text:
\section{Head Model}
\label{headM}
The different compartments of the head model are shown in figure \ref{headmodel}. The bounded domain $\Omega_c$ represent the cerebrum, which has conductivity $\sigma_c$. A shell $\Omega_f$ with conductivity $\sigma_f$, representing the cerebrospinal fluid, surrounds the domain $\Omega_c$. The cerebrospinal fluid that is surrounded by the skull is characterized by the domain $\Omega_b$ with conductivity $\sigma_b$. Finally, the skull is surrounded by the scalp that is modelled as a shell $\Omega_s$ with conductivity $\sigma_s$. Notations for the surfaces forming the boundaries of above domains are introduced in figure \ref{headmodel}. The domain exterior to the head is denoted by $\Omega_e$, and it is assumed that $\Omega_e$ is not conductive. The permeability of all domains are equal to the permeability $\mu$ of empty space.

\begin{figure}[H]
 \includegraphics[scale=0.5]{medium.eps}
  \caption{The different compartments of the head model. $\Omega_c$ denotes the cerebrum that is surrounded by three shells $\Omega_f$, $\Omega_b$, $\Omega_s$, denoting the cerebrospinal fluid, the skull, and the scalp. Their conductivities are respectively denoted by $\sigma_c$, $\sigma_f$, $\sigma_b$ and $\sigma_s$. The spaces $\Omega_c$, $\Omega_f$, $\Omega_b$ and $\Omega_s$ are bounded by the surfaces $S_c$, $(S_c,S_f)$, $(S_f,S_b)$ and $(S_b,S_s)$. }
  \label{headmodel}
\end{figure}
Table \ref{tabl4layer} presents the conductivity values of the head model as documented in  \cite{gabriel1996,manola2005,hoekema2003}.
\begin{table}[H]
\begin{center}
\begin{tabular}{ |c||c|c| }
 \hline
 Domain $\Omega$& Conductivity $\sigma$ (S/m) \\
 \hline
 Cerebrum $\Omega_c$& 0.33 \\
 \hline
 CSF $\Omega_f$ & 1.0 \\
 \hline
 Skull $\Omega_b$&  0.0042 \\
 \hline
 Scalp $\Omega_s$&  0.33 \\
 \hline
 \end{tabular}
 \end{center}
 \caption{The conductivity values for the different compartments of the head model.}
 \label{tabl4layer}
 \end{table}

The physiology of the head model is accurately characterized by the four layer compartments shown in figure \ref{headmodel}, and the results derived in \cite{fokas2009} are valid in the four layer head model. It can be observed from table \ref{tabl4layer} that $\Omega_f$ (CSF), has a higher conductivity than the remaining compartments, but it also has a very small thickness. The detailed analysis of \cite{clerk2009} shows that the brain-CSF interface has a negligible effect in the analysis. For this reason, for our numerical examples we ignore $\Omega_f$ (CSF) and restrict our analysis to the three layer head model involving the compartments $\Omega_c, \Omega_b$ and $\Omega_s$.

\section{The Measurement Equation}
\label{forwardproblem}
Let $\neurc(\boldsymbol\tau)$, $\boldsymbol\tau \in \Omega_c$, denote the primary current (neuronal current) which is assumed to be supported within the cerebrum $\Omega_c$. Under the assumption that $\Jvec(\boldsymbol\tau)$ has sufficient smoothness (continuous derivatives), we can employ the Helmholtz decomposition to express $\Jvec$ in terms of its irrotational and solenoidal components:
\begin{equation}
\Jvec(\boldsymbol\tau)=\nabla_{\boldsymbol\tau} \Psi(\boldsymbol\tau)+ \nabla_{\boldsymbol\tau} \times \Avec(\boldsymbol\tau), \qquad \boldsymbol\tau \in \Omega_c,
\label{helmddecomp}
\end{equation}
where $\Avec(\boldsymbol\tau)$ characterises  the solenoidal component and satisfies the constraint
$\nabla \cdot \Avec(\boldsymbol\tau)=0$. $\Psi(\boldsymbol\tau)$ characterises the irrotational component. Under  the assumption that $\Jvec(\boldsymbol\tau)$ vanishes on the cortical surface $S_c$, it is shown in \cite{fokas2009} that the electric potential measured on the scalp, $S_s$, is given by
\begin{equation}
\label{2ndexpress2}
u_s(\rvec)=-\frac{1}{4\pi} \int_{\Omega_c} \nabla_{\boldsymbol\tau}^{2} \Psi(\boldsymbol\tau) v_s(\rvec,\boldsymbol\tau)dV(\boldsymbol\tau), \qquad \rvec \in S_s ,
\end{equation}
where the auxiliary function $v_s(\rvec,\boldsymbol\tau)$ depends on geometry and conductivities but is independent of the current \cite{fokas2009}. It was shown further in \cite{hashemzadeh2018} that equation (\ref{2ndexpress2}) can be expressed in the form of a surface integral :
\begin{equation}
\label{2ndexpress4}
u_s(\rvec)=\frac{1}{4\pi} \int_{S_c} \nvec \cdot \biggl[ \Psi(\boldsymbol\tau) \nabla_{\boldsymbol\tau} v_s(\rvec,\boldsymbol\tau)
-v_s(\rvec,\boldsymbol\tau) \nabla_{\boldsymbol\tau} \Psi(\boldsymbol\tau)\biggr]dS(\boldsymbol\tau), \quad \rvec \in S_s.
\end{equation}
Thus,  $u_s(\rvec)$ is only affected  by the values of $\Psi$ and of $\nabla_{\boldsymbol\tau} \Psi$ on the surface of $S_c$.  It is therefore \textit{impossible} to determine the $3$-dimensional nature of $\Psi$ from the knowledge of $u_s(\rvec)$. Assuming that $v_s(\rvec,\boldsymbol\tau)$  can be computed, equation (\ref{2ndexpress2}) provides a relation between $u_s(\rvec)$ and $\nabla_{\boldsymbol\tau}^{2}\Psi(\boldsymbol\tau)$. Similarly, assuming that both $v_s(\rvec,\boldsymbol\tau)$ and $\nvec \cdot \nabla_{\boldsymbol\tau}v_s(\rvec,\boldsymbol\tau)$ can be computed, equation (\ref{2ndexpress4}) provides a relation between $u_s(\rvec)$, $\Psi(\boldsymbol\tau)$ and $\nabla_{\boldsymbol\tau} \Psi(\boldsymbol\tau)$.

\subsection{Auxiliary Functions $v_j(\rvec,\boldsymbol\tau)$}
\label{compvs}
It was shown in \cite{fokas2009} that for a given geometry, the functions $v_j(\rvec,\boldsymbol\tau)$, $j=c,f,b,s$ are defined via the following boundary value problem:

\begin{eqnarray}
\label{bveqn1}
&\nabla_{\boldsymbol\tau}^2 v_c(\rvec,\boldsymbol\tau)=0,\quad  \rvec \in \Omega_c, \boldsymbol\tau \in \Omega_c, \\
&\frac{\partial}{\partial n}\biggl[ \frac{1}{|\rvec-\boldsymbol\tau|}+v_c(\rvec,\boldsymbol\tau) \biggr]=\sigma_f \frac{\partial v_f(\rvec,\boldsymbol\tau)}{\partial n}, \quad\rvec \in S_c; \nonumber
\end{eqnarray}

\begin{eqnarray}
\label{bveqn2}
&\nabla_{\boldsymbol\tau}^2 v_f(\rvec,\boldsymbol\tau)=0, \quad \rvec \in \Omega_f, \boldsymbol\tau \in \Omega_c,\\
&v_f(\rvec,\boldsymbol\tau)=\frac{1}{\sigma_c} \biggl[ \frac{1}{|\rvec-\boldsymbol\tau|}+v_c(\rvec,\boldsymbol\tau)\biggr], \qquad \rvec \in S_c, \nonumber\\
&\sigma_f \frac{\partial v_f(\rvec,\boldsymbol\tau)}{\partial n} =\sigma_b \frac{\partial v_b(\rvec,\boldsymbol\tau)}{\partial n},\qquad \rvec \in S_f, \boldsymbol\tau \in \Omega_c;\nonumber
\end{eqnarray}

\begin{eqnarray}
\label{bveqn3}
&\nabla_{\boldsymbol\tau}^2 v_b(\rvec,\boldsymbol\tau)=0, \quad \rvec \in \Omega_b, \boldsymbol\tau \in \Omega_c,\\
&v_b(\rvec,\boldsymbol\tau)=v_f(\rvec,\boldsymbol\tau), \quad \rvec \in S_f, \nonumber \\
&\sigma_b \frac{\partial v_b(\rvec,\boldsymbol\tau)}{\partial n} =\sigma_s \frac{\partial v_s(\rvec,\boldsymbol\tau)}{\partial n},\qquad \rvec \in S_b, \boldsymbol\tau \in \Omega_c;\nonumber
\end{eqnarray}

\begin{eqnarray}
\label{bveqn4}
&\nabla_{\boldsymbol\tau}^2 v_s(\rvec,\boldsymbol\tau)=0, \quad \rvec \in \Omega_s, \boldsymbol\tau \in \Omega_c,\\
&v_s(\rvec,\boldsymbol\tau)=v_b(\rvec,\boldsymbol\tau), \quad \rvec \in S_b,\nonumber\\
&\frac{\partial v_s(\rvec,\boldsymbol\tau)}{\partial n} =0, \quad \rvec \in S_s.\nonumber
\end{eqnarray}

Equations (\ref{bveqn1}) - (\ref{bveqn4}) are \textit{independent} of the current $\Jvec(\boldsymbol\tau)$ and depend only on the geometry and on the conductivities $\sigma_c$, $\sigma_b$, $\sigma_f$ and $\sigma_s$.\par
It is shown in \cite{fokas2009} that the functions $v_j(\rvec,\boldsymbol\tau)$ can be related to the functions  $u_j(\rvec,\boldsymbol\tau)$ with unit Volts, $\rvec \in \Omega_j, ~~\boldsymbol\tau \in \Omega_c, ~~j \in \{c,f,b,s \}$. These functions are defined in terms of a single dipole with moment $\boldsymbol{Q}(\boldsymbol\tau)$ with unit Coulomb-meter, located at the position vector $\boldsymbol\tau$ via the following equations:

\begin{eqnarray}
\label{bv1}
&\sigma_c \nabla^2_{\boldsymbol\tau} u_c(\rvec,\boldsymbol\tau)=\nabla_{\boldsymbol\tau} \cdot \boldsymbol{Q} \delta (\rvec-\boldsymbol\tau), \quad \rvec \in \Omega_c,\\
&\sigma_c \frac{\partial u_c(\rvec,\boldsymbol\tau)}{\partial n}=\sigma_f \frac{\partial u_f(\rvec,\boldsymbol\tau)}{\partial n}, \quad \rvec \in S_c;\nonumber
\end{eqnarray}

\begin{eqnarray}
\label{bv2}
&\nabla_{\boldsymbol\tau}^2 u_f(\rvec,\boldsymbol\tau)=0, \quad \rvec \in \Omega_f, \\
&u_f(\rvec,\boldsymbol\tau)=u_c(\rvec,\boldsymbol\tau), \quad \rvec \in S_c,\nonumber\\
&\sigma_f \frac{\partial u_f(\rvec,\boldsymbol\tau)}{\partial n}=\sigma_b \frac{\partial u_b(\rvec,\boldsymbol\tau)}{\partial n}, \quad \rvec \in S_f;\nonumber
\end{eqnarray}
\begin{eqnarray}
\label{bv3}
&\nabla_{\boldsymbol\tau}^2u_b(\rvec,\boldsymbol\tau)=0,  \quad \rvec \in \Omega_b,\\
&u_b(\rvec,\boldsymbol\tau)=u_f(\rvec,\boldsymbol\tau), \quad \rvec \in S_f,\nonumber\\
&\sigma_b \frac{\partial u_b(\rvec,\boldsymbol\tau)}{\partial n}=\sigma_s \frac{\partial u_s(\rvec,\boldsymbol\tau)}{\partial n}, \quad \rvec \in S_b;\nonumber
\end{eqnarray}
\begin{eqnarray}
\label{bv4}
&\nabla_{\boldsymbol\tau}^2u_s(\rvec,\boldsymbol\tau)=0,  \quad \rvec \in \Omega_s,\\
&u_s(\rvec,\boldsymbol\tau)=u_b(\rvec,\boldsymbol\tau), \quad \rvec \in S_b,\nonumber\\
&\frac{\partial u_s(\rvec,\boldsymbol\tau)}{\partial n}=0, \quad \rvec \in S_s.\nonumber
\end{eqnarray}

