelerate up to the first moment of the angular flux \cite{JapanFPSA}.
In fact, higher-order moments carry important information in problems with highly forward-peaked scattering and can be used to further accelerate convergence \cite{japanDiss}.

This paper focuses on solution methods for the monoenergetic, steady-state transport equation in homogeneous slab geometry.
Under these conditions, the transport equation is given by
\begin{subequations}\label[pluraleq]{eq1}
\begin{equation}
\label{t1}
\mu\frac{\partial}{\partial x} \psi(x,\mu) + \sigma_t \psi(x,\mu) = \int_{-1}^{1} d\mu' \sigma_s(\mu,\mu') \psi(x,\mu') + Q(x, \mu), \,\,\, x\in [0, X],-1\leq\mu\leq 1  ,\\
\end{equation}
with boundary conditions
\begin{align}
\label{t2}
\psi(0,\mu) &= \psi_L(\mu), \quad \mu > 0,\\
\label{t3}
\psi(X,\mu) &= \psi_R(\mu), \quad \mu < 0.
\end{align}
\end{subequations}
Here, $\psi(x,\mu)$ represents the angular flux at position $x$ and direction $\mu$, $\sigma_t$ is the macroscopic total cross section, $\sigma_s(\mu,\mu')$ is the differential scattering cross section, and $Q$ is an internal source.

New innovations have paved the way to better solve this equation in systems with highly forward-peaked scattering.
For instance, work has been done on modified $P_L$ equations and modified scattering cross section moments to accelerate convergence of anisotropic neutron transport problems \cite{khattab}.
In order to speed up the convergence of radiative transfer in clouds, a quasi-diffusion method has been developed \cite{aristova}.
In addition, the DSA-multigrid method was developed to solve problems in electron transport more efficiently \cite{trucksin}.

One of the most recent convergence methods developed is Fokker-Planck Synthetic Acceleration (FPSA) \cite{JapanFPSA,japanDiss}.
FPSA accelerates up to $N$ moments of the angular flux and has shown significant improvement in the convergence rate for the types of problems described above.
The method returns a speed-up of several orders of magnitude with respect to wall-clock time when compared to DSA  \cite{JapanFPSA}.

In this paper, we introduce a new acceleration technique, called \textit{Nonlinear Fokker-Planck Acceleration} (NFPA).
This  method  returns  a  modified  Fokker-Planck (FP) equation  that  preserves  the  angular moments of the flux given by the transport  equation.
This preservation of moments is particularly appealing for applications to multiphysics problems \cite{multiphysics}, in which the coupling between the transport physics and the other physics can be done through the (lower-order) FP equation.
To our knowledge, this is the first implementation of a numerical method that returns a Fokker-Planck-like equation that is discretely consistent with the linear Boltzmann equation.

This paper is organized as follows.
\Cref{sec2} starts with a brief description of FPSA.
Then, we derive the NFPA scheme.
In \cref{sec3}, we discuss the discretization schemes used in this work and present numerical results.
These are compared against standard acceleration techniques.
We conclude with a discussion in \cref{sec4}.

\section{Fokker-Planck Acceleration}\label{sec2}
\setcounter{equation}{0}
In this section we briefly outline the theory behind FPSA, describe NFPA for monoenergetic, steady-state transport problems in slab geometry, and present the numerical methodology behind NFPA.
The theory given here can be easily extended to higher-dimensional problems.
Moreover, extending the method to energy-dependence shall not lead to significant additional theoretical difficulties.

To solve the transport problem given by \cref{eq1} we approximate the in-scattering term in \cref{t1} with a Legendre moment expansion:
\begin{equation}
\label{transport1}
\mu\frac{\partial}{\partial x} \psi(x,\mu) + \sigma_t \psi(x,\mu) = \sum_{l=0}^L \frac{(2l+1)}{2} P_l(\mu) \sigma_{s,l} \phi_l(x) + Q(x, \mu),
\end{equation}
with
\begin{equation}
\label{transport2}
\phi_l(x) =  \int_{-1}^{1} d\mu P_l(\mu) \psi(x,\mu).
\end{equation}
Here, $\phi_l$ is the $l^{th}$ Legendre moment of the angular flux, $ \sigma_{s,l}$ is the $l^{th}$ Legendre coefficient of the differential scattering cross section,  and $P_l$ is the $l^{th}$-order Legendre polynomial.
For simplicity, we will drop the notation $(x,\mu)$ in the remainder of this section.

The solution to \cref{transport1} converges asymptotically to the solution of the following Fokker-Planck equation in the forward-peaked limit \cite{pomraning1}:
\begin{equation}
\label{fp1}
\mu\frac{\partial \psi}{\partial x} + \sigma_a \psi = \frac{\sigma_{tr}}{2}\frac{\partial }{\partial \mu} (1-\mu^2) \frac{\partial \psi}{\partial \mu} + Q\,,
\end{equation}
where $\sigma_{tr}= \sigma_{s,0} -\sigma_{s,1}$ is the momentum transfer cross section and $\sigma_a = \sigma_t-\sigma_{s,0}$ is the macroscopic absorption cross section.

Source Iteration \cite{adamslarsen} is generally used to solve \cref{transport1}, which can be rewritten in operator notation:
\begin{equation}
\label{si1}
\mathcal{L} \psi^{m+1} = \mathcal{S} \psi^{m} + Q\,,
\end{equation}
where
\begin{equation}
\mathcal{L} = \mu \frac{\partial}{\partial x} + \sigma_t,
   \quad
\mathcal{S} = \sum_{l=0}^L \frac{(2l+1)}{2} P_l(\mu) \sigma_{s,l} \int_{-1}^{1}d\mu P_l(\mu) ,
\label{trans1}
\end{equation}
and $m$ is the iteration index.
This equation is solved iteratively until a tolerance criterion is met. The FP approximation shown in \cref{fp1} can be used to accelerate the convergence of \cref{transport1}.

\subsection{FPSA: Fokker-Planck Synthetic Acceleration}\label{FPSA}

In the FPSA scheme \cite{JapanFPSA,japanDiss}, the FP approximation is used as a preconditioner to synthetically accelerate convergence when solving \cref{transport1} (cf. \cite{adamslarsen} for a detailed description of synthetic acceleration).
When solving \cref{si1}, the angular flux at each iteration $m$ has an error associated with it.
FPSA systematically follows a predict, correct, iterate scheme.
A transport sweep, one iteration in \cref{si1}, is made for a prediction.
The FP approximation is used to correct the error in the prediction, and this iteration is performed until a convergence criterion is met.
The equations used are:
\begin{subequations}
\label{fpsaeq}
\begin{align}
\label{predict}
\mathrm{Predict}&: \mathcal{L} \psi^{m+\frac{1}{2}} = \mathcal{S} \psi^{m} + Q\,,\\
\label{correct}
\mathrm{Correct}&: \psi^{m+1} =  \psi^{m+\frac{1}{2}} + \mathcal{P}^{-1} \mathcal{S} \left( \psi^{m+\frac{1}{2}} -  \psi^{m}\right),
\end{align}
\end{subequations}
where we define $\mathcal{P}$ as
\begin{equation}
\label{FPSAsi1}
\mathcal{P} = \mathcal{A}-\mathcal{F} =\underbrace{\left(\mu\frac{\partial}{\partial x} + \sigma_a\right)}_\mathcal{A} - \underbrace{\left(\frac{\sigma_{tr}}{2}\frac{\partial }{\partial \mu} (1-\mu^2) \frac{\partial }{\partial \mu}\right)}_\mathcal{F},
\end{equation}
In this synthetic acceleration method, the FP approximation is used to correct the error in each iteration of the high-order (HO) equation (\ref{predict}).
Therefore, there is no consistency between the angular moments of the flux in the HO and low-order (LO) equations.

