prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the proposed approach in this research paper?

Read the following text and answer briefly.

Context 1: \section{Introduction}
\label{sec:introduction}

Probabilistic models have proven to be very useful in a lot of applications in signal processing where signal estimation is needed \cite{rabiner1989tutorial,arulampalam2002tutorial,ji2008bayesian}  Some of their advantages are that 1) they force the designer to specify all the assumptions of the model, 2) they provide a clear separation between the model and the algorithm used to solve it, and 3) they usually provide some measure of uncertainty about the estimation 

On the other hand, adaptive filtering is a standard approach in estimation problems when the input is received as a stream of data that is potentially non-stationary  This approach is widely understood and applied to several problems such as echo cancellation \cite{gilloire1992adaptive}, noise cancellation \cite{nelson1991active}, and channel equalization \cite{falconer2002frequency} 

Although these two approaches share some underlying relations, there are very few connections in the literature  The first important attempt in the signal processing community to relate these two fields was the connection between a linear Gaussian state-space model (i e  Kalman filter) and the RLS filter, by Sayed and Kailath \cite{sayed1994state} and then by Haykin \emph{et al } \cite{haykin1997adaptive}  The RLS adaptive filtering algorithm emerges naturally when one defines a particular state-space model (SSM) and then performs exact inference in that model  This approach was later exploited in \cite{van2012kernel} to design a kernel RLS algorithm based on Gaussian processes 

A first attempt to approximate the LMS filter from a probabilistic perspective was presented in \cite{park2014probabilistic}, focusing on a kernel-based implementation  The algorithm of \cite{park2014probabilistic} makes use of a Maximum a Posteriori (MAP) estimate as an approximation for the predictive step  However, this approximation does not preserve the estimate of the uncertainty in each step, therefore degrading the performance of the algorithm 

In this work, we provide a similar connection between state-space models and least-mean-squares (LMS)  Our approach is based on approximating the posterior distribution with an isotropic Gaussian distribution  We show how the computation of this approximated posterior leads to a linear-complexity algorithm, comparable to the standard LMS  Similar approaches have already been developed for a variety of problems such as channel equalization using recurrent RBF neural networks \cite{cid1994recurrent}, or Bayesian forecasting \cite{harrison1999bayesian}  Here, we show the usefulness of this probabilistic approach for adaptive filtering 

The probabilistic perspective we adopt throughout this work presents two main advantages  Firstly, a novel LMS algorithm with adaptable step size emerges naturally with this approach, making it suitable for both stationary and non-stationary environments  The proposed algorithm has less free parameters than previous LMS algorithms with variable step size \cite{kwong1992variable,aboulnasr1997robust,shin2004variable}, and its parameters are easier to be tuned w r t  these algorithms and standard LMS  Secondly, the use of a probabilistic model provides us with an estimate of the error variance, which is useful in many applications 

Experiments with simulated and real data show the advantages of the presented approach with respect to previous works  However, we remark that the main contribution of this paper is that it opens the door to introduce more Bayesian machine learning techniques, such as variational inference and Monte Carlo sampling methods \cite{barber2012bayesian}, to adaptive filtering \\


\section{Probabilistic Model}

Throughout this work, we assume the observation model to be linear-Gaussian with the following distribution,

\begin{equation}
p(y_k|{\bf w}_k) = \mathcal{N}(y_k;{\bf x}_k^T {\bf w}_k , \sigma_n^2),
\label{eq:mess_eq}
\end{equation}
where  $\sigma_n^2$ is the variance of the observation noise, ${\bf x}_k$ is the regression vector and ${\bf w}_k$ is the parameter vector to be sequentially estimated, both $M$-dimensional column vectors 


In a non-stationary scenario, ${\bf w}_k$ follows a dynamic process  In particular, we consider a diffusion process (random-walk model) with variance $\sigma_d^2$ for this parameter vector:


\begin{equation}
p({\bf w}_k|{\bf w}_{k-1})= \mathcal{N}({\bf w}_k;{\bf w}_{k-1}, \sigma_d^2 {\bf I}),
\label{eq:trans_eq}
\end{equation}
where $\bf I$ denotes the identity matrix  In order to initiate the recursion, we assume the following prior distribution on ${\bf w}_k$

\begin{equation}
p({\bf w}_0)= \mathcal{N}({\bf w}_0;0, \sigma_d^2{\bf I}) \nonumber
\end{equation}

\section{Exact inference in this model: Revisiting the RLS filter}

Given the described probabilistic SSM, we would like to infer the posterior probability distribution $p({\bf w}_k|y_{1:k})$ 
Since all involved distributions are Gaussian, one can perform exact inference, leveraging the probability rules in a straightforward manner  The resulting probability distribution is
\begin{equation}
p({\bf w}_k|y_{1:k}) =  \mathcal{N}({\bf w}_k;{\bf\boldsymbol\mu}_{k}, \boldsymbol\Sigma_{k}), \nonumber
\end{equation}
in which the mean vector ${\bf\boldsymbol\mu}_{k}$ is given by
\begin{equation}
{\bf\boldsymbol\mu}_k = {\bf\boldsymbol\mu}_{k-1} + {\bf K}_k (y_k - {\bf x}_k^T {\bf\boldsymbol\mu}_{k-1}){\bf x}_k, \nonumber
\end{equation}
where we have introduced the auxiliary variable
\begin{equation}
{\bf K}_k = \frac{ \left(\boldsymbol\Sigma_{k-1} + \sigma_d^2 {\bf I}\right)}{{\bf x}_k^T  \left(\boldsymbol\Sigma_{k-1} + \sigma_d^2 {\bf I}\right)  {\bf x}_k + \sigma_n^2}, \nonumber
\end{equation}
and the covariance matrix $\boldsymbol\Sigma_k$ is obtained as
\begin{equation}
\boldsymbol\Sigma_k = \left( {\bf I} -  {\bf K}_k{\bf x}_k {\bf x}_k^T \right) ( \boldsymbol\Sigma_{k-1} +\sigma_d^2), \nonumber
\end{equation}
Note that the mode of $p({\bf w}_k|y_{1:k})$, i e  the maximum-a-posteriori estimate (MAP), coincides with the RLS adaptive rule
\begin{equation}
{{\bf w}}_k^{(RLS)} = {{\bf w}}_{k-1}^{(RLS)} + {\bf K}_k (y_k - {\bf x}_k^T {{\bf w}}_{k-1}^{(RLS)}){\bf x}_k  
\label{eq:prob_rls}
\end{equation}
This rule is similar to the one introduced in \cite{haykin1997adaptive} 