The functions $u_j$ and $v_j$ are related by the equation
\begin{eqnarray}
\label{connection}
&u_j(\rvec,\boldsymbol\tau)=\frac{1}{4\pi}\boldsymbol{Q}(\boldsymbol\tau)\cdot \nabla_{\boldsymbol\tau} v_j(\rvec,\boldsymbol\tau), \\
& j\in \{f,b,s\}, \rvec \in \Omega_j, \boldsymbol\tau \in \Omega_c\nonumber.
\end{eqnarray}
It was shown in \cite{dassios2017,dassios2009,fokas2009} that for the the particular case of the spherical head model, $v_s(\rvec,\boldsymbol\tau)$ is given by
\begin{equation}
v_s(\rvec,\boldsymbol\tau)=\sum_{n=1}^{\infty} H_n \frac{2n+1}{4\pi}\frac{\tau^n}{r^{n+1}} P_n(\hat{\boldsymbol\tau} \cdot \rvecunit),
\label{vsansatz}
\end{equation}
where the  coefficients $H_n$ depend only on the conductivities and the geometry and do not depend on the neuronal current. Equation (\ref{vsansatz}) shows that the auxiliary function $v_s(\rvec,\boldsymbol\tau)$ depends on the three variables $(\tau,r,\rvecunit \cdot \hat{\boldsymbol\tau})$. It is worth noting that in the particular case of the spherical head model, $v_s(\rvec,\bold{0})=0$. \par

In the case of a realistic head model, OpenMEEG \cite{openmeeg2010} accurately solves the boundary value problem described by equations (\ref{bv1})-(\ref{bv4}) but not equations (\ref{bveqn1})-(\ref{bveqn4}). However, we can employ \textit{the fundamental theorem of line integrals} to approximate $v_s(\rvec,\boldsymbol\tau)$ from solutions of $u_s(\rvec,\boldsymbol\tau)=\frac{1}{4\pi}\boldsymbol{Q} \cdot \nabla_{\boldsymbol\tau}v_s(\rvec,\boldsymbol\tau)$ computed by OpenMEEG. \par
For completeness, we recall the \textit{fundamental theorem of line integrals}. Suppose that $C$ is a smooth curve given by $\underline{\mathbf{r}}(t)$, $a \leq t \leq b$ and let $f(\underline{\mathbf{r}}(t))$ be a function whose gradient $\nabla f(\underline{\mathbf{r}}(t))$ is continuous on $C$. Then,
\begin{equation}
f(\underline{\mathbf{r}}(b))-f(\underline{\mathbf{r}}(a))=\int_{C} \nabla f(\underline{\mathbf{r}}(t)) \cdot d\underline{\mathbf{r}}(t),
\label{fundlineinteg}
\end{equation}
where $\underline{\mathbf{r}}(a)$ and $\underline{\mathbf{r}}(b)$ represent the initial and final points on $C$, respectively (the above theorem holds in any number of dimensions). \par
If we let $f=v_s(\rvec,\boldsymbol\tau)$ and $\underline{\mathbf{r}}(t)=\hat{\boldsymbol\tau} t$, $0\leq t\leq \tau$ in equation (\ref{fundlineinteg}), then we find
\begin{equation}
v_s(\rvec,\boldsymbol\tau)-v_s(\rvec,\bold{0})=4\pi\int_{0}^{\tau} u_s(\rvec,t \hat{\boldsymbol\tau},\hat{\boldsymbol\tau}) dt,
\label{vslineinteg}
\end{equation}
where $v_s(\rvec,\bold{0})$ is a constant and $4\pi u_s(\rvec,\boldsymbol\tau,\hat{\boldsymbol\tau})=\hat{\boldsymbol\tau} \cdot \nabla_{\boldsymbol\tau}v_s(\rvec,\boldsymbol\tau)$. It is straightforward to compute numerically  the one-dimensional integral of equation (\ref{vslineinteg}). In the case of the spherical head model, $v_s(\rvec,\bold{0})=0$ (see equation (\ref{vsansatz})). If the centroid of the brain mesh of a realistic head model is denoted by $\bold{c}$, then $v_s(\rvec,\bold{c})$ is an additive constant and has no effect on the inversion formulas of equations (\ref{2ndexpress2}) and (\ref{2ndexpress4}). If we estimate  $\bold{c}$ from the triangulated surface mesh of the brain by fitting a sphere to its nodes, then a reasonable assumption is $v_s(\rvec,\bold{c})\approx 0$. The line integral of equation (\ref{vslineinteg}) has to be computed numerically.  \par
The accuracy of reconstructing $\Psi(\boldsymbol\tau)$ using the volume integral of equation (\ref{2ndexpress2}) is dependent on $v_s(\rvec,\boldsymbol\tau)$. One approach is to employ a large number of sources $\boldsymbol{Q}(\boldsymbol\tau)$, to generate the required data set $u_s(\rvec,\boldsymbol\tau,\hat{\boldsymbol\tau})$ and to approximate the line integral of equation (\ref{vslineinteg}), using a simplified integration method. However, this approach \textit{might be} computationally expensive and requires certain level of memory management. Furthermore, it is also well known that sources placed too close to the surface of cerebrum $S_c$, can result in large numerical errors in $u_s(\rvec,\boldsymbol\tau,\hat{\boldsymbol\tau})$, see \cite{openmeeg2010,openmeeg2011}. This in turn, will introduce errors in the upper limits of the line integral of equation (\ref{vslineinteg}). A computationally tractable alternative, that
can mitigate these errors involves  \textit{a carefully} constructed regression model of $u_s(\rvec,\boldsymbol\tau,\hat{\boldsymbol\tau})$ using data generated via OpenMEEG. The details of this construction are discussed in section \ref{surrogatemodel}.

\subsection{Regression Model of $u_s(\rvec,\boldsymbol\tau,\hat{\boldsymbol\tau})$ For A Realistic Head Model}
\label{surrogatemodel}
Regression models constructed using data generated by PDE solvers are referred to as \textit{surrogate models} \cite{forrester2008,simpson2008,forrester2009,grihon2013, pruett2016,raul2018}. They are essentially machine learning models, that \textit{approximate} a mapping between a set of inputs and outputs of a simulation. Suppose we are given a finite sample pairs of data, $(\bold{x}_n,\bold{y}_n)_{n=1}^{N}$ (the training data), where $\bold{x} \in \Reals{d_{in}}$ and $\bold{y} \in \Reals{d_{out}}$. These pairs of data represent the inputs and outputs of a computationally expensive simulation model $\bold{y}=f(\bold{x})$. Our goal is to construct a function $\hat{f}: \Reals{d_{in}} \rightarrow \Reals{d_{out}}$, which is as close as possible to the true function $f$. In this study, function $f=u_s(\rvec,t \hat{\boldsymbol\tau},\hat{\boldsymbol\tau})$ on the right hand side of equation (\ref{vslineinteg}). A vital step in constructing an accurate surrogate model is the correct identification of the inputs to the model. In this connection we use our insight from the analysis of the spherical head model: equation (\ref{vsansatz}) shows that $v_s(\rvec,\boldsymbol\tau)$ is a function of $r$, which is the radial distance from the center of the coordinate system to the sensor position of $\tau$, which is the radial distance from the center of the coordinate system to the source position vector, and of $(\rvecunit \cdot \hat{\boldsymbol\tau})$, which is the cosine of the angle between the source and observation unit vectors.  Thus, the necessary inputs to the surrogate model are $(r,\tau, \rvecunit\cdot \hat{\boldsymbol\tau})$. An accurate \textit{learning} of the function $u_s(\rvec,t \hat{\boldsymbol\tau},\hat{\boldsymbol\tau})$  depends on a dense sampling of the inputs $(r,\tau, \rvecunit\cdot \hat{\boldsymbol\tau}) \in [r_{min},r_{max}]\times [\tau_{min},\tau_{max}] \times [-1,1]$. The position vectors of the observation points and source points from the centroid $\bold{c}$ of the brain are denoted by  $\rvec$ and $\boldsymbol\tau$ respectively. The position vectors of the nodes of the brain with respect to $(0,0,0)$ are denoted by $\{\bold{x}_j \in \Reals{3}: 1\leq j\leq V_b\}$; similarly the position vectors of the nodes of the scalp are denoted by $\{\bold{y}_j \in \Reals{3}:1\leq j\leq V_s\}$. Recall that OpenMEEG solves the boundary value problem described by equations (\ref{bv1})-(\ref{bv4}). For any source-observation  pair $(\boldsymbol\tau,\rvec)$, OpenMEEG employs a current dipole source with moment $\boldsymbol{Q}(\boldsymbol\tau)$ Coulomb-meter. This solution, which has units Volts, will be denoted by $u_s(\rvec,\boldsymbol\tau,\boldsymbol{Q}(\boldsymbol\tau))$. The steps needed for generating the data points required for constructing the surrogate model are given below:

\begin{algorithm}
 \renewcommand{\thealgorithm}{}
\caption{: Steps for generating the data set $u_s(\rvec,\boldsymbol\tau,\boldsymbol{Q}(\boldsymbol\tau))$}
\label{array-sum}
\textbf{(1)} Fit a sphere or an ellipsoid to the realistic head model to estimate the centroid (centre).
 The position vector of this centroid is be denoted by $\bold{c}$.\\
\textbf{(2)} Translate the coordinate system from $(0,0,0)$, i.e,  $\forall j$, let $\boldsymbol\tau_j=\bold{x}_j-\bold{c}$, $\rvec_j=\bold{y}_j-\bold{c}$.\\
\textbf{(3)} For all the nodes of the brain mesh, compute $\hat{\boldsymbol\tau}_j:=\frac{\boldsymbol\tau_j}{|\boldsymbol\tau_j|}$. This is needed to define radially oriented sources.\\
\textbf{(4)} Find the minimum and maximum distance of the scalp nodes from the centroid $\bold{c}$, i.e., $r_{\textrm{min}}=\min\{|\rvec_j|:1\leq j\leq V_s\}$, $r_{\textrm{max}}=\max\{|\rvec_j|:1\leq j\leq V_s\}$.\\
\textbf{(5)}  Select $M$ nodes on the scalp (observation vectors) $\rvec_k$ in the interval $r \in [r_{\textrm{min}},r_{\textrm{max}}]$. \\
\textbf{(6)} It is well known that sources placed too close to the surface of cerebrum $S_c$, can result in inaccuracies in the numerical computation \cite{openmeeg2010,openmeeg2011}. Choose $V_b$  position vectors $3$ millimetres from each node of the brain mesh as source position vectors $\{\boldsymbol\tau^{s}_j:=\boldsymbol\tau_j-0.003\hat{\boldsymbol\tau}_j:j\leq V_b\}$ and set the corresponding dipole moment for every node as $\boldsymbol{Q}(\boldsymbol\tau_j)=\hat{\boldsymbol\tau}_j$ to compute the corresponding electric potential, $\{u(\rvec_i,\boldsymbol\tau^{s}_j,\hat{\boldsymbol\tau}_j): i\leq M, j\leq V_b\}$. \\
\textbf{(7)} Add $V_b$ sources at the centroid  $\bold{c}$, with the dipole moment $\boldsymbol{Q}(\bold{c})=\hat{\boldsymbol\tau}_j$, to compute $\{u(\rvec_i,\bold{c},\hat{\boldsymbol\tau}_j): i\leq M, j\leq V_b\}$.    \\
\textbf{(8)} Add $V_b$ sources at the position vectors $\frac{\tau_j}{2}\hat{\boldsymbol\tau}_j$ with dipole moments $\boldsymbol{Q}(\frac{\boldsymbol\tau_j}{2})=\hat{\boldsymbol\tau}_j$, to compute $\{u(\rvec_i,\frac{\boldsymbol\tau_j}{2},\hat{\boldsymbol\tau}_j): i\leq M, j\leq V_b\}$.
\end{algorithm}