\subsection{NFPA: Nonlinear Fokker-Planck Acceleration}\label{NFPA}

Similar to FPSA, NFPA uses the FP approximation to accelerate the convergence of the solution.
We introduce the additive term $\hat{D}_F$ to \cref{fp1}, obtaining the modified FP equation
\begin{equation}
\label{mfp1}
\mu\frac{\partial \psi}{\partial x} + \sigma_a \psi = \frac{\sigma_{tr}}{2}\frac{\partial }{\partial \mu} (1-\mu^2) \frac{\partial \psi}{\partial \mu} + \hat{D}_F + Q\,.
\end{equation}
The role of $\hat{D}_F$ is to force the transport and modified FP equations to be consistent.
Subtracting \cref{mfp1} from \cref{transport1} and rearranging, we obtain the consistency term
\begin{equation}
\label{dfp}
\hat{D}_F = \sum_{l=0}^L \frac{(2l+1)}{2} P_l \sigma_l \phi_l - \frac{\sigma_{tr}}{2}\frac{\partial}{\partial \mu} (1-\mu^2) \frac{\partial \psi}{\partial \mu} - \sigma_{s,0} \psi\,.
\end{equation}

The NFPA method is given by the following equations:
\begin{subequations}\label[pluraleq]{holocons}
\begin{align}
\label{HO1}
\text{HO}&: \mu\frac{\partial \psi_{HO}}{\partial x} + \sigma_t \psi_{HO} = \sum_{l=0}^L \frac{(2l+1)}{2} P_l \sigma_l \phi_{l, LO} + Q\,,\\
\label{LO11}
\text{LO}&: \mu\frac{\partial \psi_{LO}}{\partial x} + \sigma_a \psi_{LO} = \frac{\sigma_{tr}}{2}\frac{\partial }{\partial \mu} (1-\mu^2) \frac{\partial \psi_{LO}}{\partial \mu} + \hat{D}_F + Q\,,\\
\label{con1}
\text{Consistency term}&: \hat{D}_F = \sum_{l=0}^L \frac{(2l+1)}{2} P_l \sigma_l \phi_{l, HO}^m - \frac{\sigma_{tr}}{2}\frac{\partial }{\partial \mu} (1-\mu^2) \frac{\partial \psi_{HO}}{\partial \mu} - \sigma_{s,0} \psi_{HO}\,,
\end{align}
\end{subequations}
where $\psi_{HO}$ is the angular flux obtained from the HO equation and $\psi_{LO}$ is the angular flux obtained from the LO equation.
The nonlinear HOLO-plus-consistency system given by \cref{holocons} can be solved using any nonlinear solution technique \cite{kelley}. Note that the NFPA scheme returns a FP equation that is consistent with HO transport.
Moreover, this modified FP equation accounts for large-angle scattering which the standard FP equation does not.
The LO equation (\ref{fp1}) can then be integrated into multiphysics models in a similar fashion to standard HOLO schemes \cite{patelFBR}. To solve the HOLO-plus-consistency system above, we use Picard iteration \cite{kelley}:
\begin{subequations}
\begin{align}
\label{H1}
\text{Transport Sweep for HO}&:
\mathcal{L} \psi_{HO}^{k+1} = \mathcal{S} \psi_{LO}^{k} + Q, \\
\label{L1}
\text{Evaluate Consistency Term}&: \hat{D}_F^{k+1} = \left(\mathcal{S} - \mathcal{F} - \sigma_{s,0}\mathcal{I}\right) \psi_{HO}^{k+1}, \\
\label{c1}
\text{Solve LO Equation}&: \psi_{LO}^{k+1} = \mathcal{P}^{-1} \left(\hat{D}_F^{k+1} + Q\right),
\end{align}
\end{subequations}
where $\mathcal{L}$ and $\mathcal{S}$ are given in \cref{trans1}, $\mathcal{P}$ and $\mathcal{F}$ are given in \cref{FPSAsi1}, $\mathcal{I}$ is the identity operator, and $k$ is the iteration index.
Iteration is done until a convergence criterion is met.

The main advantage of setting up the LO equation in this fashion is that the stiffness matrix for LO needs to be setup and inverted \textit{only once}, just as with FPSA \cite{JapanFPSA, japanDiss}. This has a large impact on the method's performance.
A flowchart of this algorithm is shown in \cref{Nalgorithm}.

\begin{figure}[H]
\centering
\begin{tikzpicture}[node distance = 3cm, auto]

    \node [block] (init) {Initial guess of flux moments};
    \node [cloud_HO, right of=init, node distance=4cm] (HOm) {HO};
    \node [cloud_LO, below of=HOm, node distance=2cm] (LOm) {LO};
    \node [HO, below of=init] (transport) {One sweep in transport equation};
    \node [decision, below of=transport,node distance=4cm] (decide) {Flux moments converged?};
    \node [LO, left of=decide, node distance=4cm] (dterm) {Solve for consistency term};
    \node [LO, left of=dterm, node distance=3cm] (MFP) {Solve for FP angular flux};
    \node [LO, above of=MFP, node distance=4cm] (moments) {Convert angular flux to moments};
    \node [block, right of=decide, node distance=4cm] (stop) {Stop};

    \path [line] (init) -- (transport);
    \path [line] (transport) -- (decide);
    \path [line] (decide) -- node {no} (dterm);
    \path [line] (dterm) -- (MFP);
    \path [line] (MFP) -- (moments);
    \path [line] (moments) -- (transport);
    \path [line] (decide) -- node {yes}(stop);
\end{tikzpicture}
\caption{NFPA algorithm}
\label{Nalgorithm}
\end{figure}

\section{Numerical Experiments}\label{sec3}

In \cref{sec31} we describe the discretization methods used to implement the algorithms.
In \cref{sec32} we provide numerical results for 2 different choices of source $Q$ and boundary conditions.
For each choice we solve the problem using 3 different scattering kernels, applying 3 different choices of parameters for each kernel.
We provide NFPA numerical results for these 18 cases and compare them against those obtained from FPSA and other standard methods.

All numerical experiments were performed using MATLAB.
Runtime was tracked using the tic-toc functionality \cite{matlab17}, with
only the solver runtime being taken into consideration in the comparisons.
A 2017 MacBook Pro with a 2.8 GHz Quad-Core Intel Core i7 and 16 GB of RAM was used for all simulations.

\subsection{Discretization}\label{sec31}

The Transport and FP equations were discretized using linear discontinuous finite element discretization in space \cite{mpd1}, and discrete ordinates (S$_N$) in angle \cite{landm}.
The Fokker-Planck operator $\mathcal{F}$ was discretized using moment preserving discretization (MPD) \cite{mpd1}.
Details of the derivation of the linear discontinuous finite element discretization can be seen in \cite{japanDiss,martin}.
The finite element discretization for the  Fokker-Planck equation follows the same derivation.

A brief review for the angular discretization used for the FP equation is given below.
First, we use Gauss-Legendre quadrature to discretize the FP equation in angle:
\begin{equation}
\mu_n\frac{\partial \psi_n(x)}{\partial x} + \sigma_a \psi_n(x) - \frac{\sigma_{tr}}{2}\nabla^2_n \psi_n(x) =  Q_n(x),
\end{equation}
for $n=1,..,N$.
Here, $\nabla^2_n$ term is the discrete form of the angular Laplacian operator evaluated at angle $n$.

The MPD scheme is then shown as
\begin{equation}
\nabla^2_n \psi_n = M \psi_n = V^{-1} L V \psi_n,
\end{equation}
where $M$ is the MPD discretized operator defined by
\begin{subequations}
\begin{equation}
V_{i,j} = P_{i-1}(\mu_j)w_j,
\end{equation}
and
\begin{equation}
L_{i,j} = -i(i-1),
\end{equation}
\end{subequations}
for $i,j=1,...,N$.
Here, $P_l(\mu_j)$ are the Legendre polynomials evaluated at each angle $\mu_j$ and $w_j$ are the respective weights.
$M$ is defined as a (N x N) operator for a vector of $N$ angular fluxes $ \psi(x)$, at spatial location $x$.

In summary, if we write the FP equation as
\begin{equation}
\mathcal{H} \frac{\partial \psi}{\partial x}(x) + \sigma_a \psi(x) - \mathcal{F} \psi(x) =  Q(x),
\end{equation}
then $\mathcal{H}$ is Diag$(\mu_n)$ for $n=1,...,N$, $Q(x)$ is a vector of source terms $Q_n(x)$, and $\mathcal{F}$ is represented by $\frac{\sigma_{tr}}{2}M$.

\subsection{Numerical Results}\label{sec32}

It is shown that for slowly converging problems, typical convergence methods like $L_\infty$ suffer from false convergence \cite{adamslarsen}.
To work around this issue, the criterion is modified to use information about the current and previous iteration:
\begin{equation}
\label{falseconverge}
\frac{|| \phi^{m}_0(x) - \phi^{m-1}_0(x) ||_2}{1-\frac{|| \phi^{m+1}_0(x) - \phi^{m}_0(x) ||_2}{|| \phi^{m}_0(x) - \phi^{m-1}_0(x) ||_2}} < 10^{-8}.
\end{equation}

Two problems were tested using 200 spatial cells, $X$ = 400, $\sigma_a = 0$, $L$ = 15, and $N$ = 16.
Problem 1 has vacuum boundaries and a homogeneous isotropic source $Q$ for $0 < x < X$.
Problem 2 has no internal source and an incoming beam at the left boundary. The source and boundary conditions used are shown in \cref{parameters}.
\begin{table}[H]
\begin{center}
\scalebox{0.9}{
\begin{tabular}{c | c | c} \hline
& Problem 1 & Problem 2 \\ \hline \hline
Q(x) & 0.5 & 0 \\
$\psi_L$ & 0 & $\delta(\mu - \mu_N)$  \\
$\psi_R$ & 0 & 0 \\
\end{tabular}}
\end{center}
\caption{Problem Parameters}
\label{parameters}
\end{table}
We consider three scattering kernels in this paper: Screened Rutherford \cite{pomraning1}, Exponential \cite{pomraning2}, and Henyey-Greenstein \cite{HGK}.
Three cases for each kernel were tested.
The results obtained with NFPA are compared with those obtained using GMRES, DSA, and FPSA with the MPD scheme.

