text:
\section{Introduction\label{sec:intro}}

Model reduction is an essential requirement in almost all areas of science and technology to obtain efficient multi-parameter simulations and, in particular, optimization and control methods, see e.g.
\cite{Ant05,BenGW15,HesRS16}.
Often the full-order model (FOM) arises from a semi-discretization in space of a partial differential equation (PDE) and the state dimension scales with the number of grid points which is typically large.
However, one  is  usually not interested in a detailed description of the complete dynamics but often only in a low-dimensional manifold where the solution of interest approximately evolves.

Model reduction for nonlinear dynamical systems is often based on mode decomposition techniques as the proper orthogonal decomposition (POD) \cite{BenGW15,BerHL93,Vol01} or the dynamic mode decomposition \cite{KutBBP16,SchS08}.
Standard mode decomposition techniques are based on the concept of representing the unknown solution as a linear combination of modes.
More precisely, let $q$ be a function in space $x$ and time $t$ representing the state of the dynamical system, then a common model reduction ansatz is an approximation
\begin{equation}
	\label{eq:LCmodes}
	q\left(x,t\right) \approx \sum\limits_{k=1}^r {\alpha}_k\left(t\right){\psi}_k\left(x\right)
\end{equation}
with space-dependent modes ${\psi}_k$, time-dependent coefficients, or amplitudes, ${\alpha}_k$,
and $r$ is the number of modes.

While the amplitudes typically become the unknowns of the reduced-order model, the modes have to be determined in advance. To determine the modes, one typically simulates the system
and computes space- and time-discrete snapshots of a numerical approximation $q_m$ which are stored in a snapshot matrix $X\in\mathbb{R}^{m\times n}$, {i.\,e.}, $[X]_{ij} = q_m(x_i,t_j) \approx q(x_i,t_j)$ for $i=1,\ldots,m$ and $j=1,\ldots,n$. With the coefficients of the snapshot matrix one obtains a
discrete analogue of  \eqref{eq:LCmodes} as
\begin{equation}
	\label{eq:snapshotDecomposition}
	\left[X\right]_{j} \approx \sum\limits_{k=1}^r {a}_{k,j}{w}_k
\end{equation}
for $j=1,\ldots,n$, where $\left[X\right]_{j}$ denotes the $j$th column of $X$, ${w}_k\in\mathbb{R}^m$ are coefficient vector representations of the modes ${\psi}_k$, and ${a}_{k,j}$ are the corresponding amplitudes at time point $t_j$.

A classical way to obtain modes and amplitudes is the POD which is based on a singular value decomposition (SVD) of the snapshot matrix $X$. The POD  representation is optimal in the sense that it minimizes the residual in the discrete representation \eqref{eq:snapshotDecomposition}. The resulting reduced-order model is obtained as projection onto the span of the so obtained modes.

A common assumption is that POD delivers a good approximation of the form \eqref{eq:LCmodes} or \eqref{eq:snapshotDecomposition} with a small number $r$.
In many applications this assumption is valid and model reduction schemes like POD  lead to models with dimensions that are orders of magnitude smaller than those of the full-order model \cite{HolLB96}.