Finally, note that the covariance matrix $\boldsymbol\Sigma_k$ is a measure of the uncertainty of the estimate ${\bf w}_k$ conditioned on the observed data $y_{1:k}$  Nevertheless, for many applications a single scalar summarizing the variance of the estimate could prove to be sufficiently useful  In the next section, we show how such a scalar is obtained naturally when $p({\bf w}_k|y_{1:k})$ is approximated with an isotropic Gaussian distribution  We also show that this approximation leads to an LMS-like estimation 
 


\section{Approximating the posterior distribution: LMS filter }

The proposed approach consists in approximating the posterior distribution $p({\bf w}_k|y_{1:k})$, in general a multivariate Gaussian distribution with a full covariance matrix, by an isotropic spherical Gaussian distribution 

\begin{equation}
\label{eq:aprox_post}
\hat{p}({\bf w}_{k}|y_{1:k})=\mathcal{N}({\bf w}_{k};{\bf \hat{\boldsymbol\mu}}_{k}, \hat{\sigma}_{k}^2 {\bf I} ) 
\end{equation}

In order to estimate the mean and covariance of the approximate distribution $\hat{p}({\bf w}_{k}|y_{1:k})$, we propose to select those that minimize the Kullback-Leibler divergence with respect to the original distribution, i e , 

\begin{equation}
\{\hat{\boldsymbol\mu}_k,\hat{\sigma}_k\}=\arg \displaystyle{  \min_{\hat{\boldsymbol\mu}_k,\hat{\sigma}_k}} \{ D_{KL}\left(p({\bf w}_{k}|y_{1:k}))\| \hat{p}({\bf w}_{k}|y_{1:k})\right) \}  \nonumber
\end{equation}

The derivation of the corresponding minimization problem can be found in Appendix A  In particular, the optimal mean and the covariance are found as
\begin{equation}
{\hat{\boldsymbol\mu}}_{k} = {\boldsymbol\mu}_{k};~~~~~~ \hat{\sigma}_{k}^2 = \frac{{\sf Tr}\{ \boldsymbol\Sigma_k\} }{M} 
\label{eq:sigma_hat}
\end{equation}


We now show that by using \eqref{eq:aprox_post} in the recursive predictive and filtering expressions we obtain an LMS-like adaptive rule  First, let us assume that we have an approximate posterior distribution at $k-1$, $\hat{p}({\bf w}_{k-1}|y_{1:k-1}) =  \mathcal{N}({\bf w}_{k-1};\hat{\bf\boldsymbol\mu}_{k-1}, \hat{\sigma}_{k-1}^2 {\bf I} )$  Since all involved distributions are Gaussian, the predictive distribution
is obtained as %
\begin{eqnarray}
\hat{p}({\bf w}_k|y_{1:k-1}) &=& \int p({\bf w}_k|{\bf w}_{k-1}) \hat{p}({\bf w}_{k-1}|y_{1:k-1}) d{\bf w}_{k-1} \nonumber\\
&=& \mathcal{N}({\bf w}_k;{\bf\boldsymbol\mu}_{k|k-1}, \boldsymbol\Sigma_{k|k-1}), 
\label{eq:approx_pred}
\end{eqnarray}
where the mean vector and covariance matrix are given by
\begin{eqnarray}
\hat{\bf\boldsymbol\mu}_{k|k-1} &=& \hat{\bf\boldsymbol\mu}_{k-1} \nonumber \\
\hat{\boldsymbol\Sigma}_{k|k-1} &=& (\hat{\sigma}_{k-1}^2 + \sigma_d^2 ){\bf I}\nonumber 
\end{eqnarray}

From \eqref{eq:approx_pred}, the posterior distribution at time $k$ can be computed using Bayes' Theorem and standard Gaussian manipulations (see for instance \cite[Ch  4]{murphy2012machine})  Then, we approximate the posterior $p({\bf w}_k|y_{1:k})$ with an isotropic Gaussian,
\begin{equation}
\hat{p}({\bf w}_k|y_{1:k}) =  \mathcal{N}({\bf w}_k ; {\hat{\boldsymbol\mu}}_{k}, \hat{\sigma}_k^2 {\bf I} ),\nonumber
\end{equation}
where 
\begin{eqnarray}
{\hat{\boldsymbol\mu}}_{k} &= & {\hat{\boldsymbol\mu}}_{k-1}+ \frac{ (\hat{\sigma}_{k-1}^2+ \sigma_d^2)  }{(\hat{\sigma}_{k-1}^2+ \sigma_d^2)  \|{\bf x}_k\|^2 + \sigma_n^2} (y_k - {\bf x}_k^T {\hat{\boldsymbol\mu}}_{k-1}){\bf x}_k  \nonumber  \\
&=& {\hat{\boldsymbol\mu}}_{k-1}+ \eta_k (y_k - {\bf x}_k^T {\hat{\boldsymbol\mu}}_{k-1}){\bf x}_k   
\label{eq:prob_lms}
\end{eqnarray}
Note that, instead of a gain matrix ${\bf K}_k$ as in Eq ~\eqref{eq:prob_rls}, we now have a scalar gain $\eta_k$ that operates as a variable step size 