\subsubsection{SRK: Screened Rutherford Kernel}

The Screened Rutherford Kernel \cite{pomraning1, JapanFPSA} is a widely used scattering kernel for modeling scattering behavior of electrons \cite{SRK}.
The kernel depends on the parameter $\eta$, such that
\begin{equation}
\sigma^{SRK}_{s,l} = \sigma_s  \int_{-1}^{1} d\mu P_l(\mu) \frac{\eta (\eta+1)}{(1+2\eta-\mu)^2}.
\end{equation}
The SRK has a valid FP limit as $\eta$ approaches 0 \cite{patelFBR}. Three different values of $\eta$ were used to generate the scattering kernels shown in \cref{SRK}.
GMRES, DSA, FPSA, and NFPA all converged to the same solution for problems 1 and 2. \Cref{SRK_plots} shows the solutions for SRK with $\eta = 10^{-7}$.
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.1,angle=0]{SRK.jpg}
  \caption{Screened Rutherford Kernels}
  \label{SRK}
\end{center}
\end{figure}
\begin{figure}[H]
    \centering
    \subfloat[Problem 1]{{\includegraphics[width=7cm]{s7_iso.jpg} }}
    \qquad
    \subfloat[Problem 2]{{\includegraphics[width=7cm]{s7_beam.jpg} }}
    \caption{Results for SRK Problems with $\eta = 10^{-7}$}
    \label{SRK_plots}
\end{figure}

\begin{table}[H]
\begin{center}
\scalebox{0.8}{
\begin{tabular}{c || c || c || c} \hline
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$\eta = 10^{-5}$} & GMRES & 98.8 & 12 \\
& DSA & 2380 & 53585 \\
& FPSA & 1.21 & 26 \\
& NFPA & 1.39 & 26 \\ \hline
\multirow{4}{*}{$\eta = 10^{-6}$} & GMRES & 208 & 84 \\
& DSA & 3040 & 69156 \\
& FPSA & 0.747 & 16 \\
& NFPA & 0.857 & 16 \\ \hline
\multirow{4}{*}{$\eta = 10^{-7}$} & GMRES & 174 & 124 \\
& DSA & 3270 & 73940 \\
& FPSA & 0.475 & 10 \\
& NFPA & 0.542 & 10 \\ \hline
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 1 with SRK}
\label{SRKresults1}
\end{table}
\begin{table}[H]
\begin{center}
\scalebox{0.8}{
\begin{tabular}{c || c || c || c} \hline
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$\eta = 10^{-5}$} & GMRES & 52.4 & 187 \\
& DSA & 1107 & 25072 \\
& FPSA & 0.953 & 20 \\
& NFPA & 1.14 & 20 \\ \hline
\multirow{4}{*}{$\eta = 10^{-6}$} & GMRES & 108 & 71 \\
& DSA & 1434 & 32562  \\
& FPSA & 0.730 & 14 \\
& NFPA & 0.857 & 14 \\ \hline
\multirow{4}{*}{$\eta = 10^{-7}$} & GMRES & 94.1 & 185 \\
& DSA & 1470 & 33246 \\
& FPSA & 0.438 & 8 \\
& NFPA & 0.484 & 8 \\ \hline
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 2 with SRK}
\label{SRKresults2}
\end{table}

The results of all solvers are shown in \cref{SRKresults1,SRKresults2}.
We see that NFPA and FPSA tremendously outperform GMRES and DSA in runtime for all cases.
FPSA is a simpler method than NFPA, requiring less calculations per iteration; therefore, it is expected that it outperforms NFPA in runtime.
We see a reduction in runtime and iterations for FPSA and NFPA as the FP limit is approached, with DSA and GMRES requiring many more iterations by comparison as $\eta$ approaches 0.

An advantage that NFPA offers is that the angular moments of the flux in the LO equation will remain consistent with those of the transport equation even as a problem becomes less forward-peaked.
On the other hand, the moments found using only the FP equation and source iteration lose accuracy.
To illustrate this, Problem 1 was tested using different Screened Rutherford Kernels with increasing $\eta$ parameters.
The percent errors (relative to the transport solution) for the scalar flux obtained with the LO equation and with the standard FP equation at the center of the slab are shown in \cref{momcomp}.
It can be seen that the percent relative errors in the scalar flux of the FP solution is orders of magnitude larger than the error produced using the LO equation.
The same trend can be seen when using the exponential and Henyey-Greenstein kernels.

\begin{figure}[H]
\begin{center}
  \includegraphics[scale=0.15,angle=0]{relerrorlog.jpg}
  \caption{Log Scale of $\%$ Relative Error vs $\eta$ for Problem 1 at the Center of the Slab with SRK}
  \label{momcomp}
\end{center}
\end{figure}

\subsubsection{EK: Exponential Kernel}

The exponential kernel \cite{pomraning2, JapanFPSA} is a fictitious kernel made for problems that have a valid Fokker-Planck limit \cite{pomraning1}.
The zero$^{\text{th}}$ moment, $\sigma^{EK}_{s,0}$, is chosen arbitrarily; we define $\sigma^{EK}_{s,0}$ as the same zero$^{\text{th}}$ moment from the SRK.
The $\Delta$ parameter determines the kernel: the first and second moments are given by
\begin{subequations}
\begin{align}
\sigma^{EK}_{s,1} &= \sigma^{EK}_{s,0} (1-\Delta),\\
\sigma^{EK}_{s,2} &= \sigma^{EK}_{s,0} (1-3\Delta+3\Delta^2),
\end{align}
and the relationship for $l\geq 3$ is
\begin{equation}
\sigma^{EK}_{s,l} = \sigma^{EK}_{s,l-2} - \Delta(2l+1) \sigma^{EK}_{s,l-1}.
\end{equation}
\end{subequations}
As $\Delta$ is reduced, the scattering kernel becomes more forward-peaked.

The EK has a valid FP limit as $\Delta$ approaches 0 \cite{patelFBR}.
Three different values of $\Delta$ were used to generate the scattering kernels shown in \cref{EXP}.
The generated scattering kernels are shown in \cref{EXP}.
GMRES, DSA, FPSA, and NFPA all converged to the same solution for problems 1 and 2.
\Cref{EK_plots} shows the solutions for EK with $\Delta = 10^{-7}$.
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.1,angle=0]{EXP.jpg}
  \caption{Exponential Kernels}
  \label{EXP}
\end{center}
\end{figure}
\begin{figure}[H]
    \centering
    \subfloat[Problem 1]{{\includegraphics[width=7cm]{dta7_iso.jpg} }}
    \qquad
    \subfloat[Problem 2]{{\includegraphics[width=7cm]{dta7_beam.jpg} }}
    \caption{Results for EK Problems with $\Delta = 10^{-7}$}
    \label{EK_plots}
\end{figure}

The runtimes and iterations for GMRES, DSA, FPSA, and NFPA are shown in \cref{Expresults1,Expresults2}.
We see a similar trend with the EK as seen with SRK.
Smaller $\Delta$ values lead to a reduction in runtime and iterations for NFPA and FPSA, which greatly outperform DSA and GMRES in both categories.

\begin{table}[h]
\begin{center}
\scalebox{0.8}{
\begin{tabular}{c || c || c || c} \hline
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$\Delta = 10^{-5}$} & GMRES & 196 & 142 \\
& DSA & 3110 & 70140 \\
& FPSA & 0.514 & 11 \\
& NFPA & 0.630 & 11 \\\hline
\multirow{4}{*}{$\Delta = 10^{-6}$} & GMRES & 156 & 132 \\
& DSA & 3120 & 70758 \\
& FPSA & 0.388 & 7 \\
& NFPA & 0.393 & 7 \\ \hline
\multirow{4}{*}{$\Delta = 10^{-7}$} & GMRES & 81 & 127 \\
& DSA & 3120 & 70851  \\
& FPSA & 0.292 & 6 \\
& NFPA & 0.318 & 6 \\ \hline
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 1 with EK}
\label{Expresults1}
\end{table}
\begin{table}[h]
\begin{center}
\scalebox{0.8}{
\begin{tabular}{c || c || c || c} \hline
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$\Delta = 10^{-5}$} & GMRES & 110 & 73 \\
& DSA & 1455 & 33033 \\
& FPSA & 0.492 & 10 \\
& NFPA & 0.613 & 10 \\ \hline
\multirow{4}{*}{$\Delta = 10^{-6}$} & GMRES & 82.7 & 79 \\
& DSA & 1470 & 33309 \\
& FPSA & 0.358 & 7 \\
& NFPA & 0.431 & 7 \\ \hline
\multirow{4}{*}{$\Delta = 10^{-7}$} & GMRES & 56.8 & 90 \\
& DSA & 1470 & 33339 \\
& FPSA & 0.273 & 5 \\
& NFPA & 0.319 & 5 \\ \hline
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 2 with EK}
\label{Expresults2}
\end{table}