However, when the dynamics of the system features structures with high gradients that are propagating through the domain, then schemes of the form \eqref{eq:LCmodes} typically need a large number of modes to approximate the dynamics well, and hence model order reduction is not very effective.
To overcome this difficulty, recently, there have been several suggestions for model reduction of such transport-dominated systems.
In \cite{OhlR13} the authors use ideas of symmetry reduction to decompose the solution into a frozen profile and a translation group accounting for the transport.
The advantages over standard model reduction schemes are demonstrated by means of the Burgers' equation.
In \cite{RimML17_ppt} the authors present a method which is able to decompose multiple transport phenomena. The main ingredients are SVDs of several shifted snapshot matrices combined with a greedy algorithm. The method is cheap to apply but it often needs more shifted modes than necessary, as illustrated with results for the linear wave equation.
For further references on model reduction for transport-dominated problems, see \cite{AbgAC16,CagMS16_ppt,GerL14,IolL14,MojB17,SesS16}.
Most of these approaches consider transport-dominated systems with only one transport velocity and assume periodic boundary conditions.
However, in many applications, multiple transport velocities are encountered, {e.\,g.}, by different waves propagating through the domain.
To deal with such phenomena,  in \cite{ReiSSM17_ppt} the shifted POD (sPOD) method has been proposed  to obtain mode decompositions suitable for multiple transport phenomena. This new technique differs from \eqref{eq:LCmodes} by shifting the modes in space into different reference frames according to the different transports of the system, {i.\,e.},
\begin{equation}
\label{eq:shiftedSnapshotDecomposition}
q(x,t)\approx  \sum\limits_{\ell=1}^{{N_\mathrm{s}}} {\mathcal{T}_{\mathrm{per}}}\left({\Delta}^{\ell}(t)\right) \sum\limits_{k=1}^{r^\ell} {\alpha}^{\ell}_{{k}}(t){\psi}^{\ell}_{{k}}(x)
\end{equation}
where ${\mathcal{T}_{\mathrm{per}}}(\cdot)$ is a shift operator defined on a periodic domain $[0,L]$ via
\begin{equation*}
	\label{eq:periodicShiftOperator}
	{\mathcal{T}_{\mathrm{per}}}\left({\Delta}(t)\right)f\left(x,t\right)\vcentcolon=f\left(\left(x+{\Delta} (t)\right)\,\text{mod}\, L,t\right),
\end{equation*}
${N_\mathrm{s}}$ denotes the number of shifted reference frames, $\text{mod}$ denotes the \emph{modulo operator} reflecting the periodicity of the domain,
and ${\Delta}^\ell (t)$ are time-dependent shifts which track the locations of, {e.\,g.}, different wave profiles over time.
Similar as in the POD, one obtains a discrete analogue of \eqref{eq:shiftedSnapshotDecomposition} via
\begin{equation}
\label{eq:shiftedSnapshotDecomposition_discrete}
\left[X\right]_{j} \approx  \sum\limits_{\ell=1}^{{N_\mathrm{s}}} {T_{\mathrm{per}}}\left({d}^{\ell}_{j}\right) \sum\limits_{k=1}^{r^\ell} {a}^{\ell}_{k,j}{w}^{\ell}_{k}
\end{equation}
for $j=1,\ldots,n$, where ${T_{\mathrm{per}}}$ is a discrete approximation of ${\mathcal{T}_{\mathrm{per}}}$ and ${d}^{\ell}_{j}$ are shifts at discrete time points $t_j$.
In \cite{ReiSSM17_ppt} a heuristic algorithm is proposed to compute a decomposition of the form \eqref{eq:shiftedSnapshotDecomposition_discrete} in an iterative procedure, and it has been demonstrated that this approach is very successful for several examples including two separating vortex pairs and the linear wave equation. In the latter case the method needs less modes than other methods such as {e.\,g.}\ \cite{RimML17_ppt} and also retrieves the known analytic solution.

In this paper, we propose an optimization procedure to compute an \emph{optimal} decomposition of the form \eqref{eq:shiftedSnapshotDecomposition_discrete}. To this end, we generalize the cost functional
that is used to obtain the optimality of the POD method to the sPOD setting. We first consider
the optimization on the infinite-dimensional level, see Sec. \ref{sec:optimalspod}, and then present an algorithm which computes the decomposition in the fully discrete setting, see Sec.\ \ref{sec:optimization} and \ref{sec:algorithm}.
The computational cost is higher than for the method in \cite{ReiSSM17_ppt} but the obtained approximations are locally optimal in the sense that a  residual is minimized.

The focus of our work is on obtaining an optimal  mode decomposition which then can be used for the  construction of a reduced-order model, {e.\,g.}, by a Galerkin projection. A rigorous treatment of non-periodic boundary conditions is also discussed elsewhere.

To demonstrate the efficiency of the new approach, we present results for a pulsed detonation combuster (PDC). The snapshot data originate from a data assimilation, cf. \cite{GraLRPSM17}, and exhibit multiple transport phenomena which interact nonlinearly with each other and with the boundary.