Finally, to obtain the posterior variance, which is our measure of uncertainty, we apply \eqref{eq:sigma_hat} and the trick ${\sf Tr}\{{\bf x}_k{\bf x}_k^T\}= {\bf x}_k^T{\bf x}_k= \|{\bf x}_k \|^2$,

\begin{eqnarray}
\hat{\sigma}_k^2 &=& \frac{{\sf Tr}(\boldsymbol\Sigma_k)}{M} \\
&=& \frac{1}{M}{\sf Tr}\left\{ \left( {\bf I} -  \eta_k {\bf x}_k {\bf x}_k^T \right) (\hat{\sigma}_{k-1}^2 +\sigma_d^2)\right\} \\
&=& \left(1 - \frac{\eta_k \|{\bf x}_k\|^2}{M}\right)(\hat{\sigma}_{k-1}^2 +\sigma_d^2) 
\label{eq:sig_k}
\end{eqnarray}
If MAP estimation is performed, we obtain  an adaptable step-size LMS estimation

\begin{equation}
{\bf w}_{k}^{(LMS)} = {\bf w}_{k-1}^{(LMS)} + \eta_k (y_k - {\bf x}_k^T {\bf w}_{k-1}^{(LMS)}){\bf x}_k, 	
\label{eq:lms}
\end{equation}
with
\begin{equation}
\eta_k = \frac{ (\hat{\sigma}_{k-1}^2+ \sigma_d^2)  }{(\hat{\sigma}_{k-1}^2+ \sigma_d^2)  \|{\bf x}_k\|^2 + \sigma_n^2} \nonumber
\end{equation}
At this point, several interesting remarks can be made:

\begin{itemize}

\item The adaptive rule \eqref{eq:lms} has linear complexity since it does not require us to compute the full matrix $\boldsymbol\Sigma_k$ 

\item For a stationary model, we have $\sigma_d^2=0$ in \eqref{eq:prob_lms} and \eqref{eq:sig_k}  In this case, the algorithm remains valid and both the step size and the error variance, $\hat{\sigma}_{k}$, vanish over time $k$  

\item Finally, the proposed adaptable step-size LMS has only two parameters, $\sigma_d^2$ and $\sigma_n^2$, (and only one, $\sigma_n^2$, in stationary scenarios) in contrast to other variable step-size algorithms \cite{kwong1992variable,aboulnasr1997robust,shin2004variable}  More interestingly, both $\sigma_d^2$ and $\sigma_n^2$ have a clear underlying physical meaning, and they can be estimated in many cases  We will comment more about this in the next section  
\end{itemize}



\section{Experiments}
\label{sec:experiments}

We evaluate the performance of the proposed algorithm in both stationary and tracking experiments  In the first experiment, we estimate a fixed vector ${\bf w}^{o}$ of dimension $M=50$  The entries of the vector are independently and uniformly chosen in the range $[-1,1]$  Then, the vector is normalized so that $\|{\bf w}^o\|=1$  Regressors $\boldsymbol{x}_{k}$ are zero-mean Gaussian vectors with identity covariance matrix  The additive noise variance is such that the SNR is $20$ dB  We compare our algorithm with standard RLS and three other LMS-based algorithms: LMS, NLMS \cite{sayed2008adaptive}, VSS-LMS \cite{shin2004variable} \footnote{The used parameters for each algorithm are: for RLS $\lambda=1$, $\epsilon^{-1}=0 01$; for LMS $\mu=0 01$; for NLMS $\mu=0 5$; and for VSS-LMS $\mu_{max}=1$, $\alpha=0 95$, $C=1e-4$ } The probabilistic LMS algorithm in \cite{park2014probabilistic} is not simulated because it is not suitable for stationary environments 

In stationary environments, the proposed algorithm has only one parameter, $\sigma^2_n$  We simulate both the scenario where we have perfectly knowledge of the amount of noise (probLMS1) and the case where the value $\sigma^2_n$ is $100$ times smaller than the actual value (probLMS2)  The Mean-Square Deviation (${\sf MSD} = {\mathbb E} \| {\bf w}_0 - {\bf w}_k \|^2$), averaged out over $50$ independent simulations, is presented in Fig  \ref{fig:msd_statationary} 



\begin{figure}[htb]
\centering
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[width=\textwidth]{results_stationary_MSD}}
\end{minipage}
\caption{Performance in terms of MSD of probabilistic LMS with both optimal (probLMS1) and suboptimal (probLMS2) compared to LMS, NLMS, VS-LMS, and RLS }
\label{fig:msd_statationary}
\end{figure}

The performance of probabilistic LMS is close to RLS (obviously at a much lower computational cost) and largely outperforms previous variable step-size LMS algorithms proposed in the literature  Note that, when the model is stationary, i e  $\sigma^2_d=0$ in \eqref{eq:trans_eq},  both the uncertainty $\hat{\sigma}^2_k$, and the adaptive step size $\eta_k$, vanish over time  This implies that the error tends to zero when $k$ goes to infinity  Fig  \ref{fig:msd_statationary} also shows that the proposed approach is not very sensitive to a bad choice of its only parameter, as demonstrated by the good results of probLMS2, which uses a $\sigma^2_n$ that is $100$ times smaller than the optimal value  


\begin{figure}[htb]
\centering
\begin{minipage}[b]{\linewidth}
  \centering
  \centerline{\includegraphics[width=\textwidth]{fig2_final}}
\end{minipage}
\caption{Real part of one coefficient of the measured and estimated channel in experiment two  The shaded area represents two standard deviations from the prediction {(the mean of the posterior distribution)} }
\label{fig_2}
\end{figure}


\begin{table}[ht]
\begin{footnotesize}
\setlength{\tabcolsep}{2pt}
\def1 5mm{1 5mm}
\begin{center}
\begin{tabular}{|l@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|c@{\hspace{1 5mm}}|}
\hline
Method &  LMS &  NLMS & LMS-2013 & VSSNLMS & probLMS & RLS \\
\hline
\hline
MSD (dB) &-28 45 &-21 07 &-14 36 &-26 90 &-28 36 &-25 97\\
\hline                                                                     
\end{tabular}
\end{center}
\caption{Steady-state MSD of the different algorithms for the tracking of a real MISO channel }
\label{tab:table_MSD}
\end{footnotesize}