The above steps will result in a total of $M \times 3 V_b$ data points. For the purpose of constructing the surrogate model it is convenient to organize the data as a matrix $\bold{d}\in \Reals{3V_bM \times 4}$. The first three columns ($r,\tau,\hat{\boldsymbol\tau}\cdot \rvecunit$) feature as inputs and the last column (electric potential) is the output. \par We employ the commercial software package pSeven by datadvance \cite{datadvance2018} to construct the regression model (machine learning model). A discussion of the features of pSeven and its comparison to open source software is presented in \cite{mikhail2016}. \par
A detailed and thorough investigation of machine-learning models (surrogate models) for the forward problem associated with EEG is outside the scope of this paper. It is work in progress and will be published elsewhere. It is noted that pSeven has a \textit{smart selection} option for scanning through a set of algorithms to select the best model amongst the set. It performs for each training set a numerical optimization of the technique as well as its parameters \cite{snoeck2012,bergstra2011} by minimizing the cross-validation error, see \cite{mikhail2016}. Among the algorithms scanned by pSeven are the following: ridge regression \cite{tikh1943}, stepwise regression \cite{efroymson1960}, elastic net \cite{zou2005}, Gaussian processes \cite{rasmussen2005}, sparse Gaussian processes \cite{candela2005,burnaev2015}, \textit{High Dimensional Approximation (HDA)} \cite{mikhail2016,haykin2008}, and \textit{High dimensional approximation combined with Gaussian processes (HDAGP)} (this technique is related to artificial neural networks and, more specifically, to the two-layer perceptron with a non-linear activation function\cite{haykin2008}). Two desirable features of pSeven are: (i) all data manipulation is done via graphical user interface (GUI) and (ii)  it can export the constructed surrogate model as a stand alone function in a number of scientific computing languages, including Matlab, C source for MEX, C source for stand alone program, C header for library, C source for library, functional mock-up interface (FMU) for Co-simulation $1.0$ and executable.
\section{Minimization and a Numerical Solution of The Inverse Problem}
\label{inverseproblem}
It is clear from equations (\ref{2ndexpress2}) and (\ref{2ndexpress4}) that the only component of the neuronal current $\Jvec(\boldsymbol\tau)$ that affects EEG data is $\Psi(\boldsymbol\tau)$. However, as stated earlier, even this scalar function can not be computed uniquely, unless one imposes an appropriate constraint. It was shown in \cite{fokas1996,Kurylev2012} that for a spherical conductor the $L_2$  norm of $\Jvec(\boldsymbol\tau)$ (minimal energy) yields a unique solution. Here, these results are generalized to the case of a realistic head model. Let us define the functional $E$ (energy) by
\begin{equation}
E=\int_{\Omega_c}|\Jvec|^2 dV(\boldsymbol\tau).
\label{energy}
\end{equation}
Using equation (\ref{helmddecomp}) we find that
\begin{eqnarray}
&|\Jvec|^2=\biggl( \nabla \Psi+\nabla \times \Avec\biggr)\cdot \biggl( \nabla \Psi+\nabla \times \Avec\biggr).\\
&=|\nabla \Psi|^2+ |\nabla \times \Avec|^2+2 \nabla \Psi \cdot \nabla \times \Avec. \nonumber
\label{helmddecompsq}
\end{eqnarray}
Hence,
\begin{equation}
E=\int_{\Omega_c} \biggl(|\nabla \Psi|^2+ |\nabla \times \Avec|^2+2 \nabla \Psi \cdot \nabla \times \Avec\biggr) dV(\boldsymbol\tau).
\label{energy2}
\end{equation}
However, it is shown in proposition $1$ and lemma $1$ of \cite{Cantarella2002} that
\begin{equation}
\int_{\Omega_c} \nabla \Psi(\boldsymbol\tau) \cdot \nabla \times \Avec(\boldsymbol\tau) dV(\boldsymbol\tau)=0.
\end{equation}
Thus,
\begin{equation}
E=\int_{\Omega_c} \biggl(|\nabla \Psi|^2+ |\nabla \times \Avec|^2\biggr) dV(\boldsymbol\tau).
\label{energy3}
\end{equation}
Taking into consideration that $u_s$ does not depend on $\Avec$ and that the second term of the right hand side of the above equation is positive, it follows that the minimization of (\ref{energy3}) is equivalent to minimizing
\begin{equation*}
\int_{\Omega_c} |\nabla \Psi|^2 dV(\boldsymbol\tau).
\end{equation*}
Thus, we have the following constrained minimization problem
\begin{eqnarray}
\label{optproblem}
& \min_{\Psi(\boldsymbol\tau)} \int_{\Omega_c} |\nabla \Psi(\boldsymbol\tau)|^2 dV(\boldsymbol\tau)\\
& \mathrm{s.t.} \qquad u_s(\rvec)=-\frac{1}{4\pi} \int_{\Omega_c} \nabla_{\boldsymbol\tau}^{2} \Psi(\boldsymbol\tau) v_s(\rvec,\boldsymbol\tau)dV(\boldsymbol\tau), \qquad \rvec \in S_s \nonumber,
\end{eqnarray}
The requirement of the minimum norm of $\nabla \Psi(\boldsymbol\tau)$ is a well documented choice of regularization, known as Tikhonov regularization \cite{kaipio2004,tikh1943}. It forces the estimate $\Psi \in \mathbb{H}^1(\Omega_c)$. The continuum form of the constrained optimization problem given by (\ref{optproblem}) does not take into account the impact of  measurement noise on data $u_s(\rvec)$. However, additive measurement noise is modelled in section \ref{expPsiRBFsec}.

\subsection{Expansion of $\Psi(\boldsymbol\tau)$ Using Radial Basis Functions}
\label{expPsiRBFsec}
For the numerical minimization of the constrained optimization problem given in (\ref{optproblem}), radial basis functions (RBF) \cite{fornberg2015} are employed. We consider an EEG electrode cap with $M$ electrodes and discretize the domain $\Omega_c$ using $N$ cubic voxels. The function $\Psi(\boldsymbol\tau)$ is expanded using inverse multiquadric RBFs \cite{fornberg2015}, namely
\begin{equation}
\Psi(\boldsymbol\tau)=\sum_{j=1}^{N}\lambda_j \phi(\|\boldsymbol\tau-\boldsymbol\tau_j\|),
\label{psiparam}
\end{equation}
where $\{\boldsymbol\tau_j \in \Omega_c, 1\leq j\leq N\}$ are the position vectors of the centre of each voxel, and $\{\lambda_j:1\leq j\leq N\}$ denote the RBF coefficients. The inverse multiquadric basis function $\phi(\|\boldsymbol\tau-\boldsymbol\tau_j\|)$ is defined by
\begin{equation}
\phi(\|\boldsymbol\tau-\boldsymbol\tau_j\|):=\frac{1}{\sqrt{\alpha^2+\|\boldsymbol\tau-\boldsymbol\tau_j\|^2}}.
\label{mqbasis}
\end{equation}
The coefficient $\alpha$ in equation (\ref{mqbasis}) is referred to as the \textit{shape parameter} and needs to be estimated from appropriate data. In this study we propose a robust non-linear least squares strategy for estimating $\alpha$ from a data set  $\{u(\rvec_j):1\leq j\leq M\}$ generated by a known $\Psi(\boldsymbol\tau)$ which has support in $\Omega_c$.  In this setting, a convenient representation of equation (\ref{psiparam}) takes the linear algebraic form
\begin{equation}
\Psiv= \mt{A}\boldsymbol{\lambda},
\label{psirbf}
\end{equation}
where $\boldsymbol{\lambda} \in \Reals{N}$ are the RBF coefficients. The entries of the matrix $\mt{A}\in \Reals{N \times N}$ are given by
\begin{equation}
\mt{A}[i,j]=\phi(\|\boldsymbol\tau_i-\boldsymbol\tau_j\|).
\label{Amatrix}
\end{equation}
If we assume that measurements are contaminated by additive noise, then the discretized form of equation (\ref{2ndexpress2}) is given by
\begin{equation}
\uvec=\mt{V_s} \PsiL+\mathbf{w},
\label{uvecdiscrete1}
\end{equation}
where $\uvec[i]:=u(\rvec_i)$, $\mt{V_s}[i,j]:=-\frac{v_s(\rvec_i,\boldsymbol\tau_j)}{4\pi N}$, $\PsiL[j]:=\nabla^2 \Psi(\boldsymbol\tau_j),\quad j=1,...,N$ and $\mathbf{w} \in \Reals{M}$ denotes the measurement noise vector. The only unknown in equation (\ref{uvecdiscrete1}) are the discrete Laplacian values $\PsiL \in \Reals{N}$. The action of the Laplacian operator $\mt{L}$ on the inverse multiquadradic basis function of (\ref{mqbasis}) yields
\begin{equation}
\mt{L}[i,j]=-\frac{3\alpha^2}{(\alpha^2+\|\boldsymbol\tau_i-\boldsymbol\tau_j\|^2)^{\frac{5}{2}}}, \qquad 1\leq i,j\leq N.
\label{Lapop}
\end{equation}
The optimization problem defined in (\ref{optproblem}) requires the computation of $\nabla\Psi(\boldsymbol\tau)$.  Partial derivative operators enable $\nabla \Psi(\boldsymbol\tau)$ to be computed at discrete points $\{\boldsymbol\tau_j\}_{j=1}^N$. The operators for computing $\frac{\partial \Psi}{\partial \tau_x}$, $\frac{\partial \Psi}{\partial \tau_y}$, $\frac{\partial \Psi}{\partial \tau_z}$  are respectively given by
\begin{equation}
\mt{D}_{x}[i,j]=- \frac{1}{\sqrt{N}}\frac{(\tau_{x,i}-\tau_{x,j})}{(\alpha^2+\|\boldsymbol\tau_i-\boldsymbol\tau_j\|^2)^{\frac{3}{2}}}, \quad i,j \leq N,
\label{Dx}
\end{equation}

\begin{equation}
\mt{D}_{y}[i,j]=-  \frac{1}{\sqrt{N}}\frac{(\tau_{y,i}-\tau_{y,j})}{(\alpha^2+\|\boldsymbol\tau_i-\boldsymbol\tau_j\|^2)^{\frac{3}{2}}}, \quad i,j \leq N,
\label{Dy}
\end{equation}

\begin{equation}
\mt{D}_{z}[i,j]=- \frac{1}{\sqrt{N}} \frac{(\tau_{z,i}-\tau_{z,j})}{(\alpha^2+\|\boldsymbol\tau_i-\boldsymbol\tau_j\|^2)^{\frac{3}{2}}}, \quad i,j \leq N.
\label{Dz}
\end{equation}