\subsubsection{HGK: Henyey-Greenstein Kernel}

The Henyey-Greenstein Kernel \cite{HGK,JapanFPSA} is most commonly used in light transport in clouds.
It relies on the anisotropy factor $g$, such that
\begin{equation}
\sigma^{HGK}_{s,l} = \sigma_s g^l.
\end{equation}
As $g$ goes from zero to unity, the scattering shifts from isotropic to highly anisotropic.
\begin{figure}[H]
\begin{center}
  \includegraphics[scale=0.1,angle=0]{HGK.jpg}
  \caption{Henyey-Greenstein Kernels}
  \label{HGK}
\end{center}
\end{figure}
\begin{figure}[H]
    \centering
    \subfloat[Problem 1]{{\includegraphics[width=7cm]{g099_iso.jpg} }}
    \qquad
    \subfloat[Problem 2]{{\includegraphics[width=7cm]{g099_beam.jpg} }}
    \caption{Results for HGK Problems with $g = 0.99$}
    \label{HGK_plots}
\end{figure}

The HGK does not have a valid FP limit \cite{patelFBR}.
The three kernels tested are shown in \cref{HGK}.
GMRES, DSA, FPSA, and NFPA all converged to the same solution for problems 1 and 2.
\Cref{HGK_plots} shows the solutions for HGK with $g = 0.99$.
The results of each solver are shown in \cref{HGKresults1,HGKresults2}.
\begin{table}[h]
\begin{center}
\scalebox{0.8}{
\begin{tabular}{c || c || c || c} \hline
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$g=0.9$} & GMRES & 9.88 & 76 \\
& DSA & 24.5 & 554 \\
& FPSA & 1.50 & 32 \\
& NFPA & 1.39 & 27 \\ \hline
\multirow{4}{*}{$g=0.95$} & GMRES & 12.2 & 131 \\
& DSA & 47.7 & 1083 \\
& FPSA & 1.75 & 38 \\
& NFPA & 1.83 & 35 \\ \hline
\multirow{4}{*}{$g=0.99$} & GMRES & 40.0 & 27 \\
& DSA & 243 & 5530  \\
& FPSA & 3.38 & 74 \\
& NFPA & 3.93 & 73 \\ \hline
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 1 with HGK}
\label{HGKresults1}
\end{table}
\begin{table}[h]
\begin{center}
\scalebox{0.8}{
\begin{tabular}{c || c || c || c} \hline
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$g=0.9$} & GMRES & 24.3 & 135 \\
& DSA & 14.8 & 336  \\
& FPSA & 1.15 & 23 \\
& NFPA & 1.35 & 24 \\ \hline
\multirow{4}{*}{$g=0.95$} & GMRES & 31.3 & 107 \\
& DSA & 29.7 & 675 \\
& FPSA & 1.56 & 32 \\
& NFPA & 1.90 & 33 \\ \hline
\multirow{4}{*}{$g=0.99$} & GMRES & 41.4 & 126 \\
& DSA & 146 & 3345 \\
& FPSA & 3.31 & 67 \\
& NFPA & 3.99 & 67 \\ \hline
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 2 with HGK}
\label{HGKresults2}
\end{table}

Here we see that NFPA and FPSA do not perform as well compared to their results for the SRK and EK.
Contrary to what happened in those cases, both solvers require more time and iterations as the problem becomes more anisotropic.
This is somewhat expected, due to HGK not having a valid Fokker-Planck limit.
However, both NFPA and FPSA continue to greatly outperform GMRES and DSA.
Moreover, NFPA outperforms FPSA in iteration count for problem 1.

\section{Discussion}\label{sec4}

This paper introduced the Nonlinear Fokker-Planck Acceleration technique for steady-state, monoenergetic transport in homogeneous slab geometry.
To our knowledge, this is the first nonlinear HOLO method that accelerates \textit{all $L$ moments} of the angular flux.
Upon convergence, the LO and HO models are consistent; in other words, the (lower-order) modified Fokker-Planck equation \textit{preserves the same angular moments} of the flux obtained with the (higher-order) transport equation.

NFPA was tested on a homogeneous medium with an isotropic internal source with vacuum boundaries, and in a homogeneous medium with no internal source and an incoming beam boundary.
For both problems, three different scattering kernels were used.
The runtime and iterations of NFPA and FPSA were shown to be similar.
They both vastly outperformed DSA and GMRES for all cases by orders of magnitude.
However, NFPA has the feature of preserving the angular moments of the flux in both the HO and LO equations, which offers the advantage of integrating the LO model into multiphysics models.

In the future, we intend to test NFPA capabilities for a variety of multiphysics problems and analyze its performance.
To apply NFPA to more realistic problems, it needs to be extended to include time and energy dependence.
Additionally, the method needs to be adapted to address geometries with higher-order spatial dimensions.
Finally, for the NFPA method to become mathematically ``complete", a full convergence examination using Fourier analysis must be performed.
However, this is beyond the scope of this paper and must be left for future work.

\section*{Acknowledgements}

The authors acknowledge support under award number NRC-HQ-84-15-G-0024 from the Nuclear Regulatory Commission.
The statements, findings, conclusions, and recommendations are those of the authors and do not necessarily reflect the view of the U.S. Nuclear Regulatory Commission.

J.~K. Patel would like to thank Dr.~James Warsa for his wonderful transport class at UNM, as well as his synthetic acceleration codes.
The authors would also like to thank Dr.~Anil Prinja for discussions involving Fokker-Planck acceleration.


\section{Introduction: Success of inflation}
Inflation solved the various cosmological problems like horizon,
homogeneity, flatness problem, just by assuming that the potential
energy of scalar fields dominates the expansion of the universe.
Moreover, inflation scenario naturally
explains the origin of the almost scale invariant density perturbation.
The observations in the past decades verified many of the predictions of
inflation. Almost scale invariant spectrum is now confirmed and now we
even know that the violation of the exact scale invariance is 4\% level~\cite{Komatsu:2010fb},
which is also consistent with the simplest slow roll inflation scenario.

\subsection{Density perturbation}
The amplitude of quantum fluctuation of inflaton $\phi$
during slow-roll inflation is determined by the unique relevant mass scale
at that time, e.g. the Hubble rate $H$:
\begin{equation}
 \delta\phi={H\over 2\pi}.
\end{equation}
However, on scales as large as the Horizon radius
the meaning of the amplitude of field
perturbation becomes subtle because it depends on the choice of gauge.
For the flat slicing gauge, in which the trace of the spatial curvature
is kept unperturbed, the perturbation equation for a scalar field becomes
very simple
and looks very similar to the one without gravitational
perturbation~\cite{text}.
Therefore the amplitude mentioned above can be in fact
understood as that in the flat slicing gauge.
This amplitude of perturbation can be interpreted as the dimensionless curvature
perturbation on a uniform energy density surface $\zeta$. Transformation law
is given by
\begin{equation}
 \zeta= H\delta t=H{\delta\phi\over\dot\phi}={H^2\over 2\pi\dot \phi},
\end{equation}
where $\delta t$ is the shift in time coordinate for this
transformation, which is given by $\delta\phi/\dot\phi$
using the time derivative of the background scalar field.
Writing the perturbation amplitude in terms of $\zeta$ is useful because
the curvature perturbation does not evolve on super-horizon scales
when the evolution path of the universe is unique.
This constancy of $\zeta$ simplifies the analysis of density
perturbation during inflation for a single inflaton model.
An extension of this argument to the case of multi-component inflaton is
the $\delta N$
formalism~\cite{Starobinsky:1986fx,Salopek:1990jq,Sasaki:1995aw,Sasaki:1998ug,Lyth:2004gb,Lyth:2005fi},
as we explain below.

\subsection{Further steps from observations}
Further detailed comparison between the theoretical predictions and
future observations is awaited. One important observation will be
tensor-type perturbation in cosmic microwave background. Tensor-type
perturbation is the transverse-traceless part of the spatial metric
perturbation, which is generated by the same mechanism
as the perturbation of the inflaton field. Gravitational action
has a overall prefactor $m_{pl}^2$, and hence we define the canonically
normalised metric perturbation $\psi_{\mu\nu}=m_{pl}\delta
g_{\mu\nu}$ so as to absorb this factor from the quadratic action.
Then, the quadratic action for the transverse traceless
part of $\psi_{\mu\nu}$ becomes identical to the one for a massless
scalar field. Therefore the amplitude of perturbation $\psi_{\mu\nu}$
generated through almost de Sitter inflation is also $O(H)$.
Using this relation, we have
\begin{equation}
 \left\vert{\delta T\over T}\right\vert_{tensor}=O(\delta g_{\mu\nu})
  =O\left({\psi_{\mu\nu}\over m_{pl}}\right)= O\left({H\over m_{pl}}\right).
\end{equation}
Thus we find that the CMB temperature fluctuation caused by
the tensor perturbation directly probes the value of $H$ during
inflation, i.e. the energy scale of inflation. This can be discriminated
from the scalar-type perturbation by looking at B-mode
polarisation~\cite{Mandolesi:2010yw,Baumann:2008aq,quiet}.