\end{table}
\newpage
In a second experiment, we test the tracking capabilities of the proposed algorithm with {real} data of a wireless MISO channel acquired in a realistic indoor scenario  More details on the setup can be found in \cite{gutierrez2011frequency}  Fig  \ref{fig_2} shows the real part of one of the channels, and the estimate of the proposed algorithm  The shaded area represents the estimated uncertainty for each prediction, i e  $\hat{\mu}_k\pm2\hat{\sigma}_k$  Since the experimental setup does not allow us to obtain the optimal values for the parameters, we fix these parameters to their values that optimize the steady-state mean square deviation (MSD)  \hbox{Table \ref{tab:table_MSD}} shows this steady-state MSD of the estimate of the MISO channel with different methods  As can be seen, the best tracking performance is obtained by standard LMS and the proposed method  





\section{Conclusions and Opened Extensions}
\label{sec:conclusions}

{We have presented a probabilistic interpretation of the least-mean-square filter  The resulting algorithm is an adaptable step-size LMS that performs well both in stationary and tracking scenarios  Moreover, it has fewer free parameters than previous approaches and these parameters have a clear physical meaning  Finally, as stated in the introduction, one of the advantages of having a probabilistic model is that it is easily extensible:}

\begin{itemize}
\item If, instead of using an isotropic Gaussian distribution in the approximation, we used a Gaussian with diagonal covariance matrix, we would obtain a similar algorithm with different step sizes and measures of uncertainty, for each component of ${\bf w}_k$  Although this model can be more descriptive, it needs more parameters to be tuned, and the parallelism with LMS vanishes 
\item Similarly, if we substitute the transition model of \eqref{eq:trans_eq} by an Ornstein-Uhlenbeck process, 

\begin{equation}
p({\bf w}_k|{\bf w}_{k-1})= \mathcal{N}({\bf w}_k;\lambda {\bf w}_{k-1}, \sigma_d^2), \nonumber
\label{eq:trans_eq_lambda}
\end{equation}
a similar algorithm is obtained but with a forgetting factor $\lambda$ multiplying ${\bf w}_{k-1}^{(LMS)}$ in \eqref{eq:lms}  This algorithm may have improved performance under such a kind of autoregresive dynamics of ${\bf w}_{k}$, though, again, the connection with standard LMS becomes dimmer 

\item As in \cite{park2014probabilistic}, the measurement model \eqref{eq:mess_eq} can be changed to obtain similar adaptive algorithms for classification, ordinal regression, and Dirichlet regression for compositional data  

\item A similar approximation technique could be applied to more complex dynamical models, i e  switching dynamical models \cite{barber2010graphical}  The derivation of efficient adaptive algorithms that explicitly take into account a switch in the dynamics of the parameters of interest is a non-trivial and open problem, though the proposed approach could be useful 

\item Finally, like standard LMS, this algorithm can be kernelized for its application in estimation under non-linear scenarios 

\end{itemize}


\begin{appendices}

\section{KL divergence between a general gaussian distribution and an isotropic gaussian}
\label{sec:kl}

 We want to approximate  $p_{{\bf x}_1}(x) = \mathcal{N}({\bf x}; \boldsymbol\mu_1,\boldsymbol\Sigma_1)$ by $p_{{\bf x}_2}({\bf x}) = \mathcal{N}({\bf x}; \boldsymbol\mu_2,\sigma_2^2 {\bf I})$  In order to do so, we have to compute the parameters of $p_{{\bf x}_2}({\bf x})$, $\boldsymbol\mu_2$ and $\sigma_2^2$, that minimize the following Kullback-Leibler divergence,

\begin{eqnarray}
D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) &=&\int_{-\infty}^{\infty} p_{{\bf x}_1}({\bf x}) \ln{\frac{p_{{\bf x}_1}({\bf x})}{p_{{\bf x}_2}({\bf x})}}d{\bf x} \nonumber  \\
&= &  \frac{1}{2} \{ -M + {\sf Tr}(\sigma_2^{-2} {\bf I}\cdot \boldsymbol\Sigma_1^{-1})  \nonumber \\
  & &  + (\boldsymbol\mu_2 - \boldsymbol\mu_1 )^T \sigma^{-2}_2{\bf I} (\boldsymbol\mu_2 - \boldsymbol\mu_1 )  \nonumber \\
 & &   +  \ln \frac{{\sigma_2^2}^M}{\det\boldsymbol\Sigma_1} \}   
\label{eq:divergence}
\end{eqnarray}
Using symmetry arguments, we obtain 
\begin{equation}
\boldsymbol\mu_2^{*} =\arg \displaystyle{  \min_{\boldsymbol\mu_2}} \{ D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) \} = \boldsymbol\mu_1 
\end{equation}
Then, \eqref{eq:divergence} gets simplified into 

\begin{eqnarray}
D_{KL}(p_{{\bf x}_1}\| p_{{\bf x}_2}) = \frac{1}{2}\lbrace { -M + {\sf Tr}(\frac{\boldsymbol\Sigma_1}{\sigma_2^{2}}) + \ln \frac{\sigma_2^{2M}}{\det\boldsymbol\Sigma_1}}\rbrace 
\end{eqnarray}
The variance $\sigma_2^2$ is computed in order to minimize this Kullback-Leibler divergence as

\begin{eqnarray}
\sigma_2^{2*} &=& \arg\min_{\sigma_2^2} D_{KL}(P_{x_1}\| P_{x_2}) \nonumber \\
 &=& \arg\min_{\sigma_2^2}\{ \sigma_2^{-2}{\sf Tr}\{\boldsymbol\Sigma_1\} + M\ln \sigma_2^{2} \}  
\end{eqnarray}
Deriving and making it equal zero leads to

\begin{equation}
\frac{\partial}{\partial \sigma_2^2} \left[ \frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{\sigma_2^{2}} + M \ln \sigma_2^{2} \right] = \left  {\frac{M}{\sigma_2^{2}}-\frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{(\sigma_2^{2})^2}}\right|_{\sigma_2^{2}=\sigma_2^{2*}}\left  =0 \right   
\nonumber
\end{equation}
Finally, since the divergence has a single extremum in $R_+$,
\begin{equation}
\sigma_2^{2*} = \frac{{\sf Tr}\{\boldsymbol\Sigma_1\}}{M} 
\end{equation}