\subsection{Reconstructing $\Psi(\boldsymbol\tau)$}
\label{disreconPsisec}
The implementation of the constrained optimization problem given by (\ref{optproblem}) is well documented, see \cite{kaipio2004}. The shape parameter $\alpha$ that features in equations (\ref{Amatrix})-(\ref{Dz}) for computing the operators $\mt{A}$ $\mt{L}$, $\mt{D}_x$, $\mt{Dy}$ and $\mt{Dz}$ plays a central role in the accuracy and stability of reconstructions. In fact, the shape parameter $\alpha$ can be interpreted as the regularization parameter \cite{kaipio2004}.
Extensive numerical tests suggest that the optimal $\alpha$ is not sensitive to the choice of $\Psi(\boldsymbol\tau)$ but it depends on the geometry of the head model, position vectors $\{\boldsymbol\tau_j \in \Reals{3}:j=1,...,N\}$.  Recall the Laplacian operator $\mt{L}$ given by equation (\ref{Lapop}), by letting $\PsiL=\mt{L} \lambda$ in equation (\ref{uvecdiscrete1}), we find
\begin{equation}
\uvec=\mt{V_s} \mt{L}\boldsymbol{\lambda}+\mathbf{w},
\label{uvecdiscrete2}
\end{equation}
The model given by equation (\ref{uvecdiscrete2}) is linear in $\boldsymbol{\lambda} \in \Reals{N}$, non-linear in the shape parameter $\alpha$, and importantly it is a separable model \cite{kay1992}. The least squares error may be minimized with respect to $\boldsymbol{\lambda}$ and, thus it is reduced to a function of $\alpha$ only. The discretized form of the optimization problem of equation (\ref{optproblem}) is given by
\begin{eqnarray}
\label{optproblem2}
& \qquad \min_{\alpha,\boldsymbol{\lambda}} \|\mt{G}(\alpha) \boldsymbol{\lambda}\|^2 \\
& \mathrm{s.t.} \quad \uvec=\mt{V_s} \mt{L}\boldsymbol{\lambda}+\mathbf{w} \nonumber,
\end{eqnarray}
where $\|.\|^2$ is the  Euclidean norm and $\mt{G}(\alpha) \in \Reals{3N \times N}$ is defined as
\begin{equation*}
 \mt{G}(\alpha):=
  \left[ {\begin{array}{c}
   \mt{D_x}\\
   \mt{D_y}\\
   \mt{D_z}
  \end{array} } \right]
\end{equation*}
We employ the well documented weighted least squares method \cite{kay1992} to model the impact of the measurement noise $\mathbf{w}$ on the parameters $(\alpha,\boldsymbol{\lambda})$. The WLS method involves employing the covariance matrix of the measurement noise $\mt{C}_{\mathbf{w}}=\mathbb{E}[\mathbf{w} \mathbf{w}^{T}]$ in the cost function to be minimized. For convenience, we introduce the following notation :
 \begin{equation}
   \mt{H}(\alpha)=
  \left[ {\begin{array}{cc}
   \mt{C}^{\frac{1}{2}}_{\mathbf{w}}\mt{V_s}\mt{L} \\
   \mt{D_x}\\
   \mt{D_y}\\
   \mt{D_z}
  \end{array} } \right]
   \boldsymbol{\mathrm{b}}=
  \left[ {\begin{array}{cc}
   \mt{C}_{\mathbf{w}}^{\frac{1}{2}} \uvec \\
   \boldsymbol{0}\\
   \boldsymbol{0}\\
   \boldsymbol{0}
  \end{array} } \right],
\end{equation}
where $\mt{H}(\alpha) \in \Reals{(M+3N) \times N}$, $\boldsymbol{0} \in \Reals{N}$ and $\boldsymbol{\mathrm{b}} \in \Reals{M+3N}$. The optimization problem of equation (\ref{optproblem2}) is reduced to the following system of linear equations:
\begin{equation}
\label{optproblem3}
\boldsymbol{\mathrm{b}}=\mt{H}(\alpha) \boldsymbol{\lambda}.
\end{equation}
The dependency of $\mt{H}(\alpha)$ in equation (\ref{optproblem3}) on the parameter $\alpha$ is via the RBF matrices $\mt{L}$, $\mt{D}_x$, $\mt{D}_y$ and $\mt{D}_z$, respectively given by equations (\ref{Lapop})-(\ref{Dz}). The WLS error of equation (\ref{optproblem3}) reads as
\begin{equation}
J(\alpha,\boldsymbol{\lambda})=(\boldsymbol{\mathrm{b}}-\mt{H}(\alpha) \boldsymbol{\lambda})^{T} (\boldsymbol{\mathrm{b}}-\mt{H}(\alpha) \boldsymbol{\lambda}).
\label{costlambda}
\end{equation}
The $\boldsymbol{\lambda} \in \Reals{N}$ that minimizes equation (\ref{costlambda}) for a given $\alpha$ is given by
\begin{equation}
\hat{\boldsymbol{\lambda}}=(\mt{H}^T(\alpha) \mt{H}(\alpha))^{-1} \mt{H}^T(\alpha) \boldsymbol{\mathrm{b}}.
\label{lambdaest}
\end{equation}
The resulting WLS error is given by
\begin{equation}
J(\alpha,\hat{\boldsymbol{\lambda}})=\boldsymbol{\mathrm{b}}^T\biggr[\bold{I}-\mt{H}(\alpha) \biggl( \mt{H}^{T}(\alpha) \mt{H}(\alpha)\biggr)^{-1} \mt{H}^{T}(\alpha) \biggr] \boldsymbol{\mathrm{b}},
\label{costFalpha2}
\end{equation}
where $\bold{I} \in \Reals{(M+3N) \times (M+3N)}$ is the identity matrix. Hence, the problem of finding the optimal value of the shape parameter $\alpha$ is reduced to the following 1-D maximization problem :
\begin{equation}
\alpha_{\textrm{opt}}=\arg \max_{\alpha} \biggl[ \boldsymbol{\mathrm{b}}^T \mt{H}(\alpha) \biggl( \mt{H}^{T}(\alpha) \mt{H}(\alpha)\biggr)^{-1} \mt{H}^{T}(\alpha) \boldsymbol{\mathrm{b}}\biggr].
\label{alphaopteqn}
\end{equation}
The $\boldsymbol{\lambda}$ that minimize equation (\ref{costlambda}) for the optimal shape parameter $\alpha_{\textrm{opt}}$ is given by
\begin{equation}
\hat{\boldsymbol{\lambda}}=(\mt{H}^T(\alpha_{\textrm{opt}}) \mt{H}(\alpha_{\textrm{opt}}))^{-1} \mt{H}^T(\alpha_{\textrm{opt}}) \boldsymbol{\mathrm{b}}.
\label{lambdaoptimal}
\end{equation}
The optimal shape parameter $\alpha_{\textrm{opt}}$ is employed in equation (\ref{Amatrix}) to compute the RBF matrix $\mt{A}$. Finally,  $\hPsi$ is given by
\begin{equation}
\hPsi=\mt{A}\hat{\boldsymbol{\lambda}}.
\label{Psiestimate}
\end{equation}

\section{Numerical Results}
\label{numerics}
In the interest of reproducibility and to encourage further research in this direction, the code (surrogate model and reconstruction routines), head model (triangulated meshes) and the electrode positions are made available via Github \url{https://github.com/parham1976}.  The realistic head model considered in this study, is taken from the sample data set in the OpenMEEG package. The number of nodes and triangles of the surface meshes  is shown in table \ref{nestedopenmeeg}. The centroid of the brain mesh is  $\bold{c}=[-0.0043,0.0169,0.0672]$ (this provides the center of the coordinate system for constructing the surrogate model).
\begin{table}[H]
\begin{center}
\begin{tabular}{ |c||c|c|c|| }
 \hline
 Surface meshes & Number of nodes& Number of triangles \\
 \hline
 Cerebrum: $S_c$ &2562& 5120 \\
 \hline
 Skull $S_b$ &2562& 5120 \\
 \hline
 Scalp $S_c$ &2562& 5120 \\
 \hline
 \end{tabular}
 \end{center}
 \caption{This table shows the number of nodes and triangles in the realistic head model.}
 \label{nestedopenmeeg}
 \end{table}

\subsection{Surrogate Model}
In section \ref{forwardproblem}, the steps for generating the data set required for constructing the surrogate model were described. Those steps will yield $3V_bM$ data points. In this study, we have $122$ electrodes, i.e., $M=122$ and $V_b=2562$ as given in table \ref{nestedopenmeeg}. So, a total of $3V_bM=937692$ data points are generated via OpenMEEG. The data set is split into a training and a test data set. A total of $890807$ data points were randomly selected for training and the remaining $46885$ data points are used for testing. For the problem under study and the $890807$ samples chosens as training data, the \textit{smart selection} setting in pSeven package \cite{datadvance2018}, chose a two layer neural network. We employ the error metrics relative mean distance measure (RDM)$\in [0,2]$, where minimum RDM$=0$, as well as the natural logarithm of the magnification factor ($\ln$(MAG)) \cite{vorkwek2012,Reichenbach2010}. RDM is defined by
\begin{equation}
\textrm{RDM}= \sqrt{\sum_{i=1}^n\biggl( \frac{u^{s}_i}{\sum_{j=1}^n (u^{s}_j)^2}-\frac{u^{o}_i}{\sum_{j=1}^n (u^{o}_j)^2}\biggr)^2},
\end{equation}
where $u^s_j$, $u^o_i$  respectively, denote the electric potential (Volts) estimated by the surrogate model and the OpenMEEG package (reference). The natural logarithm of the magnification factor is given by
\begin{equation}
\ln[\textrm{MAG}]=\ln\biggl[ \sqrt{\frac{\sum_{i=1}^n (u^s_i)^2}{\sum_{i=1}(u^o_i)^2}}\biggr].
\end{equation}
Figure \ref{crossvaldiationplot} depicts the comparison of the solution obtained via OpenMEEG with the surrogate model. For clarity of presentation only $1000$ points from the test data set of $46885$ have been plotted. However, the values of RDM$=0.2102$  and $\ln$[MAG]$=-0.0208$, are computed using all $46885$ samples of the test data. In Figure \ref{crossvaldiationplot}, the blue solid line with cross is the solution obtained via OpenMEEG and the red solid line with circle is the output of the surrogate model.
\begin{figure}[H]
 \includegraphics[scale=0.83]{cross_valiation.eps}
  \caption{This figure depicts the comparison of the solution obtained via OpenMEEG(BEM)  and via the surrogate model on the test data set for $1000$ sample points. It shows the comparison of the solution of $\hat{\boldsymbol\tau}_i \cdot \nabla_{\boldsymbol\tau} v_s(\rvec_i,\boldsymbol\tau_i)$ obtained via OpenMEEG and via the surrogate model. The solid blue line with cross is the OpenMEEG solution and the solid red line with circles is the surrogate model solution. The relative mean distance measure (RDM) is $0.2102$ and the natural logarithm of the magnification factor ($\ln$(MAG)) is $-0.0208$.}
  \label{crossvaldiationplot}
\end{figure}