Another important observation is the non-Gaussianity of the
temperature fluctuations~\cite{Komatsu:2001rj,Bartolo:2004if}.
The non-Gaussianity is caused by
the non-linear dynamics of cosmological perturbation.
Once we have completely understood the evolution of density perturbation
at late time, the remaining non-Gaussianity which is not accounted
for should have their origin in the earlier universe.
The results of WMAP 7 years indicate that the local-type non-Gaussianity
parameter is given as $f_{NL}=32\pm 21$ at 68\% confidence level~\cite{Komatsu:2010fb}.
For the Planck satellite, it is expected that
the window of $f_{{\rm NL}}$ is expected to be reduced to
$O(10)$~\cite{Mandolesi:2010yw}.

Recently, non-Gaussianity of the primordial
perturbation also has been studied by many authors
~\cite{Lyth:2004gb,Lyth:2005fi,Komatsu:2001rj,
Bartolo:2004if,Maldacena:2002vr,Seery:2005wm,Seery:2006vu,Seery:2005gb,Malik:2003mv,Malik:2005cy,
Yokoyama:2007uu,Yokoyama:2007dw,Byrnes:2006vq,Byrnes:2007tm,Yokoyama:2008by,
Creminelli:2003iq,Alishahiha:2004eh,Koyama:2010xj,Alabidi:2006hg,Malik:2006pm,Sasaki:2006kq,Dvali:2003em,Bernardeau:2002jy,Bernardeau:2002jf,Lyth:2005qk,Salem:2005nd,Seery:2006js,Alabidi:2006wa,ByrnesTasinato,Byrnes:2008wi,Byrnes:2010em,Cogollo,Rodriguez,Enqvist:2004ey,Jokinen:2005by,
Chambers:2007se,Chambers:2008gu,Barnaby:2006cq,Mulryne:2009ci}.
The main reason for non-Gaussianity to attract much attention
is the expectation to future observations mentioned above.
These observations may bring us valuable
information about the dynamics of inflation.
Here in this short paper, we would like to clarify the difficulties in
generating large non-Gaussianity from the non-linear dynamics
during inflation.

\section{Basic feature of generation of non-Gaussianity of local-type.}
\subsection{Local non-Gaussianity}

Primordial non-Gaussianity gives rise as non-trivial higher order correlation
functions in primordial perturbations.
In principle, various types of
non-Gaussianity are possibly
generated~\cite{Creminelli:2003iq,Alishahiha:2004eh,Koyama:2010xj},
but here we focus on
the so-called local-type non-Gaussianity.
Local-type non-Gaussianity is
characterised by the existence of one-to-one local map between
the physical curvature perturbation
$\zeta(x)$ and the
variable which follows the Gaussian statistics $\zeta_G(x)$
at respective spatial points.
Namely, we can expand the curvature perturbation as
\begin{equation}
\zeta(x)=\zeta_G(x)+{3\over 5}f_{NL}\zeta_G^2(x)+\cdots,
\end{equation}
and the Gaussian variable $\zeta_G$ satisfies ordinary Gaussian statistics:
\begin{equation}
 \left\langle \zeta^G_{{\bf k}_1}\zeta^G_{{\bf k}_2}\right\rangle
 = \delta^{(3)}({\bf k}_1 +{\bf k}_2 )P_\zeta(k_1).
\end{equation}

In this case three point function can be characterised by a single
non-linear parameter $f_{NL}$~\cite{Komatsu:2001rj} as
\begin{equation}
   \left\langle \zeta_{{\bf k}_1}\zeta_{{\bf k}_2}\zeta_{{\bf
  k}_3}\right\rangle
  = \delta^{(3)}({\bf k}_1 +{\bf k}_2 +{\bf k}_3)B_\zeta (k_1,k_2,k_3),
\end{equation}
with
\begin{eqnarray}
 \!\!\!
 B_\zeta (k_1,k_2,k_3)={6\over 5}{f_{NL}\over(2\pi)^{3/2}}
 \Biggl[P_\zeta(k_1)P_\zeta(k_2)
+P_\zeta(k_2)P_\zeta(k_3)+P_\zeta(k_3)P_\zeta(k_1)
     \Biggr].
\end{eqnarray}
In general, other types of non-Gaussianity can be generated.
However, when we focus on super-horizon dynamics,
only local-type non-Gaussianity can be produced as is explained
below.
\begin{figure}
\begin{center}
\includegraphics[width=80mm]{fig1.jpg}
\caption{Schematic picture to explain $\delta N$ formalism.}
\end{center}
\end{figure}

\subsection{Super-horizon dynamics}

Here we just present an intuitive explanation about how the density
perturbations evolve when the length scale is much larger than the
Hubble scale. Super-horizon dynamics is locally described by the
Friedmann-Robertson-Walker universe. We consider the evolution of
the universe starting with an initial
flat hypersurface at $t=t_*$, on which the initial values of the
inflaton field have a certain distribution. We evolve the
spacetime until reaching the final surface at $t=t_F$
that is characterised by a specified energy density as shown
in Fig.~1. Hence, the
final surface is a uniform energy density surface by definition.
As each horizon patch is causally disconnected from the others
in the inflating universe, its evolution is determined
as a local process.
If initial conditions are completely given in each horizon patch,
we can solve the evolution of its future.
Here, for simplicity, we assume that the
evolution of the averaged values of fields in each horizon patch
is determined by the averaged values of initial data. This assumption
is not necessarily true in general, but we have a good reason to
assume so in most cases. Initial conditions for the smaller scale
perturbations can affect the evolution of the averaged values of variables.
However, as there are so many small scale degrees of freedom,
the average of their effects will not be largely fluctuated
except for rare situations.
Of course, even if we can neglect the effect of fluctuations on small scales,
this does not mean that the backreaction due to small scale
degrees of freedom does not modify
the dynamics of the averaged values of variables. This backreaction
effect arises at the second order perturbation or higher.
Therefore the difference
in the backreaction effect between different horizon patches
is even higher order in perturbation.
Thus, we usually neglect this effect for
simplicity. Of course, in the situation like preheating the
sub-horizon scale perturbations evolve more rapidly than the
averaged values, we definitely
need to take the backreaction into account. However, even in
that case, it will not be necessary to know all the details of
small scale perturbations to understand the evolution of
the averaged variables.

Anyway, we neglect these effects originating from small scale perturbations
here. Then, the e-folding number between the initial surface and the
final one is completely determined by the initial averaged values
of the variables. In the scalar field dominant universe, such
initial conditions are specified by the values of the field components
and their time derivatives, $\phi^i(t_*,x)$ and $\dot \phi^i(t_*,x)$.
We denote $\phi^i(t_*,x)$ together with $\dot \phi^i(t_*,x)$ by
$\phi^I(t_*,x)$.
Then, the e-folding number between the initial surface
and the final one,
\begin{equation}
N(t_F,t_*,\phi^I)\equiv \int_{t_*}^{t_F}H dt
\end{equation}
is given as a function of $\phi^I(t_*,x)$.
Then, roughly speaking, the spatial metric on the final uniform energy
density surface will be given by
\begin{equation}
ds_{(3)}^2\approx e^{2N(t_F,\phi^I(t_*,x))}
\delta_{ij}dx^i dx^j,
\end{equation}
The curvature perturbation is in fact the perturbation in the above
exponent $N(t_F,t_*,\phi^I(t_*,x))$, and hence we have
\begin{equation}
\zeta(t_F)\approx \delta N(t_F,\phi^I(t_*,x)).
\label{zetadeltaN}
\end{equation}
This formula is the heart of the $\delta N$
formalism~\cite{Starobinsky:1986fx,Salopek:1990jq,Sasaki:1995aw,Sasaki:1998ug,Lyth:2004gb,Lyth:2005fi}.

In the slow roll case, the evolution equation for the scalar field
can be approximated by a first order differential equation in $t$. Therefore
the phase space of the initial conditions $\phi^I(t_*,x)$
is reduced to $\phi^i(t_*,x)$.
In this case, if there is only a single component, the trajectory
in the field configuration space, which is one dimensional, is
necessarily unique.
Then, $\delta N(t_F,\phi^i(t_*,x))$
does not depend on the choice of the
energy density of the final surface. It can be set to a representative
background value at the initial time. Namely,
\begin{equation}
 \delta N(t_F,\phi^i(t_*,x))=\delta N(t_*,\phi^I(t_*,x)).
\end{equation}
Therefore nothing non-trivial happens during the super-horizon evolution
for slow-roll single field inflation.
Even if the slow-roll conditions are violated, single field inflation
does not produce sizable $\delta N$ for modes far beyond the horizon
scale.
This is because the trajectories in the configuration space continues to
converge as the decaying perturbation mode gets irrelevant.
In order to generate non-Gaussianity through the
evolution on super horizon scales,
it is therefore essential to consider multi-component inflaton field.