\end{appendices}

\vfill
\clearpage

\bibliographystyle{IEEEbib}

Context 2: \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
Context 3: \section{Model equations} \label{sec:equations}

In drift-fluid models the continuity equation
\begin{align}
 \frac{\partial n}{\partial t} + \nabla\cdot\left( n \vec u_E  \right) &= 0 \label{eq:generala} 
\end{align}
describes the dynamics of the electron density $n$  Here
$\vec u_E := (\hat{\vec b} \times \nabla \phi)/B$ gives the electric drift
velocity in a magnetic field $\vec B := B \hat{\vec b}$ and an electric
potential $\phi$  We neglect contributions of the diamagnetic drift~\cite{Kube2016} 




Equation~\eqref{eq:generala} is closed by invoking quasineutrality, i e  the divergence of the ion polarization, 
the electron diamagnetic and the gravitational drift currents must vanish
\begin{align}
  \nabla\cdot\left( \frac{n}{\Omega} \left( \frac{\partial}{\partial t} 
  + \vec u_E \cdot\nabla  \right)\frac{\nabla_\perp \phi}{B}  + n\vec u_d - n\vec u_g\right) &=0
    
 
 
  \label{eq:generalb}
\end{align}
Here we denote 
$\nabla_\perp\phi/B := - \hat{\vec b} \times \vec u_E$, 
the electron diamagnetic drift
$\vec u_d := - T_e(\hat{\vec b} \times\nabla n ) /enB$
with the electron temperature $T_e$,
the ion gravitational drift velocity  
$\vec u_g := m_i \hat{\vec b} \times \vec g /B$
with ion mass $m_i$, and the ion gyro-frequency
$\Omega := eB/m_i$ 

Combining Eq ~\eqref{eq:generalb} with Eq ~\eqref{eq:generala} yields
\begin{align}
 \frac{\partial \rho}{\partial t} + \nabla\cdot\left( \rho\vec u_E \right) + \nabla \cdot\left( n(\vec u_\psi + \vec u_d + \vec u_g) \right) &= 0\label{eq:vorticity}
\end{align}
with the polarization charge density 
$\rho = \nabla\cdot( n\nabla_\perp \phi / \Omega B)$ 
and
$\vec u_\psi := \hat{\vec b}\times \nabla\psi /B$ 
with 
$\psi:= m_i\vec u_E^2 /2e$ 
We exploit this form of Eq ~\eqref{eq:generalb} in our numerical simulations 

Equations~\eqref{eq:generala} and \eqref{eq:generalb} respectively \eqref{eq:vorticity} have several invariants 
First, in Eq ~\eqref{eq:generala} the relative particle number 
$M(t) := \int \mathrm{dA}\, (n-n_0)$ is conserved over time
$\d M(t)/\d t = 0$  
Furthermore, we integrate 
$( T_e(1+\ln n) -T_e \ln B)\partial_t n$
as well as
$-e\phi \partial_t\rho - (m_i\vec u_E^2/2+gm_ix - T_e\ln B)\partial_t n$ 
over the domain to get, disregarding boundary contributions,
\begin{align}
  \frac{\d}{\d t}\left[T_eS(t) + H(t) \right] = 0, \label{eq:energya}\\ 
    \frac{\d}{\d t} \left[ E(t) - G(t) - H(t)\right] =  0,
    \label{eq:energyb}
\end{align}
where we define 
the entropy
$S(t):=\int \mathrm{dA}\, [n\ln(n/n_0) - (n-n_0)]$,  
the kinetic energy 
$E(t):=m_i \int \mathrm{dA}\, n\vec u_E^2/2$ 
and the potential energies
$G(t) := m_i g\int \mathrm{dA}\, x(n-n_0)$
and
$H(t) := T_e\int \mathrm{dA}\, (n-n_0) \ln (B^{-1})$ 
Note that $n\ln( n/n_0) - n + n_0 \approx (n-n_0)^2/2$ for $|(n-n_0)/n_0| \ll 1$ and $S(t)$ thus reduces to the 
local entropy form in Reference~\cite{Kube2016}  

We now set up a gravitational field $\vec g = g\hat x$ and a constant homogeneous background
magnetic field $\vec B = B_0 \hat z$ in a Cartesian coordinate system 
Then the divergences of the electric and gravitational drift velocities $\nabla\cdot\vec u_E$ and $\nabla\cdot\vec u_g$
and the diamagnetic current $\nabla\cdot(n\vec u_d)$ vanish, which makes the 
flow incompressible  Furthermore, the magnetic potential energy vanishes $H(t) = 0$ 

In a second system we model the inhomogeneous magnetic field present in tokamaks as
$\vec B := B_0 (1+ x/R_0)^{-1}\hat z$ and neglect the gravitational drift $\vec u_g = 0$ 
Then, the potential energy $G(t) = 0$  
Note that 
$H(t) = m_i \ensuremath{C_\mathrm{s}}^2/R_0\int\mathrm{dA}\, x(n-n_0) +\mathcal O(R_0^{-2}) $
reduces to $G(t)$ with the effective gravity $g_\text{eff}:= \ensuremath{C_\mathrm{s}}^2/R_0$ with $\ensuremath{C_\mathrm{s}}^2 := T_e/m_i$  
For the rest of this letter we treat $g$ and $g_\text{eff}$ as well as $G(t)$ and $H(t)$ on the same footing 
The magnetic field inhomogeneity thus entails compressible flows, which is 
the only difference to the model describing dynamics in a homogeneous magnetic field introduced above  
Since both $S(t)\geq 0$ and $E(t)\geq 0$ we further derive from Eq ~\eqref{eq:energya} and Eq ~\eqref{eq:energyb} that the kinetic energy
is bounded by $E(t) \leq T_eS(t) + E(t) = T_e S(0)$; a feature absent from the gravitational system with 
incompressible flows, where $S(t) = S(0)$  