\subsection{Reconstruction of $\Psi(\boldsymbol\tau)$}
\label{numreconPsisec}
We present numerical results using synthetic data. In order to avoid the inverse crime \cite{kaipio2004}, we generate the data using a different model to that employed for reconstruction. We assume that $\Psi(\boldsymbol\tau)$ takes the form
\begin{equation}
\Psi(\boldsymbol\tau)=\sum_{i=1}^4 \alpha_i e^{-\beta_i \|\boldsymbol\tau-\boldsymbol\tau_i\|^2},
\label{psiknown}
\end{equation}
where the values of $\{\alpha_i,\beta_i, \boldsymbol\tau_i:1\leq i\leq 4\}$ are given in the table \ref{tablpsiparam}.
\begin{table}[H]
\begin{center}
\begin{tabular}{ |c|c|c|c|c| }
 \hline
 $\alpha_i$ & $\beta_i$&$\boldsymbol\tau_{x,i}$&$\boldsymbol\tau_{y,i}$&$\boldsymbol\tau_{z,i}$\\
 \hline
  $2$&$1e3$&  $-0.0638$ &$-0.0185$ & $0.0546$\\
\hline
 $-2$&$1e3$&  -0.0424  &  0.0630  &  0.0324\\
\hline
 $2$&$1e3$&  0.0076 &  -0.0185  &  0.1139\\
\hline
 $-2$&$1e3$&  0.0433   &-0.0366 &   0.0398\\
\hline
 \end{tabular}
 \end{center}
 \caption{The table shows the values of the parameters $\{\alpha_i,\beta_i, \boldsymbol\tau_i:1\leq i\leq 4\}$ used for generating data.}
 \label{tablpsiparam}
 \end{table}
The Laplacian of equation (\ref{psiknown}) is given by
\begin{equation}
\nabla^{2}\Psi(\boldsymbol\tau)=\sum_{i=1}^4 2\alpha_i \beta_i e^{-\beta_i\|\boldsymbol\tau-\boldsymbol\tau_i\|^2}  (-3+2 \beta_i \|\boldsymbol\tau-\boldsymbol\tau_i\|^2).
\label{LapPsiknown}
\end{equation}
The steps for generating the numerical results presented in this section are given below:
\begin{enumerate}
\item The head model will have a sensor array of $M=122$ sensors. Setup a uniform Cartesian grid in the domain $\Omega_c$. Discretize $\Omega_c$ using $N=3293$ voxels. The length of the edge for each voxel is $h=0.0074$ meters.
\item Compute and store $\mt{V_s}[i,j]:=-\frac{v_s(\rvec_i,\boldsymbol\tau_j)}{4\pi N}$. The auxiliary function matrix  $\mt{V_s} \in \Reals{M \times N}$.
\item Use in equation (\ref{LapPsiknown}) the values of $\{\alpha_i,\beta_i, \boldsymbol\tau_i:1\leq i\leq 4\}$ given in table \ref{tablpsiparam}  to compute
$\{\nabla^2 \Psi(\boldsymbol\tau_j):j=1,...,N\}$. Use in equation (\ref{uvecdiscrete1}) $\mt{V_s}$ and $\PsiL[j]:=\{\nabla^2 \Psi(\boldsymbol\tau_j)\} $  to generate synthetic data, i.e., $\uvec=\mt{V_s}\PsiL+\mathbf{w}$. Here, we have employed additive white Gaussian noise (AWGN), with a signal to noise ratio of $20$ dB.
\item Solve the optimization problem of equation (\ref{alphaopteqn}) to find the optimal shape parameter, $\alpha_{\textrm{opt}}$ and estimate $\hPsiL$. The optimal shape parameter is found to be $\alpha_{\textrm{opt}}=0.0169$.
\item Use $\alpha_{\textrm{opt}}$ to compute the linear operator $\mt{A}$ given by equation (\ref{Amatrix}). The reconstruction is given by equation (\ref{Psiestimate}), i.e., $\hPsi=\mt{A}\hat{\boldsymbol{\lambda}}$.
\end{enumerate}
The matrix of auxiliary function values $\mt{V}_s \in \Reals{122 \times 3293}$. It has a rank($\mt{V}_s$)=$122$ and condition number $\kappa(\mt{V_s})=7.9804 \times 10^{3}$. The matrix $\mt{H}(\alpha_{\textrm{opt}}) \in \Reals{10001 \times 3293}$, which features in equation (\ref{optproblem2}) has a rank($\mt{H}(\alpha_{\textrm{opt}})$)$=3293$ and a condition number $\kappa(\mt{H}(\alpha_{\textrm{opt}}))=8.4017 \times 10^{7}$. \par
Figures \ref{reconstruction1}(a)-\ref{reconstruction1}(d) depict the comparison between $\Psiv$ and $\hPsi$ for position vectors inside the domain $\{\boldsymbol\tau_j \in \Omega_c: 1\leq j\leq N\}$ (internal nodes). To facilitate visualization of the function values $\Psiv$ and $\hPsi$ inside the domain $\Omega_c$, the position vectors $\boldsymbol\tau_j \in \Omega_c$  are divided into four non-intersecting sets. The blue solid line with cross are the function values $\Psiv$ given by equation (\ref{psiknown}) and red solid line with dots is the estimate $\hPsi$ given by equation (\ref{Psiestimate}).  The domain $\Omega_c$ has been discretized using a uniform Cartesian grid, i.e., $N=3293$ cubic voxels. Subplot (a) depicts the comparison of the function values  of $\Psiv$ and $\hPsi$ for the set of position vectors $\{\boldsymbol\tau_j \in \Omega_c: \tau_x>0, \tau_y>0\}$. Subplot (b) depicts the comparison of the function values of $\Psiv$ and $\hPsi$ for the set of position vectors $\{\boldsymbol\tau_j \in \Omega_c: \tau_x<0, \tau_y>0\}$. Subplot(c) depicts the comparison of the function values of $\Psiv$ and $\hPsi$ for the set of position vectors $\{\boldsymbol\tau_j \in \Omega_c: \tau_x>0, \tau_y<0\}$. Finally, subplot (d) depicts the comparison of the function values of $\Psiv$ and $\hPsi$ for the set of position vectors $\{\boldsymbol\tau_j \in \Omega_c: \tau_x<0, \tau_y<0\}$. The root mean square error  (RMSE) of $\Psiv$ and $\hPsi$ is given by
\begin{equation}
\mathrm{RMSE}=\biggl[ \frac{1}{N} \sum_{i=1}^{N} \biggl( \Psi(\boldsymbol\tau_j)-\hat{\Psi}(\boldsymbol\tau_j)\biggr)^2 \biggr]^{\frac{1}{2}}.
\label{rmse}
\end{equation}
Using the analytic expression of $\Psi(\boldsymbol\tau)$ given by equation (\ref{psiknown}) and $\hPsi$ given by equation (\ref{Psiestimate}) in equation (\ref{rmse}), we find that RMSE=$0.1122$.

\begin{figure}[H]
\includegraphics[scale=0.78]{reconstruction1.eps}
\caption{This figure depicts the comparison of equations (\ref{psiknown}) and (\ref{Psiestimate}). The blue solid line with cross is $\{\Psi(\boldsymbol\tau_j): \boldsymbol\tau_j \in \Omega_c\}$ as given by equation (\ref{psiknown}) and the red solid line with dots is $\{\hPsi(\boldsymbol\tau_j):\boldsymbol\tau_j \in \Omega_c\}$ as given by equation (\ref{Psiestimate}). Subplot (a) depicts the comparison of the function values  of $\Psiv$ and $\hPsi$ for
the set of position vectors $\{\boldsymbol\tau_j \in \Omega_c: \tau_x>0, \tau_y>0\}$. Subplot (b) depicts the comparison of the function values of $\Psiv$ and $\hPsi$ for the set of position vectors $\{\boldsymbol\tau_j \in \Omega_c: \tau_x<0, \tau_y>0\}$. Subplot(c) depict the comparison of the function values of $\Psiv$ and $\hPsi$ for the set of position vectors $\{\boldsymbol\tau_j \in \Omega_c: \tau_x>0, \tau_y<0\}$. Subplot (d) depicts the comparison of the function values of $\Psiv$ and $\hPsi$ for the set of position vectors $\{\boldsymbol\tau_j \in \Omega_c: \tau_x<0, \tau_y<0\}$. The root mean square error (RMSE)=$0.1122$.}
\label{reconstruction1}
\end{figure}
Figure (\ref{reconstruction2}) depicts the comparison of $\Psiv$ as given by equation (\ref{psiknown}) on the surface $S_c$ with $\hPsi$ given by equation (\ref{Psiestimate}) on the surface $S_c$.
\begin{figure}[H]
\includegraphics[scale=0.6]{reconstruction.eps}
\caption{This figure depicts the comparison of equations (\ref{psiknown}) and equation (\ref{Psiestimate}) on the surface of the cerebrum $S_c$. Subplots (a) and (b) depict $\Psiv$ on the surface $S_c$ as given by equation (\ref{psiknown}) and subplots (c) and (d) depict $\hPsi$ on $S_c$ as given by (\ref{Psiestimate}). The root mean square error (RMSE)=$0.1122$.}
\label{reconstruction2}
\end{figure}
The reconstructions in figure \ref{reconstruction2} are plotted using the open source fieldtrip package \cite{Fieldtrip}.
\section{Conclusion}
\label{conclusion}
OpenMEEG package solves the electric potential (Volts) $u_s(\rvec,\boldsymbol\tau)=\frac{1}{4\pi}\boldsymbol{Q}(\boldsymbol\tau)\cdot \nabla_{\boldsymbol\tau} v_s(\rvec,\boldsymbol\tau)$ for a given dipole source  but not $v_s(\rvec,\boldsymbol\tau)$. To this end we propose a strategy involving 1-D line integrals to estimate $v_s(\rvec,\boldsymbol\tau)$ by constructing a surrogate model from data generated via OpenMEEG. The inputs to the surrogate model were selected based on insight provided by the analytic equation (\ref{vsansatz}), i.e., $v_s(\rvec,\boldsymbol\tau)$  for the spherical head model. The surrogate model was constructed using the pSeven Datadvance surrogate modelling toolbox \cite{datadvance2018}.
A total of $890807$ data points were randomly selected for training and the remaining $46885$ data points are used for testing. For this construction, the smart selection routine of Datadvance chose the high dimensional approximation \cite{mikhail2016}, which is essentially a two-layer neural network. The relative mean distance measure (RDM) is $0.2102$ and the natural logarithm of the magnification factor ($\ln$(MAG)) is $-0.0208$.  \par
The irrotational component of the neuronal current denoted by the scalar function $\Psi(\boldsymbol\tau)$ has been parametrized using inverse multiquadric radial basis functions (RBFs) on a uniform Cartesian grid inside the cerebrum of a realistic head model. By employing synthetic data, the shape parameter denoted by  $\alpha$ is estimated  using a computationally efficient approach and was estimated to be $\alpha_{\textrm{opt}}=0.0169$. Furthermore, it is found that the shape parameter $\alpha$ is not sensitive to data, but depends on the configuration of the position vectors $\{\boldsymbol\tau_j \in \Omega_c: 1\leq j\leq N\}$ of the centres of the RBF expansion. The choice of RBF and $\alpha_{\textrm{opt}}$ yields the condition number $\kappa(\mt{H}(\alpha_{\textrm{opt}}))=8.4017 \times 10^{7}$ for the inversion matrix $\mt{H}(\alpha_{\textrm{opt}})$. Moreover, the regularization strategy involving minimizing the energy is in fact equivalent to Tikhonov regularization. Reconstructions are shown using synthetic data with a root mean square error (RMSE)=$0.1122$. The complete set of MATLAB files (surrogate model, inversion code and head model) as well as data set for reproducing the results in this paper are  available from \url{https://github.com/parham1976}.

\section*{Acknowledgement}
A.S. Fokas and P. Hashemzadeh are grateful to EPSRC and centre for mathematical imaging in healthcare (CMIH) for partial support. The grant number for this project is EP/N014588/1. C.B. Sch\"{o}nlieb acknowledges support from the EPSRC grant number EP/N014588/1. The authors are grateful for helpful comments, feedback and discussions with Martin Buhmann who is with the department of mathematics at the University of Giessen, as well as Jingwei Liang and No\'{e}mie Debroux who are with the Department of Applied Mathematics and Theoretical Physics, university of Cambridge.
\section*{References}


President Barack Obama got his first job scooping ice cream at a Baskin-Robbins in Honolulu. He also worked as a waiter in an assisted living facility for seniors and as a painter before going to law school. Michelle took her first job in a book-binding shop.
In a joint interview with Parade Magazine the Obamas shared how these minimum-wage jobs that paid little but required hard work helped them develop life-long work ethics.
My last year in high school, I worked at a bindery, side by side with grown-ups who had been there their entire lives. Knowing that I, as a 16-year-old, was getting the same income and doing the same work… it gave me respect for those workers. But it also gave me an understanding that more is needed for folks to be able to cobble together a decent life on minimum wage.
The Obamas also feel that their daughters Malia and Sasha should work minimum wage jobs and get a taste of what it's like to put in hard work.
We are looking for opportunities for them to feel as if going to work and getting a paycheck is not always fun, not always stimulating, not always fair. But that's what most folks go through every single day.
I agree with the Obamas. Teenagers learn many important lessons from those first jobs. Like the President, I got my start as an ice cream scooper at a gelato shop in high school. Getting to work on time, dealing with grumpy customers, closing the cash registrar, setting the alarm — these were all responsibilities that helped me develop the confidence I need to survive in the real world.
Do you think it's important for kids, even those who don't necessarily need the money, to work minimum-wage jobs in their teens?


You might soon be able to pay for that Uber ride or that AirBnB reservation using bitcoin. The Wall Street Journal reports that eBay has been working to integrate bitcoin into its Braintree payments subsidiary.
Who uses Braintree for backend payment processing? GitHub, Hailo, Fab.com, LivingSocial, OpenTable, Hotel Tonight, and many many others.