Using the formula (\ref{zetadeltaN}), we can express the
curvature perturbations on the final surface in terms of
the field perturbations on the initial surface as
\begin{equation}
\zeta(t_c)= \delta N=
 N^*_I\delta\phi^I_*+{1\over 2}N^*_{IJ}\delta\phi^I_*
 \delta\phi^J_*+\cdots,
\label{zetaexp}
\end{equation}
where the subscript $*$ indicates the quantities evaluated at
$\phi^I=\bar\phi^I(t_*)$, and the subscript $I$ associated with $N$
means the differentiation with respect to $\phi^I(t_*)$:
\begin{equation}
 N_I^*\equiv \left. {\partial N(t_c,\phi^I)\over
	      \partial\phi^I}\right\vert_{\phi=\bar \phi(t_*)}, \qquad
 N_{IJ}^*\equiv \left.
   {\partial^2 N(t_c,\phi^I)\over \partial\phi^I\partial\phi^J}\right\vert_{\phi=\bar \phi(t_*)},
\end{equation}
where $\bar\phi^I(t)$ is the background trajectory.
With the aid of Eq.~(\ref{zetaexp}),
we can write the three point correlation function of $\zeta$ at $t=t_F$
as
\begin{eqnarray}
\left\langle\zeta({\bf x}_1)\zeta({\bf x}_2)\zeta({\bf
 x}_3)\right\rangle
 & = & N^*_I N^*_J N^*_K
    \left\langle\delta\phi_*^I({\bf x}_1)\delta\phi_*^J({\bf x}_2)\delta\phi_*^K({\bf
      x}_3)\right\rangle\cr
 && +N^*_I N^*_J N^*_{K L} \Bigl[
    \left\langle\delta\phi_*^I({\bf x}_1)\delta\phi_*^J({\bf
     x}_2)\delta\phi_*^K({\bf x}_3)\delta\phi_*^L({\bf
     x}_3)\right\rangle
\cr &&\qquad\qquad\qquad\qquad
+(3 {\rm perms})\Bigr]+\cdots.
\end{eqnarray}
The first term contains only three $\delta\phi$s. If the initial
fluctuations at $t=t_*$ are Gaussian, this term vanishes. Depending
on the model of inflation, early generation of non-Gaussianity at
around horizon crossing is possible~\cite{Alishahiha:2004eh}.
However, the observed almost scale
invariant and slightly red spectrum of initial curvature perturbation
strongly suggests that the inflation is likely to have been
in the slow roll regime at
around the horizon crossing. For the slow roll inflation, it is shown that the early
production of non-Gaussianity is strongly
suppressed~\cite{Maldacena:2002vr,Seery:2005wm,Seery:2006vu}.
Therefore, it is well motivated to consider non-Gaussianity contained in
the second term (or even higher order terms), generated by the
non-linear evolution after the horizon crossing.

Generation of non-Gaussianity can be classified into two classes.
One is generation through the super horizon evolution during inflation.
The other is generation at the end of or after
inflation~\cite{Malik:2006pm,Sasaki:2006kq,Dvali:2003em,Lyth:2005qk,Salem:2005nd,Seery:2006js,Alabidi:2006wa,
Enqvist:2004ey,Jokinen:2005by}.
In either case the non-Gaussianity is of local type.
Three point correlation function is characterised by one parameter
$f_{NL}$ alone, and it is related to the e-folding number $N(t_F,\phi^I)$
as
\begin{equation}
{6\over 5}f_{NL}\approx {N^I_* N^J_* N^*_{IJ}\over (N^K_* N^*_{K})^2},
\label{fNLformula}
\end{equation}
neglecting higher order terms.

\section{Non-Gaussianity produced at the end of or after inflation}

There are various models of inflation that produces large
non-Gaussianity of local type at the end of or after inflation.
Here we briefly mention curvaton type
model~\cite{Moroi:2001ct,Lyth:2001nq} as a typical example to explain the mechanism
to generate large non-Gaussianity.
Curvaton is another scalar field that is introduced to explain the origin
of primordial curvature perturbation. During inflation, curvaton field
is almost massless but it starts to behave as a massive field later when
the Hubble rate decays down to the mass scale of the curvaton
field. Then, the curvaton starts to oscillate around the minimum of the
potential and eventually decays into radiation.
At this point, the fluctuations in the curvaton $\sigma$ are transferred into
curvature perturbations. For simplicity, we consider the case
that the density perturbations are dominated by the fluctuations
originating from the curvaton.
Neglecting a contribution of second order of
$\delta\rho_\sigma$ to $\zeta$, which turns out to yield only
$f_{NL}=O(1)$,
we have~\cite{Sasaki:2006kq}
\begin{equation}
 \zeta = {1\over 4}{\delta\rho_\sigma\over \rho}=
   {r\over 4}\left(2{\delta\sigma\over\sigma}+\left({\delta\sigma\over\sigma}\right)^2\right),
\end{equation}
where $r=\rho_\sigma/\rho_{tot}$ is the fraction of the energy density
that the decay products of $\sigma$ takes. Here, we used the estimate
$\rho_\sigma\approx m^2(\sigma+\delta\sigma)^2/2$ with $m$ being the
mass of the curvaton.

The amplitude of the power spectrum is observationally fixed as
\begin{equation}
P_\zeta=O\left(\left[r{\delta\sigma/\sigma}\right]^2\right)=O( 10^{-10}).
\end{equation}
Therefore, if $\delta\sigma/\sigma$ is not small, $r$ can be as small as
$10^{-5}$.
On the other hand, the non-linear parameter is given by
\begin{equation}
 f_{NL}=O\left({\langle \zeta^3\rangle/\langle\zeta^2\rangle^2}\right)
   =O\left({1/ r}\right),
\end{equation}
which can be as large as $10^5$. In this way, the production of
large non-Gaussianity at the end of inflation is rather easily
achieved.

{ As a mechanism for getting large non-Gaussianity,
one can also consider generation of perturbations during preheating
phase~\cite{Tanaka:2003cka,Suyama:2006rk,Enqvist:2004ey,Jokinen:2005by,Chambers:2007se,Chambers:2008gu,Barnaby:2006cq}
In this case, the background evolution is strongly coupled to the
evolution of small scale fluctuations
because the e-folding number changes depending on how fast
the energy of coherent oscillation of fields decays into the small scale
perturbations.
While in the other models
the transmutation
of homogeneous isocurvature perturbations on each Hubble patch
into adiabatic perturbations is determined just by solving the background
evolution of various homogeneous and isotropic universe models.
In this sense
the mechanism to generate density perturbation during preheating
stage is quite different from the other models.
}
\section{Non-Gaussianity produced during inflation}

Here we just present the formula for $f_{NL}$ in slow-roll inflation
with canonical kinetic term.
A systematic derivation for more general formula, including higher
order correlators, is given in Ref.~\cite{Yokoyama:2008by}.
Assuming that the initial perturbations of fields
are Gaussian in the following form
\begin{equation}
 \left\langle \delta\phi^i_{{\bf k}_1}\delta\phi^j_{{\bf k}_2}\right\rangle
 = \delta^{ij} \delta^{(3)}({\bf k}_1 +{\bf k}_2 )P_{\delta\phi}(k_1),
\end{equation}
the formula for $f_{NL}$ is written in terms of the potential of the
scalar field $V$ as
\begin{equation}
{6\over 5}f_{NL}=(N^i_* N_i^*)^{-2}
  \int_{N_*}^{N_F} dN'\, N_j(N')Q^j_{kl}(N')\Theta^k(N')\Theta^l(N').
\label{formula}
\end{equation}
Here, we neglected non-Gaussianity in the relation between $\delta N$ and $\delta\phi^i$
at $N=N_F$.
To define $N_i$ and $\Theta^i$,
it is convenient to introduce the propagator
\begin{equation}
\Lambda^i_j(N,N')=\left[T\exp\left(\int_{N'}^N
				P(N'')dN''\right)\right]{}^i_j,
\label{Tproduct}
\end{equation}
where
\begin{equation}
 P^i_j(N)\equiv\left[-{V^i_j\over V}+{V^i V_j\over V^2}
   \right]_{\phi=\bar\phi(N)},
\end{equation}
is the potential term that appears in the linear perturbation equation
when we use the e-folding number $N$ as a time coordinate. The indices
associated with $V$ represent differentiation with respect to scalar field.
$T$ in Eq.~(\ref{Tproduct}) means that the matrices $P^i_j$
are ordered in time when the exponential is expanded in power of $P^i_j$.
Using this propagator, $N_i$ and $\Theta^i$ are defined as
\begin{eqnarray}
 && N_i(N)=N_j(N_F)
\Lambda^j_i(N_F,N),\cr
&&\Theta^i(N)\equiv\Lambda^i_j(N,N_*)N^j_*.
\end{eqnarray}
$N_i(N)$ satisfies the equation of motion adjoint to the linear perturbation
equation for $\delta\phi^i$,
\begin{equation}
{d\over dN}N_i(N)=-P_i^j N_j(N),
\end{equation}
with the boundary condition $N_j(N_F)=(\partial
N(N_F,\phi^i)/\partial\phi^j)|_{\phi=\bar\phi(N_F)}$.
$\Theta^i(N)$ satisfies the same equation as $\delta\phi^i$
\begin{equation}
{d\over dN}\Theta^j(N)=P_i^j \Theta^i(N),
\end{equation}
and the boundary condition is given by
$\Theta^i(N_*)=N^i(N_*)$, where the index is raised by the inverse of
the field space
metric, which is assumed to be $\delta^{ij}$ here.
The three point interaction $Q^i_{jk}$ is given by
\begin{equation}
Q^i_{jk}(N)\equiv\left[-{V^i_{jk}\over V}+{V^i_{j}V_k\over V^2}+
            {V^i_k V_j\over V^2}+{V^i V_{jk}\over V^2}-2{V^i V_j V_k\over V^3}
                    \right]_{\phi=\bar\phi(N)}.
\end{equation}
In a naive sense, slow-roll conditions require that the
potential of the inflaton is a smooth function of $\phi^i$.
Hence higher order differentiations with respect to $\phi^i$
are suppressed. More precisely, introducing a slow-roll parameter
\begin{equation}
 \epsilon\equiv {1\over 2}{V_i V^i\over V^2},
\end{equation}
where $m_{pl}$ is set to unity, $P^i_j$ and $Q^i_{jk}$ are supposed
to be of $O(\epsilon)$ and $O(\epsilon^{3/2})$, respectively.
To obtain an almost scale-invariant spectrum, assuming this kind
of scaling is natural although it is not strictly required~\cite{Stewart:2001cd}.
In general, there might be a very massive isocurvature degree of
freedom, which does not satisfy the above simple scaling.
However, in this simple example,
fluctuations in such a massive degree of freedom
decay rapidly, and hence they can be safely neglected.