We now show that the invariants Eqs ~\eqref{eq:energya} and \eqref{eq:energyb} present restrictions on the velocity and
acceleration of plasma blobs  
First, we define the blobs' center of mass (COM) via $X(t):= \int\mathrm{dA}\, x(n-n_0)/M$ and 
its COM velocity as $V(t):=\d X(t)/\d t$  
The latter is proportional to the total radial particle flux~\cite{Garcia_Bian_Fundamensky_POP_2006, Held2016a} 
We assume
that $n>n_0$ and $(n-n_0)^2/2 \leq [ n\ln (n/n_0) - (n-n_0)]n $ to show for both systems 
\begin{align}
  (MV)^2 &= \left( \int \mathrm{dA}\, n{\phi_y}/{B} \right)^2
  = \left( \int \mathrm{dA}\, (n-n_0){\phi_y}/{B} \right)^2\nonumber\\
 
&\leq 2 \left( \int \mathrm{dA}\, \left[n\ln (n/n_0) -(n-n_0)\right]^{1/2}\sqrt{n}{\phi_y}/{B}\right)^2\nonumber\\
 
  &\leq 4 S(0) E(t)/m_i 
 
  \label{eq:inequality}
\end{align}
Here we use the Cauchy-Schwartz inequality and 
$\phi_y:=\partial\phi/\partial y$  
Note that although we derive the inequality Eq ~\eqref{eq:inequality} only for amplitudes $\triangle n >0$  we assume that the results also hold for depletions  This is justified by our numerical results later in this letter  
If we initialize our density field with a seeded blob of radius $\ell$ and amplitude $\triangle n$ as 
\begin{align}
  n(\vec x, 0) &= n_0 + \triangle n \exp\left( -\frac{\vec x^2}{2\ell^2} \right), \label{eq:inita}
 
 
\end{align}
and  
$\phi(\vec x, 0 ) = 0$,
we immediately have $M := M(0) = 2\pi \ell^2 \triangle n$, $E(0) = G(0) = 0$ and 
$S(0) = 2\pi \ell^2 f(\triangle n)$, where $f(\triangle n)$ captures the amplitude dependence of 
the integral for $S(0)$  

The acceleration for both incompressible and compressible flows can be estimated
by assuming a linear acceleration $V=A_0t$ and $X=A_0t^2/2$~\cite{Held2016a} and using 
$E(t) = G(t) = m_igMX(t)$ in Eq ~\eqref{eq:inequality}
\begin{align}
  \frac{A_0}{g} =  \mathcal Q\frac{2S(0)}{M} \approx \frac{\mathcal Q}{2} \frac{\triangle n }{n_0+2\triangle n/9} 
  \label{eq:acceleration}
\end{align}
Here, we use the Pad\'e approximation of order $(1/1)$ of $2S(0)/M $
and define a model parameter $\mathcal Q$ with $0<\mathcal Q\leq1$ to be determined by numerical simulations 
Note that the Pad\'e approximation is a better approximation than a simple 
truncated Taylor expansion especially for large relative amplitudes of order unity 
Eq ~\eqref{eq:acceleration} predicts that $A_0/g\sim \triangle n/n_0$ for small 
amplitudes $|\triangle n/n_0| < 1$ and $A_0 \sim g $ for very large amplitudes $\triangle n /n_0 \gg 1$, 
which confirms the predictions in~\cite{Pecseli2016} and reproduces the limits discussed in~\cite{Angus2014} 

As pointed out earlier for compressible flows Eq ~\eqref{eq:inequality} can be further estimated
\begin{align}
  (MV)^2  \leq 4 T_eS(0)^2/m_i  
  \label{}
\end{align}
We therefore have a restriction on the maximum COM velocity for compressible flows, which is absent for incompressible flows
\begin{align}
  \frac{\max |V|}{\ensuremath{C_\mathrm{s}}} = {\mathcal Q}\frac{2S(0)}{M} \approx \frac{\mathcal Q}{2} \frac{|\triangle n| }{n_0+2/9 \triangle n } \approx \frac{\mathcal Q}{2} \frac{|\triangle n|}{n_0} 
  \label{eq:linear}
\end{align}
For $|\triangle n /n_0|< 1$ Eq ~\eqref{eq:linear} reduces to the linear scaling derived in~\cite{Kube2016}  
Finally, a scale analysis of Eq ~\eqref{eq:vorticity} shows that~\cite{Ott1978, Garcia2005, Held2016a}
\begin{align}
  \frac{\max |V|}{\ensuremath{C_\mathrm{s}}} = \mathcal R \left( \frac{\ell}{R_0}\frac{|\triangle n|}{n_0} \right)^{1/2} 
  \label{eq:sqrt}
\end{align}
This equation predicts a square root dependence of the center of mass velocity 
on amplitude and size  





We now propose a simple phenomenological model that captures the essential dynamics
of blobs and depletions in the previously stated systems  More specifically 
the model reproduces the acceleration Eq ~\eqref{eq:acceleration} with and without
Boussinesq approximation, the square root scaling for the COM velocity 
Eq ~\eqref{eq:sqrt} for incompressible flows as well as the relation between the 
square root scaling Eq ~\eqref{eq:sqrt} and the linear scaling 
Eq ~\eqref{eq:linear} for compressible flows  
The basic idea is that the COM of blobs behaves like 
the one of an infinitely long plasma column immersed in an ambient plasma  
The dynamics of this column reduces to the one of a two-dimensional ball 
This idea is similar to the analytical ``top hat'' density solution for
blob dynamics recently studied in~\cite{Pecseli2016} 
The ball is subject to buoyancy as well as linear and nonlinear friction
\begin{align}
  M_{\text{i}} \frac{d V}{d t} = (M_{\text{g}} - M_\text{p}) g - c_1 V  - \mathrm{sgn}(V ) \frac{1}{2}c_2 V^2



ground_truth: This research paper proposed an approach based on approximating the posterior distribution with an isotropic Gaussian distribution.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How do the runtimes and iteration counts of NFPA and FPSA compare to GMRES and DSA in the numerical experiments?