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text:
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The Startups businesses make progress at snail's pace during their first days . They toil hard to meet their workforce and other costs which are frequently paid such as rent , salaries , utility bills etc . although ,wise startups always prefer to save some balance amount to cover the costs of routine business.
This is very important for startups in order to survive for long time with huge or average investment . Then , as the company grows and sales increase , it starts support its running costs easily . Physical address is considered positive sign for any company since it increases the chances of reliability and results positive feedback than totally virtual companies .
The physical address gives unique exposure and identity to your business unlike search engine king and influential leader , Google whose physical address is also unique , Mountain view ,CA or popularly known as silicon valley . Means , your address should be so impressive that it names the street as your company not the people get direction of popular buildings or offices . The physical address has more value in business so the people always value such practices .
The website at same time gives your business a good exposure and you can display your services , Products and portfolio easily on the website which increases the chances for your revenue and your company gives a professional look . professional developed website is the great identity of your company and you can cash the same by selling the products online , online orders placement or free quotes for the customers who approach your for any specific service or product .
ALSO READ : Five Tips for Startup Owners to Launch their Startups Professionally
Avoid making free websites which are not trusted and contain limited services which may not satisfy you or your business customers since it does not contain unique domain name along with your company logo which are supposed to be necessary for your business . Always engage a professional Web design firm or It consultancy firm to make your website more user centered and friendly so that the customers may experience very easy and smooth navigation and find the services of their choice and place orders for the business .
Related Topics:addresspostal addresswebsite
Five Reasons Why Startups should Choose .Com Domains for their Startups sites to maximize visibility
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Social media has offered amazing tools and apps that have revolutionized the lifestyle of people. Social networks always keep you connected with your friends, colleagues and family. Some creative apps have become very popular in recent times especially among the enthusiastic youth who create funny clips with the help of TikTok.
Tiktok is the greatest platform for creating short mobile videos. These videos contribute to inspire creativity and bring smiles on the faces of people even amid pandemic and lockdowns. The app has become very popular even in Pandemic and as some of the best informative and educational content went Viral.
Tiktok is the subsidiary of Grand Tech Startup Byetdance -a leading Chinese Startup with mushrooming growth. TikTok has always been expanding its reach worldwide due to its aggressive expansion programs. At present, Tiktok has offices in Los Angeles, New York, London, Paris, Berlin, Dubai, Mumbai, Singapore, Jakarta, Seoul, and Tokyo.
Tiktok was founded in 2012 by Zhang Yiming. Video sharing is the most popular segment among youth and the company has tapped tremendous growth. The Company is also called douyin in China but its global version is called Tiktok. The continual growth empowered it to acquire its American rival app musical.ly. The Company has expanded its outreach globally.
New Dimensions i.e. Online Education
Continuing Growth amid Pandemic
Hiring than Firing amid Pandemic: A big surprise
Increasing Visibility of Byetdance Globally
Will the bubble burst or Sustain Market Shocks?
The covid-19 has affected the Education badly throughout the world and the online digital education has been getting ground. Most of the Schools, Colleges, Universities, Coaching centres and Academies have anchored to Online Education such as live lectures, quizzes, Presentations and Webinars. Tiktok has announced to enter the online Education in India as some of the content uploaded on Tiktok related to Education and learning has become viral earning millions of views and generating great revenue prompting Tiktok to enter the world of Education.
ALSO READ : Four Reasons why you should develop A Mobile APP for Your Startup Company
Tiktok has already interacted with content creators and firms in India to Provide innovative Learning material as all the educational institutions are closed for an indefinite period and Tiktok finds it the perfect time to enter the world of Education.
According to statistics that social Media app used by more than 200 million users every month in India and Tiktok will cover the range of Science and Math related topics to help students learn from this service.
It has partnered with tech startups Vedanta, Toppr, Made Easy and Gradeup that will produce educational content for TikTok. It is also collaborating with social enterprises Josh Talks and the Nudge Foundation to mentor 5,000 people across India.
According to reports of Forbes and Bloomberg, Byetdance surpasses Uber growth as it has become the top Startup with value at $78 Billion. Byetdance is the parent company of Tiktok that runs various services. Byetdance has made London as its strong base owing to trade war of the US with China in the aftermath of the Wuhan covid-19 outbreak. Trump Administration has imposed limitations on Chinese companies to pressurize China to accept US' demands. Byetdance has attracted many professionals from Google, Yahoo, Microsoft and Apple to establish a strong hub at London to expand its outreach in Europe.
It is really surprising that when the biggest companies and Franchises are constantly laying off their employees owing to prolonged lockdowns due to Covid-19, The Parent company of Tiktok, the Byetdance has gone on a hiring spree and plans to recruit 10000 professionals in engineering, software and marketing fields that has stunned the world as even tech giants Google, Yahoo, Microsoft have laid off their staff and closed their offices over the covid-19 outbreak advising their employees to work from home.
ALSO READ : Five Hiring Mistakes Startups Should Avoid to escape Disaster
"Hiring spree on the part of Byetdance amid pandemic is a pleasant surprise for the entrepreneurs, Tech giants and Experts globally. "
With the ambitious journey of growth and reaping the enormous benefits, the company has no more remained a secret for the world as its subsidiary Tiktok has broken all the records of usability and popularity exploring new frontiers with successful footprint. The recent leap of Tiktok to launch its Education initiative in India will further its grip in the Asian markets.
The Byetdance has established footprints in Asia and Europe building strong basis and marketing its services, it is estimated that it can sustain the jolts of pandemic and will retain its position as some reports regarding bubble reputation will fade away forever as the popularity graph of the company maintains upward trend and keeps going.
The majority of the world population is youth so if Tiktok focuses on youth-related interests by adopting AI practices, it has already achieved the milestone.
The Tech experts argue that TikTok has foreseen opportunity amid pandemic when the entire markets have crashed, Offices, Business, schools, colleges have been closed and stock markets crashed, even in such circumstances, if a company keeps hiring that means it established its strong bases so the bubble is not going to burst likely in post-Covid-19 World provided that it adheres to its policies of customer retention, market intelligence and the most important Artificial intelligence to explore the interests of users and offering more relevant content.
Besides profits, Tiktok pledged a huge amount for Vaccine development for Africa under its Tiktok for Good endeavours.
Social Media has witnessed a mushroom growth that has impacted the discourse of political, cultural and religious systems by providing the equal opportunity of freedom of expression and of sharing an opinion or viewpoint on any issue.
Social Media has played a greater role in rightwing politics that paved the way for populist politicians to reach out to their voters.
Though Social Media has connected the people around the world, at the same time, it is causing division or disintegration and facilitating the social media lobbyists to polarize the communication so much the people support the arguments or opinions or political tirades against leaders without fact checking.
The world has been rocked by social networks such as Twitter, Facebook, LinkedIn and YouTube because the users of these social networks have already surpassed millions and are growing, but due to the limited regulatory framework, fake news, cyberbullying, extremism and terrorism has reached new heights.
The influencers' interactions with social media users have polarized the political and social circles that led to the formation of online echo chambers, a tactic of social media recommendations by politicians to influence the users to strengthen their narrative regardless of facts that whether it is right or wrong, but it is knit in a manner that echo chambers appear to be normal for the people thus changing the way of thinking and increasing the supporters base.
The echo chambers post such content that we enjoy a lot while reading, while keeping us away from the content that may stir a debate or arguments or expression of our disagreement over certain social, economic, religious, cultural or policy matters.
The echo chambers keep ideas flowing on a regular basis to the one side –the positive side of the argument, while its negative side is deliberately concealed to hide the facts.
This is the big issue that might be discussed with students, who should be informed and taught at school, college or degree level so that positive and constructive use of this digital medium could be ensured.
Social media connectivity around the world is undoubtedly amazing but the problems it created for both people and students are very dangerous, and need to be addressed and regulated.
Digital media, especially social media, for a long time remained unbridled to post any content that may fall in the purview of hatred, racism, bullying, derogatory remarks, character assault, rebellion, or anti-state elements instigating people to create chaos using the olive branch of social media. Since it was not monitored given the type of content being circulated and the information of people posting such content, including their motives behind such content as may have a constant flow of information that is recommended by the users pretending to be professionals, experts and mentors.
Alarmingly, people without fact-checking go on sharing the content and thus creating hype for such an issue as does not have a basis and was the result of a conspiracy to exert pressure on some person or leaders to get some favours.
"The schools and colleges should chalk out the initiative to create awareness and practice of fair use so that social platforms should not post single or one-sided opinions or arguments but should welcome both so that positive use could be ensured. This will further the importance of social media."
Sometimes, the active echo chambers or volcanoes erupt over such issues as may be beneficial for the wider circle of society, but this finds serious criticism masterminded by people with an individual approach. They term it detrimental for them since it did not serve their interests, though it might have been beneficial for general people.
Humans have an intuitional inclination to communicate with like-minded people or the people with the same choices on social media, thus the echo chambers get even stronger. The discourse that discourages dissenting voices always comes under fire from the thinkers and intellectuals of the nation since it is tantamount to concealing the facts from the people.
The world is coming closer after the covid-19 pandemic that engulfed the entire world, bringing misery and economic crisis. The social media debates go with biased and unbiased approaches regarding getting the vaccine. Some optimists termed it great, while some pessimists or biased approaches raised questions.
The echo chambers try to influence their opinions and win support.Similarly, echo chambers of both Democrats and Republicans only canvassed for their candidates but the leaders instigated the masses to attack Capitol Hill through social media and emotional speeches using their influence on social media, especially Twitter .
Some researchers have found that echo chambers influence people to win support but social media influencers have a follower-base of millions, and may polarize the public and politics, since influencers share success stories like internet marketers compelling you to buy their products.
The opinion leaders and experts reveal that for the last two decades, the percentage of American people having consistently holding liberal or conservative beliefs— rather than a blend of the two, which is the case for most people— has increased from 10 percent to over 20. Beliefs about the other side are becoming more negative creating an alarming situation.
Since 1994, the number of people who see the opposing political party as a threat to "the nation's well-being or security" has doubled, which heralds how deepening polarization has predictable results raising eyebrows.
The government shutdowns, violent protests, scathing attacks on elected officials, the capitol Hill incident –all indicate that social media have been polarizing the political system and divided the nation each passing day due to echo chambers of both mainstream parties regarding the issues of health, security, human rights, employment and environment.
According to evidence from empirical research conducted on US politicians, politicians following extreme ideologies attracted a larger public audience than those who were moderate, which is the real example of echo chambers.
Usually, messages containing extreme thoughts, emotional instigation, patriotism, or criticizing the opposition received a warm welcome on social media platforms than those messages reflecting merely information.
The polarization of issues and political ideology will have serious repercussions in future that should be addressed on a timely basis so that societies could be saved from further isolation and disintegration.
The issues could be resolved through educating the people and creating awareness about the fair and professional use of social media channels for advocacy for winning support for legislation, an issue, a policy or employment.
Instead of showing the dark side of the issue, we have to educate people regarding both pros and cons of matters of importance for the general public rather than misleading the people with biased approaches to oppose the initiative of the government, community or civil society for the general benefit.
ALSO READ : Email Marketing: why and How?
Social Media network operators are the bigwigs and the most influential in this.They cannot be controlled but only be regulated by framing such laws that may ensure fair use of this digital platform and are careful so that users may not face any legal issue by discussing sensitive issues such as desecration of holy places, religions, personalities, anti-state tirades, rebellion, terrorism, extremism and so on. The sensitive issue may stir protests and cause legal proceedings, if the cyber-regulatory laws are implemented to discourage the influence of one-sided opinions or campaigns.
We cannot deny the importance of social media platforms as they provide equal opportunity to people to express and share their thoughts or opinions, but safe and thoughtful use of these platforms will encourage healthy discourse and prove pivotal in promoting dialogue to discuss social, economical and cultural issues peacefully without hitting the sentiments of other communities.
Thus, echo chambers will be less powerful to build a narrative that is appealing but the readers will be engaging themselves in the debates that need interaction.
Undoubtedly, social media platforms have changed the political and social approach of the people and enabled face-to-face digital interaction through video and audio conference calls, so their fair use will mitigate their cons manifold.
The social media networks apply algorithms to connect you with like-minded people but it is up to you whether your engagement in communication is either unbiased or biased.
Social media should be promoting positive discourse to explore the political and economic solutions to the problems of people by reflecting the two sides of the coin so that the discourse be constructive and problem-solving. Social media has given a voice to marginalized people and governments are compelled to resolve the issues of people on a priority basis due to trending topics of social media.
Even the populist leaders may face resistance from social media as the public makes the most of the freedom of expression and criticizes the role of politicians for their failure to address the issues of the public.
In recent times, Facebook, YouTube and Twitter have been creating history in highlighting issues of both public and national interest but at the same time such platforms have been taken by storm by those influencers who continue to feed the news about their vested interests and create psychosocial hype for any issue that may jolt the power corridors of Government.
The politicians always take to Twitter for their precise viewpoint and policy debate and are usually followed by millions of followers. Even the print and electronic media take the tweets as policy statements or the narrative of some party or group.
WHATSAPP Privacy Concerns Affecting Public Data -MOIT&T Pakistan
Startuspro News
Reference to on-going news threads by National Media and social media platforms regarding change in privacy terms & conditions of WhatsApp, Ministry of IT & Telecom is monitoring the current developments and clarifications provided by Facebook Inc. in this regard.
It is brought to notice that subject changes in privacy are applicable on WhatsApp business account only, while regular non-business/ individual profiles/ accounts are not affected.
MOIT&T would like to emphasize here to all such digital social media platforms including WhatsApp administration to adhere by privacy rights of citizens of Pakistan. In this regard, all such digital platforms need to strengthen their engagements with the Government of Pakistan so that concerns of General Public and businesses can be well addressed by all means.
ALSO READ : Which is better for Startups : Free or Paid Website