Duration of the inflation is roughly estimated as
\begin{equation}
 N=O\left({V\over dV/dN}\right)=O\left( {H V\over V'\dot\phi}\right)=O\left(\epsilon^{-1}\right).
\end{equation}
Hence, the exponent in the propagator $\Lambda^i_j$ is estimated to be $O(1)$.
As it is the exponent, it is crucial whether the amplitude of this
factor is slightly bigger
or smaller than unity. In principle, therefore $\Lambda^i_j$ itself can
be much larger or much smaller than unity.
The magnitudes of $N_i$ and $\Theta^j$ depend on the behaviour of the
propagator, but the value of $N_i$ at $N=N_F$ is given by definition
so as to satisfy
\begin{equation}
 V(\phi(N_F+\delta N))=V(\bar\phi(N_F)).
\end{equation}
Expanding this equation up to first order, we obtain
\begin{equation}
N_i(N_F)=O\left({V V_i/ V^j V_j}\right)
=O(\epsilon^{-1/2}).
\end{equation}
If $\Lambda^i_j$ stays of $O(1)$, one can say that $N_i$ and $\Theta^j$
also stay $O(\epsilon^{-1/2})$. Substituting these estimates into
Eq.~(\ref{formula}), we find that
$
 f_{NL}=O(\epsilon),
$
as the general argument as given in Refs.~\cite{Maldacena:2002vr,Seery:2005wm,Seery:2006vu} tells.

However, if amplitude of $\Lambda^i_j$ largely deviates from unity,
$N_i$ and hence $\Theta^i$ are also largely enhanced or suppressed,
resulting in large enhancement or suppression of the integrand in
Eq.~(\ref{formula}). Then, the order of magnitude of $f_{NL}$ is
not necessarily suppressed like $O(\epsilon)$. Furthermore, slow roll conditions are
violated, we have more chance to generate large non-Gaussianity.
However, it is not so easy to construct a viable model which
generates large non-Gaussianity from the super-horizon dynamics
during inflation as we shall see below.

\section{Is there any model which produces large non-Gaussianity
during inflation?}

\begin{figure}
\begin{center}
\includegraphics[width=60mm]{fig2.jpg}
\label{Fig:trajectory}
\caption{A background trajectory in double inflation
 model. Taken from Ref.~\cite{Yokoyama:2007dw}.}
\end{center}
\end{figure}

\subsection{Temporal slow-roll violation is not sufficient}
\label{doubleinf}
First we consider double inflation model~\cite{Silk:1986vc}, which is a simple two-field
model. The potential is given by
\begin{equation}
 V(\phi,\chi)={1\over 2}m^2_\phi \phi^2+{1\over 2}m^2_\chi\chi^2.
\end{equation}
There is no direct coupling between two fields, $\phi$ and $\chi$,
except for gravitational one.

Here we follow the analysis of Ref.~\cite{Yokoyama:2007dw}.
We show the background trajectory for $m_\chi/m_\phi=20$
in Fig.~2 taken from the same reference.
Since the mass of $\chi$-field is much
larger, the field moves rapidly in $\chi$-direction first.
Arriving at around the minimum in $\chi$-direction, the slow roll
conditions are violated and the field oscillates several times.
This violation of slow-roll conditions at the intermediate step
is possible when the mass ratio is large enough.
Otherwise, the trajectory becomes just a smooth curve and
slow roll conditions are not violated in the middle.
One may expect that this temporal violation of slow roll produces
large non-Gaussianity.
However, this is not the case.

During the period when $\chi$-field is moving rapidly, $\phi$-field
stays almost constant because the mass of $\phi$-field is small
compared with the Hubble rate which is dominantly determined by the term
$m^2_\chi\chi^2$ in the potential. Hence, the total e-folding
number is simply given by the sum of e-foldings for two stages of
inflation:
\begin{equation}
 N=N^{(\phi)}+N^{(\chi)}\approx \phi_*^2+\chi_*^2.
\end{equation}
Applying the formula (\ref{fNLformula}), we find
\begin{equation}
{6\over 5}f_{NL}={N_i N_j N^{ij}\over (N_k N^k)^2}={2\over
 \phi_*^2+\chi_*^2}=O\left({1\over N}\right).
\end{equation}
Hence, as in the usual slow-roll case, the non-Gaussianity is
suppressed by the slow-roll parameter $\epsilon=O(1/N)$.

\subsection{Simple hybrid inflation does not give large non-Gaussianity}
\label{hybrid}
As a second example, let's consider hybrid inflation model~\cite{Linde:1993cn}
whose potential is given by
\begin{equation}
 V(\phi,\chi)={\lambda\over 4}(\chi^2-v^2)^2
   +{m^2\over 2}\phi^2+{g^2\over 2}\phi^2\chi^2.
\end{equation}
Usually $\phi$ is the inflaton field and $\chi$ stays at $\chi=0$
during inflation. The effective mass of the $\chi$-field is
a function of $\phi$ given by
\begin{equation}
 m^2_\chi=-\lambda v^2+g^2\phi^2,
\end{equation}
assuming $\chi=0$. The mass of $\chi$-field changes its sign
at some point, which we denote by $N=N_c$. Then, tachyonic
instability in $\chi$-direction occurs, leading to the end
of inflation.

To analyse the dynamics of $\chi$-field at this critical point
analytically, we approximate the mass of $\chi$-field as
\begin{equation}
 m^2_\chi=-\mu^2(N-N_c),
\end{equation}
where $\mu^2$ is a parameter that depends on the choice of the model
potential.

If $\mu^2\gtrsim H^2$, $\chi$-field is massive during $\phi$-field
inflation. Thus, the fluctuation of $\chi$-field decays on large scales.
If $\mu^2\lesssim H^2$, $\chi$
field can stay nearly massless during $\phi$-field
inflation. In this case, the fluctuation of $\chi$-field may play an
important role. As the mass scale of $\chi$ is small, we can apply the
slow roll approximation to $\chi$-field, too. Then, we have
\begin{equation}
3H^2\partial_N \chi=\mu^2(N-N_c)\chi.
\end{equation}
This equation can be solved to obtain
\begin{equation}
\ln {\chi\over\chi_*}={\mu^2\over 6H^2}\left[
   (N-N_c)^2-(N_*-N_c)^2 \right].
\end{equation}
Applying the formula $(6/5)f_{NL}\approx N_{\chi\chi}/N_\chi^2$, which is
valid when the fluctuation of $\chi$-field dominates, we have
\begin{equation}
{6\over 5}f_{NL}\approx {\mu^2\over 3H^2}(N_F-N_c).
\label{estimatehybrid}
\end{equation}
Since we are restricted to the case with $\mu^2\lesssim H^2$ and
$N_F-N_c$ cannot be very large, $f_{NL}$ cannot be larger than
20 or so. If the fluctuations of $\phi$-field also contribute,
which is usually required in order to produce almost scale invariant
spectrum, the value of $f_{NL}$ becomes even smaller.