Read the following text and answer briefly.

Context 2: 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 




Context 3: 
  \label{eq:ball}
\end{align}
The gravity $g$ has a positive sign in the coordinate system; sgn$(f)$ is the sign function  
The first term on the right hand side is the buoyancy, where 
$M_{\text{g}} := \pi \ell^2 (n_0 + \mathcal Q \triangle n/2)$ 
is the gravitational mass of the ball with radius $\ell$ and 
$M_\mathrm{p} := n_0 \pi \ell^2 $ 
is the mass of the displaced ambient plasma 
Note that if $\triangle n<0$ the ball represents a depletion and the buoyancy term has a negative sign, i e  the depletion will rise  
We introduce an inertial mass 
$M_{\text{i}} := \pi\ell^2 (n_0 +2\triangle n/9)$ 
different from the gravitational mass $M_{\text{g}}$ in order to 
recover the initial acceleration in Eq ~\eqref{eq:acceleration}  
We interpret the parameters $\mathcal Q$ and $2/9$ as geometrical factors 
that capture the difference of the actual blob form from the idealized
``top hat'' solution  
Also note that the Boussinesq approximation appears in the model as a neglect of inertia, $M_{\text{i}} = \pi\ell^2n_0$ 

The second term is the linear friction term with coefficient $c_1(\ell)$, which
depends on the size of the ball 
If we disregard the nonlinear friction, $c_2=0$, Eq ~\eqref{eq:ball} directly yields a 
maximum velocity $c_1V^*=\pi \ell^2 n g \mathcal Q\triangle n/2$ 
From our previous considerations $\max V/\ensuremath{C_\mathrm{s}}=\mathcal Q \triangle n /2n_0$, we thus identify 
\begin{align}
  c_1 = \pi\ell^2 n_0 g/\ensuremath{C_\mathrm{s}}  
  \label{}
\end{align}
The linear friction coefficient thus depends on the gravity and the size of the
ball  

The last term in \eqref{eq:ball} is the nonlinear friction  The sign of the force depends on whether
the ball rises or falls in the ambient plasma  
If we disregard linear friction $c_1=0$, we have the maximum velocity 
$V^*= \sigma(\triangle n)\sqrt{\pi \ell^2|\triangle n| g\mathcal Q/c_2}$, 
which must equal 
$\max V= \sigma(\triangle n) \mathcal R \sqrt{g \ell |\triangle n/n_0|}$ 
and thus
\begin{align}
  c_2 = {\mathcal Q\pi n_0\ell }/{\mathcal R^2} 
  \label{}
\end{align}
Inserting $c_1$ and $c_2$ into Eq ~\eqref{eq:ball}
we can derive the maximum absolute velocity in the form 
\begin{align}
  \frac{\max |V|}{\ensuremath{C_\mathrm{s}}} = 
        \left(\frac{\mathcal R^2}{\mathcal Q}\right) \frac{\ell}{R_0} \left( 
        \left({1+\left( \frac{\mathcal Q}{\mathcal R} \right)^{2} \frac{|\triangle n|/n_0 }{\ell/R_0}}\right)^{1/2}-1 \right)
  \label{eq:vmax_theo}
\end{align}
and thus have a concise expression for $\max |V|$ that captures both the linear
scaling \eqref{eq:linear} as well as the square root scaling \eqref{eq:sqrt} 
With Eq ~\eqref{eq:acceleration} and Eq ~\eqref{eq:sqrt} respectively Eq ~\eqref{eq:vmax_theo} we 
finally arrive at an analytical expression for the time at which the maximum velocity is reached via 
$t_{\max V} \sim \max V/A_0$  Its inverse $\gamma:=t_{\max V}^{-1}$ gives the
global interchange growth rate, for which an empirical expression was
presented in Reference~\cite{Held2016a} 


We use the open source library FELTOR 
to simulate 
Eqs ~\eqref{eq:generala} and \eqref{eq:vorticity} with and without 
drift compression 
For numerical stabilty we added small diffusive terms on the right hand 
sides of the equations 
The discontinuous Galerkin methods employ three polynomial coefficients and a minimum of $N_x=N_y=768$ grid cells  The box size is $50\ell$ in order to mitigate 
influences of the finite box size on the blob dynamics  
Moreover, we used the invariants in Eqs  \eqref{eq:energya} and \eqref{eq:energyb} as consistency tests to verify the code and repeated simulations 
also in a gyrofluid model  
No differences to the results presented here were found  
Initial perturbations on the particle density field are given by Eq ~\eqref{eq:inita},
where the perturbation amplitude $\triangle n/n_0$ was chosen between $10^{-3}$ and $20$ for blobs and $-10^0$ and $ -10^{-3}$ for depletions  
Due to computational reasons we show results only for $\triangle n/n_0\leq 20$  


For compressible flows we consider two different cases $\ell/R_0 = 10^{-2}$ and
$\ell /R_0 = 10^{-3}$  
 For incompressible flows Eq ~\eqref{eq:generala} and \eqref{eq:vorticity}
 can be normalized such that the blob radius is absent from the equations~\cite{Ott1978, Kube2012}  
 The simulations of incompressible flows can thus be used for both sizes  
The numerical code as well as input parameters and output data can be found 
in the supplemental dataset to this contribution~\cite{Data2017} 

\begin{figure}[htb]
    \includegraphics[width=\columnwidth]{com_blobs}
    \caption{
      The maximum radial COM velocities of blobs for compressible and incompressible flows are shown  
      The continuous lines show Eq ~\eqref{eq:vmax_theo} while the 
      dashed line shows the square root scaling Eq ~\eqref{eq:sqrt} with 
      $\mathcal Q = 0 32$ and $\mathcal R=0 85$ 
    }
    \label{fig:com_blobs}
\end{figure}
In Fig ~\ref{fig:com_blobs} we plot the maximum COM velocity for blobs 
with and without drift compression 
For incompressible flows blobs follow the square root scaling almost 
perfectly  Only at very large amplitudes velocities are slightly below
the predicted values  
For small amplitudes we observe that the compressible blobs follow
a linear scaling  When the amplitudes increase there is a transition to the
square root scaling at around $\triangle n/n_0 \simeq 0 5$ for 
$\ell/R_0=10^{-2}$ and $\triangle n/n_0 \simeq 0 05$ for $\ell/R_0=10^{-3}$, which is consistent with Eq ~\eqref{eq:vmax_theo} and Reference~\cite{Kube2016}  
In the transition regions the simulated velocities are slightly larger than the predicted ones from Eq ~\eqref{eq:vmax_theo} 
Beyond these amplitudes
the velocities of compressible and incompressible blobs align  