redpajama_set_name: RedPajamaCommonCrawl


FL Technics has successfully completed a cooperation project launched together with the United States Armed Forces. Under the project, FL Technics provided support services as well as secured appropriate conditions for the maintenance of heavy-lift helicopters CH-47 Chinook conducted by the U.S. Army 160th SOAR (A).
The U.S. Armed Forces chose Lithuania as its base for carrying out maintenance works on its military MH-47Gs (one of the latest versions of CH-47 Chinook) while preparing for the annual Exchange Training JCET 2016 in Europe. Operated and serviced by specialists from the 160th SOAR (A), MH-47Gs reached Kaunas International Airport in the beginning of April.
The 160th Special Ops Aviation Regiment (Airborne), also known as the Night Stalkers, is a special operations force of the United States Army that provides helicopter aviation support for general purpose forces and special operations forces. With a fleet of over 180 rotorcraft, including MH-47G Chinooks, A/MH-6M Little Birds and MH-60M Black Hawks, the 160th SOAR (A) is engaged in attack, assault, and reconnaissance missions.


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Q: Issue with Devise on Heroku I am facing issues after deploying my rails app on Heroku. My app essentially is an online cab booking platform. I have used Devise to authenticate users. I have several users in the Users table in the SQLite DB in the dev env. As such, in the dev env, when I try to log into my app using any of these users, it works perfectly. However, when I deploy the same on Heroku, and try to login using the same user credentials, it shows me "Invalid email or password" error.
I checked the Heroku logs, can't understand where I am going wrong. Please help!
2014-12-05T03:43:18.023059+00:00 app[web.1]: Processing by Devise::SessionsController#new as HTML
2014-12-05T03:43:18.017343+00:00 app[web.1]: Started GET "/users/sign_in" for 114.79.138.136 at 2014-12-05 03:43:18 +0000
2014-12-05T03:43:18.031662+00:00 app[web.1]:   Rendered vendor/bundle/ruby/2.0.0/gems/devise-3.4.1/app/views/devise/shared/_links.html.erb (0.4ms)
2014-12-05T03:43:18.031791+00:00 app[web.1]:   Rendered vendor/bundle/ruby/2.0.0/gems/devise-3.4.1/app/views/devise/sessions/new.html.erb within layouts/application (6.7ms)
2014-12-05T03:43:18.033007+00:00 app[web.1]: Completed 200 OK in 10ms (Views: 8.3ms | ActiveRecord: 0.0ms)
2014-12-05T03:43:18.395813+00:00 heroku[router]: at=info method=GET path="/favicon.ico" host=www.taxibol.in request_id=e7c11827-dbf9-4b6b-a208-e6c9c3b0cca2 fwd="114.79.138.136" dyno=web.1 connect=1ms
service=6ms status=304 bytes=133
2014-12-05T03:43:46.219043+00:00 app[web.1]: Started POST "/users/sign_in" for 114.79.138.136 at 2014-12-05 03:43:46 +0000
2014-12-05T03:43:46.226183+00:00 app[web.1]: Processing by Devise::SessionsController#new as HTML
2014-12-05T03:43:46.226214+00:00 app[web.1]:   Parameters: {"utf8"=>"???", "authenticity_token"=>"+EssZfe0ztWQvwNXt0u7/j8ntpBIDdM/l/DusVTXRZI=", "user"=>{"email"=>"raju.s@merucabs.com", "password"=>"[
FILTERED]", "remember_me"=>"0"}, "commit"=>"Log in"}
2014-12-05T03:43:46.331970+00:00 app[web.1]:   Rendered vendor/bundle/ruby/2.0.0/gems/devise-3.4.1/app/views/devise/shared/_links.html.erb (0.4ms)
2014-12-05T03:43:46.332041+00:00 app[web.1]:   Rendered vendor/bundle/ruby/2.0.0/gems/devise-3.4.1/app/views/devise/sessions/new.html.erb within layouts/application (3.2ms)
2014-12-05T03:43:46.332936+00:00 app[web.1]: Completed 200 OK in 107ms (Views: 4.5ms | ActiveRecord: 0.0ms)
2014-12-05T03:43:46.221273+00:00 app[web.1]: Processing by Devise::SessionsController#create as HTML
2014-12-05T03:43:46.221311+00:00 app[web.1]:   Parameters: {"utf8"=>"???", "authenticity_token"=>"+EssZfe0ztWQvwNXt0u7/j8ntpBIDdM/l/DusVTXRZI=", "user"=>{"email"=>"raju.s@merucabs.com", "password"=>"[
FILTERED]", "remember_me"=>"0"}, "commit"=>"Log in"}
2014-12-05T03:43:46.224795+00:00 app[web.1]: Completed 401 Unauthorized in 3ms

A: I don't believe the data in the local dev SQLite DB will be imported into your production database on Heroku.
When you deploy that site, you're going to get an empty database which has migrations run on it so the app functions, but no data is populated unless you create seed data (more on that later).
You can either manually create those users using rails console via heroku's command line tools (https://devcenter.heroku.com/articles/getting-started-with-rails4#console) or sign up as a new user on production than give yourself admin access via the console.
The last way you can do it is by seeding your database with an admin user and storing the username / password in configuration variables. Basically you setup a seed (http://edgeguides.rubyonrails.org/active_record_migrations.html#migrations-and-seed-data) with an admin user and setup the username / password via config vars (https://devcenter.heroku.com/articles/config-vars). You want to use config vars because putting usernames and passwords in a seed file is insecure.
I've never had this specific issue, but I've deal with similar ones before, and I think this will work for you.
--Corey

redpajama_set_name: RedPajamaStackExchange


just wanted to know if anybody was going to ride out there?
ill be on a white exodus mtb about 6:30 am if anyone wants to join.
Ahhh, Fully Loop....there used to be a guy named Stephen that 'owned that loop...maybe he will chime in.


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The Trust's Data Policy
Hogarth's House
Hogarth and Chiswick
Sergeant Painter to the King
Letters Patent, 1757, purchased
Letters Patent, 1757, Text
Replicas of the Garrick urns
Hogarth's House Centenary 2004
The Hogarth Birthday Parties
Hogarth's statue
Statue picture gallery
High Road House sponsors information panel
Pug statue for Hogarth School
Hogarth's House restoration 2009-11
Rosalind Elliott (1933-2020)
We are saddened to report the death on 19 December 2020 of one of our long-serving supporters, the actor Rosalind Elliott. She was one of the local residents who enthusiastically raised money to pay for the millennium statue of Hogarth in Chiswick High Road before our charity existed. She joined our executive committee in 2010, becoming a trustee in 2013. She was serving her second term of office in this role at the time of her death. We shall miss her contrib-utions to our meetings – she was funny, thoughtful and sometimes quite forthright.
Rosalind with one of the replica urns on the day they were installed
Rosalind was one of the performers at our special event at St Nicholas Church to mark the 250th anniversary of Hogarth's death in 2014. She was also the main fund-raiser for the Trust's project to reinstate urns on the gate piers at Hogarth's House.
While a grant paid for the restoration of the originals Rosalind persuaded theatrical friends and local residents to fund the creation of handsome replicas to fix to the gate piers.
At events she was a wonderful hostess, ensuring glasses were topped up and guests were enjoying themselves. We shall miss her enormously.
New website for Hogarth's House
As the "supporters' club" for Hogarth we have used our own website to focus on Trust activities and the work we have done in partnership with the team at Hogarth's House and in Hounslow Council's arts and heritage section.
Now we are delighted that Hogarth's House has launched a website here: https://hogarthshouse.org/ This is part of the Mulberry Garden Project which has received funding from @HeritageFund UK, Hounslow Council, our trust and many other trusts and foundations. We eagerly await the re-opening.
With wonderful images and learning materials it will be the best source of information about the House, the Mulberry Garden, the new learning studio and future activities – do sign up for news from the website and follow @HogarthsHouse on Twitter


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News, Rescue Horses
Rescue horses can come in all shapes and sizes and backgrounds. But when people think about rescue horses, it's usually a starved and neglected horse that comes to mind. Obviously that is a worst-case scenario, and those horses should always be under the care and advice of a treating veterinarian who will closely monitor their health. On the flip side, for rescue horses that are healthier, they may just need some help to optimize their health before being sent on to a new home.
We talked to two veterinarians and the founder of a rescue to determine the conditions they see the most, how they help cases of starvation or neglect, when fish oil is wise to use, and when adding any fat source, including fish oil, is not recommended.
Caring for a Rescue Horse
Regardless of the reason a horse ends up unwanted, it is crucial to put a neglected horse under the care of a good veterinarian.
"First, you need to review the basics to make sure that proper dentition, proper deworming, and making sure there's no underlying disease," said Fernando Cardenas, DVM, who also recommends doing bloodwork.
Cardenas, who has been a veterinarian for 19 years, is the owner of 3H Veterinary Services and Rehabilitation Facility in New Hill, NC. He has been a veterinarian at top international FEI competitions and is also a top grand prix show jumper with his Oldenburg stallion, Quincy Car.
Harry Werner, VMD, owner of Werner Equine in North Granby, CT, has had a lifelong passion for animal care and welfare. He has treated horses in southern New England and eastern New York since he opened his clinic in 1979. He is the American Veterinary Medical Association's liaison to the Unwanted Horse Coalition, has worked with the American Association of Equine Practitioners to affect equine welfare and public policy, and speaks internationally on the topic of equine welfare.
Werner also stressed the importance of shelter, routine wellness care, and vaccinations.
"Routine wellness care is usually not present," Werner said. "If the previous owner wasn't feeding them, chances are they weren't deworming them or vaccinating them or providing dental care."
"When they've been starved and neglected their thermoregulation capabilities are not normal," continued Werner. "Obviously the skinny horse is going to get a lot colder in the wintertime in the cold weather. Adequate shelter, clean fresh water 24/7, a trace mineral salt block…those are all things that should be provided."
He also stresses the importance of farrier care since many times neglected horses have not had adequate foot care and are at risk of laminitis or other foot-originated lamenesses.
It will take time for a neglected horse to recover from the health conditions that may plague him. He could be suffering from parasites, overgrown hooves, skin conditions, and illnesses. Quarantining them is highly recommended. Keep in mind that sometimes it may be better for the horse to deal with increasing his weight and getting gut function back to normal before challenging the horse's system with deworming and vaccinations. Consult a veterinarian to create a plan on how to get the horse well.
Refeeding a Starved or Neglected Horse
Once these horses are rescued, the phrase "kill them with kindness" can often be unintentionally taken literally. Oftentimes people will want to feed these horses as much as they can and/or give them free choice of whatever feed will make them gain weight the fastest. However, this can be detrimental and sometimes fatal.
Werner has only seen a limited number of starvation cases himself. When he does need to rehabilitate a horse, he follows the refeeding guidelines created by the University of California-Davis. Just as in humans suffering from starvation, refeeding done incorrectly can lead to heart and respiratory problems, kidney failure, and even death.
The guidelines recommend feeding high-quality alfalfa hay in small meals initially, and then gradually increasing meal size while decreasing the number of meals fed over time. Weight gain usually begins after a month of refeeding, since in severe cases normal metabolism and gut function must be restored first. No treats or grain are recommended until the horse is further along in his recovery, because it can delay the return to normal metabolic function or cause death. It usually takes three to five months for the horse to gain enough weight to be a normal body weight, depending on his size.
When to Use Fish Oil
While healthy horses are often given a fat source to help them gain weight, Werner recommends against adding any fat source to the diet of a neglected horse for the first four to six weeks. After that, fat can be added gradually.
"I would not include a fat source in the beginning," said Werner. "The reason for that, extracted from the data from the University of California-Davis, where they showed a significant drop in serum magnesium and phosphorus when the horses were fed in this case corn oil. In my world, I just don't go there with any fat substance early in the process."
Once a horse is in a healthier state, or if the horse was not starved in the first place, fish oil can often be a great addition to the diet. Cardenas agrees with that thought.
"Using fish oil as a source of fat for weight gain is a healthy way to do it, because you are avoiding the carbohydrate overload that can sometimes cause a drop in the acid in the GI system. It's what we call acid gut," said Cardenas, who added that ulcers and colic can be caused by acid gut. "A fat works very differently, and so it is safe to use and also has some benefits in preventing or treating ulcers. And it gives them a slow release of energy, and therefore that energy will be converted into weight gain as well. It will do it without disrupting the GI system, unless you do too much. If you overdo it in fat, your horse will start having a loose stool."
Cardenas noted that most feeds don't have enough DHA and EPA for what horses need. "Fish oils are going to be better than plant-sourced oils," he added. "The plant source is going to have more of the omega-6s in it. The omega-6s are inflammatory. With fish oil, you have a lot more of the omega-3s, and a better balance between the omega-3s and the omega-6s."
While Werner said he has never fed fish oil, he said that the omega-3 fatty acids could be beneficial due to their purported ability to reduce inflammation once the horse has reached a body condition score of at least a four. A score of 4 on the widely used Henneke Body Condition Scoring System means the horse has reached "Moderately Thin" with a "slight ridge along the back; faint outline of the ribs discernible; and withers, shoulders, and neck not obviously thin."
Cardenas recommends Wellpride to his clients, uses it on his horse, and takes Wellpride himself. He added that fish oil can help those horses that also suffer from skin problems and who may need to have sustained energy.
Brent Glover, founder of Orphan Acres in Viola, ID, also uses fish oil, but more so for the benefits of the omega-3 fatty acids than as a fat source for starved horses. Orphan Acres has had more than 4,200 horses placed there (rehoming almost 3,000 of those horses) since 1975, and Glover was helping horses in need even before he founded Orphan Acres.
"I've used fish oil with the horses to help with arthritic problems, because the omega-3 fatty acids are also supposed to help with the arthritis, and it's supposed to help lubricate the joints," said Glover.
Glover starts his refeeding process with lower grades of hay and gradually builds up the amount of protein and eventually switches to a higher-quality hay. "It's a process," he added. "You just can't go from 0 to 1,000 in one day."
Rehabilitating a severely neglected horse should be done under the care of a veterinarian. The UC-Davis refeeding guidelines are a good place to start. Once the horse has reached a body condition score of 4 or more, then a fat source, such as Wellpride, can be added for further weight gain as well as the benefits of the EPA and DHA omega-3s.
With 270 kCals/ounce, 4.5 grams of EPA, 3 grams of DHA, and 2 grams of other omega-3s, along with vitamin E, Wellpride has been proven as a beneficial supplement for horses with a variety of ailments and can also be used to help optimize a horse's health.
Please Note: Those who are quoted in this article are not representatives of Wellpride and do not necessarily use the brand, or fish oil in general. However, we have noted those who do within the article.
« A Natural Aid for Horses with Gastric Ulcers
What the Studies Say About Equine Asthma and Omega-3 »