Furthermore, a more stringent constraint can be set by considering the
following condition. If $\bar\chi(N_c)\gtrsim H$
is not satisfied,
fluctuations at the horizon scale modes at that time determine
on which side the field rolls down.
Then, the tachyonic instability does not keep the coherence
of $\chi$-field on the super horizon scales,
leading to the so-called spinodal decomposition.
As a result, long wavelength fluctuation is, roughly speaking,
completely erased. Hence, there is no chance to generate
large non-Gaussianity originating from the fluctuation of
$\chi$-field. The condition $\bar\chi(N_c)\gtrsim H$ becomes
\begin{equation}
 {\mu^2\over 6H^2}(N_F-N_c)^2\approx \ln {\chi_F\over \chi_c}
   \lesssim \ln {v \over H},
\end{equation}
where we used the estimate $\chi_F\approx v$.
Using this inequality, the above estimate for $f_{NL}$
(\ref{estimatehybrid}) is bounded by
\begin{equation}
{6\over 5}f_{NL}\lesssim {\mu\over H}\sqrt{{2\over 3}\ln{m_{pl}\over H}},
\end{equation}
where we used $H^2=O(\lambda v^4/m^2_{pl})$ and $|m_\chi^2|_{\phi=0}
=O( \lambda v^2)> O(H)$. The latter condition is required to avoid
topological inflation happens at $\phi=\chi=0$.
The factor $\sqrt{{2\over 3}\ln{m_pl\over H}}$ is $\approx 7$ even if we reduce
the energy scale of inflation $\sqrt{m_{pl} H}$ to TeV scale. Here,
as $\mu^2\ll H^2$ is also required, we conclude that $f_{NL}$ cannot
be much larger than unity.

\subsection{Modular inflation}
\label{KS}
\begin{figure}
\begin{center}
\includegraphics[width=70mm]{fig3.jpg}
\caption{Contour plot of the potential of Kadota-Stewart model
and a typical trajectory.}
\end{center}
\end{figure}

In the preceding subsection we have observed that large mass scale
for the isocurvature modes
is necessary to generate large non-Gaussianity. However, the mass
squared should not be large positive at the early stage of inflation
when the interesting scales exit the horizon. This requirement
is not easily compromised unless we invent a very artificial form of
potential just to produce large non-Gaussianity.

Here we given one rather natural model which satisfies our requirement.
This is the model that was proposed by Kadota and
Stewart~\cite{Kadota:2003fs}.
The basic model potential takes the form
\begin{equation}
 V=V_0-m_\phi^2|\Phi|^2
     +{1\over 3}m_\Phi^2(\Phi^3+h.c.)+m_\Phi^2|\Phi|^4,
\end{equation}
where again we set $m_{pl}=1$. $\Phi=(\phi+i\chi)/\sqrt{2}$ is
a complex scalar field. This potential is modified near $\Phi=0$, the point
with enhanced symmetry, due to loop correction of the Coleman-Weinberg
type:
$ V_{loop}=-\beta m_\phi^2|\Phi|^2\ln\left|{\Phi/\Phi_*}\right|$.
This correction produces ring-shaped maximum around $\Phi=0$, where
topological inflation occurs. From the edges of the region of
topological inflation, the field slowly rolls down the hill.

Contour plot of the potential is given in Fig.~3.
We focus on the trajectories which pass near the saddle point
as shown in the same figure with a red curve.
In this model mass squared in the $\chi$-direction is negative
from the beginning. However, since it is the direction of approximate
$U(1)$ symmetry, the trajectories in any direction are almost equally
preferred at this stage. As $\phi$ increases, the fluctuation in
$\chi$-direction
is enhanced by the ratio of $\phi$ compared with
its initial value. After the direction of trajectory changes,
this fluctuation in $\chi$-direction is transferred to the
curvature perturbation.
In this way the fluctuations in $\chi$-direction
can rather easily dominate the final density perturbations.
{ Then, the amplitude of fluctuations in $\chi$-direction stays
almost constant near the ring-shaped top of potential, and
therefore scale invariant spectrum is realised. }

Near the saddle point, the equation in $\chi$-direction can
be approximated by
\begin{equation}
3H^2\partial_N\chi=-m_\chi^2 \chi,
\end{equation}
with $m_\chi^2$ being the mass squared of $\chi$-field evaluated
at the saddle point. Using the approximation $H^2=$constant,
this equation can be solved to give
\begin{equation}
 N={3H^2\over -m_\chi^2}\ln{\chi\over\chi_*},
\end{equation}
where $\chi_*$ is the initial value of $\chi$.
Applying the formula $(6/5)f_{NL}\approx N_{\chi\chi}/N_\chi^2$,
we obtain
\begin{equation}
{6\over 5}f_{NL}\approx {m_\chi^2 \over 3H^2}.
\end{equation}
This can be large if $m_\chi^2 \gg H^2$.
However, to achieve this by natural form of potential, the
potential minima as well as the saddles points should be
located close to the origin in Planck unit. Then, inflation
does not occur except for the top of the potential
or near the saddle points. To sustain a sufficiently large
e-folding number, inflation in these specific regions should last long.
However, this is not allowed. If inflation lasts long near the
top of the potential, which is ring-shaped, the motion in the
radial direction becomes very slow. As a result, the
amplitude of curvature perturbation originating from the
fluctuation in $\phi$ direction dominates. For inflation to
last long near the saddle point, the trajectory should pass
very close to the saddle point. However, in such cases
significant fraction of the universe will fall into the
other side of the saddle point. As a result
domain walls are formed, which leads to
the problematic domain wall dominated universe.

Even if we can avoid the domain wall formation (by considering
models with higher symmetry), it is not easy to construct
models which explain the naturalness of such a special trajectory.
The probability distribution of the universes with different values of
$\chi_*$ will be affected by the volume expansion factor during
inflation, which is proportional to $e^{3N}$. If ${3H^2/ |m_\chi^2|}$
is small as is requested for large non-Gaussianity, $\chi_*$ must be
extremely small to gain a large e-folding there.
Then, the probability of having such a small value of $\chi_*$ is
extremely small. As long as we consider models which realize
sufficiently large e-foldings for the natural choice of
the fiducial value of $\chi_*$,
$f_{NL}$ will not largely exceed unity.
Further variation of this model is possible by introducing loop correction
at around the saddle points, but the final conclusion does not change
much. We will report detailed analysis about this model
in our future publication.

\section{Conclusion}

Multi-field inflation generates entropy perturbation as well as the
adiabatic one. This entropy perturbation in general affects the
evolution of the curvature perturbation even after the horizon crossing.
Possible mechanisms of production of non-Gaussianity in the super-horizon
regime can be classified into two cases. One is the production of
non-Gaussianity during inflation and the other is that at the end of
or after inflation. Although we could not give a general proof,
it seems very difficult to produce large non-Gaussianity
during inflation as far as we consider potential without fine-tuning,
although there are several claims that suggest it
possible
\footnote{
In the examples given in Refs.~\cite{Byrnes:2010em,Cogollo},
it is not clear if we
should say that the non-Gaussianity is generated during super-horizon
evolution during inflation
because distribution in field space stays Gaussian.
In fact, the origin of non-Gaussianity is concentrated in the term
neglected in Eq.~(\ref{formula}).}.

{
Here are several subtle issues.
In this paper we claimed it difficult to construct a
natural model in which
large non-Gaussianity originating from the non-linear evolution
is produced before inflation terminates.
By contrast, generation of large non-Gaussianity
at the end of or after inflation is rather easy.
However, it may not be so trivial in general to identify when
non-Guassianity is generated. We gave an example of double inflation in
\S~\ref{doubleinf}. Although we did not mention it in the text,
in this model the non-linear parameter $f_{NL}$ as a function of
$t_F$ becomes very large temporally when the background trajectory
changes its direction, but this large $f_{NL}$ does not persist
long. This means that the time dependence of $f_{NL}$ is not
always appropriate to read when non-Gaussianity is mainly generated.

Another important point to note is the limitation of
neglecting small scale perturbations.
The small scale perturbations often become important for the field
components that become massive during inflation.
A typical example is the hybrid
inflation model discussed in \S~\ref{hybrid}.
To take into account the effect of small scale perturbations,
$\delta N$ formalism
based on the background evolution for spatially homogeneous spacetime
is not sufficient. We need to extend $\delta N$ formalism incorporating
collective variables that characterise statistical state such as
the averaged magnitude of small scale fluctuation.

Finally we should mention the naturalness of
the initial conditions. It might be possible to obtain large
non-Gaussianity by tuning the initial conditions, but it will not be
realised in reality if they are extremely fine-tuned.
As an example, we discussed Kadota-Stewart model in \S~\ref{KS}.
However, once we start to discuss the likeliness of the chosen
initial conditions, the long-standing measure problem arises.
At the moment we are very far from a conclusive answer
on this issue.
}

\ack
This work is supported by the JSPS through Grants Nos. 19540285, 21244033.
We also acknowledge the support of the Grant-in-Aid for the Global COE Program ``The Next
Generation of Physics, Spun from Universality and Emergence'' and
the Grant-in-Aid for Scientific Research on Innovative Area N.21111006
from the
Ministry of Education, Culture, Sports, Science and Technology (MEXT) of
Japan.

\section*{References}