\begin{figure}[htb]
    \includegraphics[width=\columnwidth]{com_holes}
    \caption{
      The maximum radial COM velocities of depletions for compressible and incompressible flows are shown  
      The continuous lines show Eq ~\eqref{eq:vmax_theo} while the 
      dashed line shows the square root scaling Eq



ground_truth: NFPA and FPSA greatly outperform GMRES and DSA.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the relationship between the maximum velocity and the amplitude of the blob or depletion?

Read the following text and answer briefly.

Context 3: ~\eqref{eq:sqrt} with 
      $\mathcal Q = 0 32$ and $\mathcal R=0 85$ 
      Note that small amplitudes are on the right and amplitudes close to unity are on the left side 
  }
    \label{fig:com_depletions}
\end{figure}
In Fig ~\ref{fig:com_depletions} we show the maximum radial COM velocity 
for depletions instead of blobs 
For relative amplitudes below $|\triangle n|/n_0 \simeq 0 5$ (right of unity in the plot) the velocities
coincide with the corresponding blob velocities in Fig ~\ref{fig:com_blobs}  
 For amplitudes larger than $|\triangle n|/n_0\simeq 0 5$ the 
velocities follow the square root scaling 
We observe that for plasma depletions beyond $90$ percent the velocities 
in both systems reach a constant value that is very well predicted by the
square root scaling  

\begin{figure}[htb]
    \includegraphics[width=\columnwidth]{acc_blobs}
    \caption{
      Average acceleration of blobs for compressible and incompressible flows are shown 
      The continuous line shows the acceleration in Eq ~\eqref{eq:acceleration} 
      with $\mathcal Q=0 32$
      while the dashed line is a linear reference line, which corresponds to the Boussinesq approximation  
  }
    \label{fig:acc_blobs}
\end{figure}
In Fig ~\ref{fig:acc_blobs} we show the average acceleration of blobs 
for compressible and incompressible flows computed
by dividing the maximum velocity $\max V$ by the time  
to reach this velocity $t_{\max V}$  
We compare the simulation results
to the theoretical predictions Eq ~\eqref{eq:acceleration} of our model with and without inertia  
The results of the compressible and incompressible systems coincide and fit very
well to our theoretical values  
For amplitudes larger than unity the acceleration deviates significantly from the prediction with Boussinesq approximation 

\begin{figure}[htb]
    \includegraphics[width=\columnwidth]{acc_holes}
    \caption{
      Average acceleration of depletions for compressible and incompressible flows are shown 
      The continuous line shows the acceleration in Eq ~\eqref{eq:acceleration} 
      with $\mathcal Q=0 32$
      while the dashed line is a linear reference line, which corresponds to the Boussinesq approximation  
    }
    \label{fig:acc_depletions}
\end{figure}
In Fig ~\ref{fig:acc_depletions} we show the simulated acceleration of depletions in the
compressible and the incompressible systems  We compare the simulation results
to the theoretical predictions Eq ~\eqref{eq:acceleration} of our model with and without inertia 
Deviations from our theoretical prediction Eq ~\eqref{eq:acceleration} are visible for amplitudes smaller than $\triangle n/n_0 \simeq -0 5$ (left of unity in the plot)  The relative deviations are small at around $20$ percent  
As in Fig ~\ref{fig:com_depletions} the acceleration reaches a constant values
for plasma depletions of more than $90$ percent 
Comparing Fig ~\ref{fig:acc_depletions} to Fig ~\ref{fig:acc_blobs} the asymmetry between blobs and depletions becomes 
apparent  While the acceleration of blobs is reduced for large 
amplitudes compared to a linear dependence the acceleration 
of depletions is increased  In the language of our simple buoyancy 
model the inertia of depletions is reduced but increased for blobs  



In conclusion  
  we discuss the dynamics of seeded blobs and depletions in a 
  compressible and an incompressible system 
  With only two fit parameters our theoretical results reproduce the 
  numerical COM velocities and accelerations over five orders of magnitude 
  We derive the amplitude dependence of the acceleration of blobs and depletions from 
  the conservation laws of our systems in Eq ~\eqref{eq:acceleration}  
  From the same inequality a linear regime is derived in the compressible system for 
  ratios of amplitudes to sizes smaller than a critical value 
   In this regime 
  the blob and depletion velocity depends linearly on the initial amplitude and 
  is independent of size  The regime is absent from the system with incompressible flows 
  Our theoretical results are verified by numerical simulations for all 
  amplitudes that are relevant in magnetic fusion devices 
  Finally, we suggest a new empirical blob model that captures the detailed dynamics of more complicated models  
  The Boussinesq approximation is clarified as the absence of inertia and a thus altered acceleration of blobs and depletions 
  The maximum blob velocity is not altered by the Boussinesq approximation 

The authors were supported with financial subvention from the Research Council of Norway under grant
240510/F20  M W  and M H  were supported by the Austrian Science Fund (FWF) Y398   The computational
results presented have been achieved in part using the Vienna Scientific Cluster (VSC)  Part of this work was performed on the Abel Cluster, owned by the University of Oslo and the Norwegian metacenter
for High Performance Computing (NOTUR), and operated by the Department for Research Computing at USIT,
the University of Oslo IT-department 
This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 under grant agreement No 633053  The views and opinions expressed herein do not necessarily reflect those of the European Commission 



ground_truth: The maximum velocity scales with the square root of the amplitude.