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Co-op Credit Union
CCU SECURE EMAIL
Club Savings Accounts
Cub Account for Kids
Members Money Savings
IRA Deposit Accounts
WINcentive Savings Account
Agriculture Business Loans
Virtual Branch
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OriginalChloeAoki 6 stories
Chloe is saved from abuse by Rascal. Romance blossoms.
Fantasy / Romance
OriginalChloeAoki
(PLEASE DON'T READ IF YOU GET TRIGGERED BY THINGS SUCH AS ABUSE AND RAPE!!! YOU HAVE BEEN WARNED. Also, I'm using the AU where Chloe didn't have parents or a brother and instead had an abusive Uncle. I read it in another person's story but I can't remember who the author was! Sorry! Also, there will be a fair bit of backstory before we get to the main Chloe x Rascal bit.)
Rascal the joker/jester. He was cunning, manipulative, sadistic. But, he had a soft side for someone. Someone who... made him feel happy and warm. Chloe Aoki. She made him... complete. But, she had a secret. A very dangerous secret that could get her hurt.
Long story short (and to have a bit of backstory), Chloe was a victim of abuse. Chloe never had parents. She was kicked out of her house and put up for adoption as a baby. Even then, no-one wanted her. Chloe's uncle adopted her and treated her like a slave. She has acknowledged this and, instead of complaining and whining, she is dilligent and is subservient to her uncle even when he throws his alcohol bottles at her. She thought everyone went through this and, therefore, began to look it up. You see, she was fortunate enough that her uncle went out to the pub every day so she could sneak onto his computer and look things up.
She was shocked when she found out this was not normal and instead was something called "abuse." Even when a website called ChildLine told her to contact someone if this went on, she didn't and continued as normal because she was afraid of her uncle.
To rub salt on the wound, Chloe was forced to wear a maid's dress when serving her uncle and she was to call him "master". If she was ever disobediant, she would be punished which usually included being whipped and used as a target whenever her uncle was throwing alcohol bottles at the wall.
Chloe was also a victim of rape. She would've called the police YEARS ago but... this man had... her secrets. Dirty secrets. And... the fact that Chloe was a Glitter Force Warrior. To understand why this is so bad, we need to go back...
Once, somebody spread a rumour that Chloe was a magical girl. Of course, people didn't believe the rumour but they began to ridicule and bully her. They humiliated her. They wrote things on her desk like: "KILL YOURSELF!" "I dare you to go jump off the roof!" "DIE- from the whole class!" and other things. However, Chloe didn't look depressed on the outside. Instead she hit back at the rumours with comebacks such as "Jump off? I'll do a friggin BACKFLIP off the roof AND LAND IT!" (Bijuu Mike? Is that you in a blue wig?)
But, Chloe began to slowly feel more and more depressed and just decided to move schools and transfer to Rainbow Hills High.
Anyway, enough of the backstory of the backstory.
Chloe always tried to fight the rapist but her efforts led to nothing. He always made her be on birth control so she wouldn't get pregnant with a "bastard child" as he called them. The rape started when she was seven years old. She was now fourteen years old. Seven years... SEVEN YEARS this had been going on. And, to make it worse, she had a crush on her arch-nemesis Rascal.
Of course, he would never love her back. If anything, he would laugh in her face and begin to mock her. Or so she thought.
Okay, so now we move onto the main story. Everybody BUCKLE UP cause here we go!
Rascal was looking for an opportunity to cause chaos when he felt a surge of negative energy. He smiled evily and went straight towards the negative energy. However, what he found was not what he expected at all.
It was Chloe, crying in the middle of the woods. Rascal felt this... strange feeling... sympathy and pity but also.... sadness. He felt sad for this girl, clearly hurt, crying and sobbing into her knees.
Rascal approached Chloe with great caution. "Breeze?" He asked, "What's wrong?"
Chloe shot up with fear in her puffy, red eyes. "R-Rascal!!! Don't come any closer or... or I'll transform!"
"Hey! I just wanna know what's wrong." Rascal convinced her to sit down and talk about it.
(You guys already know Chloe's backstory!)
"A-And I managed to escape his house but I have nowhere to go! A-And to make matters worse, my maid's dress is getting quite tight around my chest and-" Chloe stopped when she noticed a blush creeping up both of their faces. "N-Nevermind."
Rascal couldn't stop staring at Chloe's tight maid dress. He's 17! Jeez! Give him a break!
"Chloe... I didn't realise! I'm so sorry." Rascal said.
"It's fine... there's nothing you could do about it." Chloe said, sadly, "Well, I better get some rest."
"Wait! Stay with us in the Shadow Realm! We won't hurt you, I promise. And if one of those idiots does hurt you, I'll kill them." Rascal proposed.
"O-Okay. But... I have to ask my childhood friend first."
"Who would that be?"
"April."
Rascal's eyes widened. "Y-You don't mean Glitter Spring, do you?"
"Yes. I do. But, if she is really my friend, she won't judge me on this." Chloe smiled sweetly.
"Okay. Meet me in this exact spot in a few hours." Rascal put down a two sticks in the shape of an X so they could remember where the spot was.
"Okay." And so, Chloe ran. She ran to April.
"APRIL!" She yelled, "I need your opinion!"
"What's up Frost Queen?" April asked.
"Y-You know how I live with my uncle?" April nodded and Chloe continued, "I ran away because of... things. A-And... Rascal has offered to have me stay in the Shadow Realm. He promised to not let anybody hurt me."
April was silent for a moment before smiling. "Of course you can go with him! And, from the way you look at him during our battles, I think you have a crush on him!"
Chloe smiled and hugged April. "Thank you so much April!!!! I promise I won't turn evil." She paused for a second, "Could you lend me some clothes to change into?"
"Of course." April handed Chloe some clothes and Chloe changed into them.
"Thanks April!"
Chloe then left and met Rascal at that same spot.
"So... what did Spring say?" Rascal asked.
"She said yes!" Chloe beamed and hugged Rascal, "Thank you. Thank you for being their for me!"
Rascal didn't know how to respond to this but he did blush and hug back. There was a warm feeling in his chest. Love... that's what it was called. Love.
When Rascal and Chloe teleported to the Shadow Realm, the other generals gasped.
"Look~ Clown Boy's got a girlfriend~ A Glitter Girl~" Cooed Ulric as both Chloe and Rascal blushed.
The other generals laughed as Rascal took Chloe to his room.
"Sorry we don't have a spare room. You'll have to sleep with me." Rascal said, smiling nervously.
"I-I wouldn't mind that..." Chloe muttered, red-faced.
"Awww~" They both heard. It was Brute, Brooha, Ulric and, surprisingly, Nogo in his smaller form. Nogo was human-sized now but he still had his mask on with his purple outfit.
Rascal closed the door in annoyance and went back over to Chloe. "So... what do you want to do first?"
"The windows. I'll scrub them first so that way I won't drip when-" Chloe started but Rascal cut her off by chuckling.
"Princess, you don't need to do anything like that! You're living here, not working here."
Chloe giggled. "Sorry, I'm just so used to working and now... it's gonna be so strange! Just to be clear, I'm still gonna fight against you guys so don't expect me to go easy on you."
"It's fine. I might go easy on you though." Rascal hugged Chloe and caressed her face, "you're beautiful."
Chloe blushed and looked away shyly.
Rascal and Chloe went to the dining area with the other generals and Nogo. She was surprised they didn't have much to eat.
"U-Um..." She said shyly, "I see you don't have a lot to eat here so..." And then she brought out oen of her famous batches of cupcakes. "Here. I hope you enjoy them." She smiled sweetly.
All the generals, including Rascal and Nogo, took a cupcake and enjoyed them. Chloe beamed as they let out surprised gasps and grunts of happiness. Her eyes sparkled when everyone complimented her on her cooking.
Nogo pulled her to one side.
"So... Chloe... You like to cook?" Nogo asked.
"Yeah. I guess it's just become second nature after... my uncle forced me to do it."
"Oh... I'm so sorry. You'll be safe here."
"Thank you. I um... I don't know how to say this but... I think I'm in love with Rascal." She blurted out.
Nogo smirked and chuckled. "Oh, we all know. We also know that he is head over heals in love with you."
Chloe blushed. "Really?"
Nogo told Chloe the plan and smirked as she made her way to his bedroom. When he entered, she sat there in a dress covered with hearts. Blue and Purple hearts.
She pulled Rascal close to her and kissed him with tongue. Rascal kissed back and it soon turned into a make-out session.
They soon pulled away because the heard wolf-whistles from the other generals and Nogo.
"R-Rascal... will you be my boyfriend?" Chloe asked.
"Yes, I will... princess~" Rascal then pulled her into another kiss and shut the door.
It was going to be an interesting night.
Did you enjoy my story? Please let me know what you think by leaving a review! Thanks, OriginalChloeAoki
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