\section{Optimal sPOD Approximation}\label{sec:optimalspod}

As a model problem for a partial differential equation whose solution features multiple transport velocities we consider the linear acoustic wave equation
\begin{equation}
\begin{aligned}
\partial_t \rho  +  {\rho_{\text{ref}}}   \partial_x u  &=& 0, \\
\partial_t  u   +  c^2/{\rho_{\text{ref}}} \partial_x \rho  &=& 0 ,
\end{aligned}
\label{eq:linearWave}
\end{equation}
on a one-dimensional spatial domain $\Omega= (0,1) $ with periodic boundary conditions.
Here, $u$ is the velocity, $\rho$ the density, ${\rho_{\text{ref}}}$ a reference density, and $c$ the speed of sound.
The analytic solution of \eqref{eq:linearWave} can be expressed as
\begin{equation}
\label{eq:analyticSolution}
\begin{bmatrix}
\rho\left(x,t\right)\\
u\left(x,t\right)
\end{bmatrix}
=
q_{-}\left(x+ct\right)
\begin{bmatrix}
{\rho_{\text{ref}}}\\
-c
\end{bmatrix}
+q_{+}\left(x-ct\right)
\begin{bmatrix}
{\rho_{\text{ref}}}\\
c
\end{bmatrix},
\end{equation}
where $q_{-}$ and $q_{+}$ are the Riemann invariants which are uniquely determined by the initial conditions. In the following, we use ${\rho_{\text{ref}}}=1$ and $c=1$ and we consider the initial conditions
\begin{equation*}
	\rho(x,0)=\rho_0(x)=\exp\left(-\left(\frac{x-0.5}{0.01}\right)^2\right),\qquad u(x,0)=u_0(x)\equiv 0,
\end{equation*}
which represent a pressure pulse with large gradients.
Due to these large gradients, the analytic solution, see Fig.\ \ref{fig:linearWave_FOM_snapshots2}, is very hard to approximate by a classical POD approach, since the singular values of the snapshot matrix, that is obtained by sampling the analytic solution, are decaying very slowly, cf.\ Fig.\ \ref{fig:svDecay}.
\begin{figure}[htb]
	\centering
	\includegraphics [width=0.325\linewidth]{linearWave_density}
	\includegraphics [width=0.325\linewidth]{linearWave_velocity}
	\caption{Linear wave equation: snapshots of the full-order solution for the density (left) and the velocity (right).}
	\label{fig:linearWave_FOM_snapshots2}
\end{figure}
\begin{figure}[htb]
	\centering
	\includegraphics [width=0.325\linewidth]{linearWave_singularValueDecay}
	\caption{Linear wave equation: singular value decay of the snapshot matrix.}
	\label{fig:svDecay}
\end{figure}
To demonstrate the difficulties that POD has for this problem consider the \emph{relative approximation error}
\begin{equation}
	\label{eq:errorMeasure}
	\left.\left(\sum_{j=1}^n \left\lVert \left[X\right]_j-\left[\tilde X\right]_j\right\rVert^2\right)\right/\left(\sum_{j=1}^n \left\lVert \left[X\right]_j\right\rVert^2\right)
\end{equation}
of an approximation $\tilde X$ of the snapshot matrix $X$, $\left\lVert\cdot\right\rVert$ being the Euclidean norm.

In this model problem, to obtain a relative error of less than $1\%$, the POD needs $124$ modes (cf.\ dashed lines in Fig.\ \ref{fig:svDecay}) although the analytic solution is simply represented by the sum of two shifted functions.
Indeed, the analytic solution \eqref{eq:analyticSolution} can be formulated within the more general representation \eqref{eq:shiftedSnapshotDecomposition} with only two modes and
\begin{alignat*}{2}
	&{N_\mathrm{s}}=2,\quad r_1=r_2=1,\quad &&{\Delta}^{1}(t)=-{\Delta}^{2}(t)=t,\quad{\alpha}^{1}_{1}(t)={\alpha}^{2}_{1}(t)\equiv 0.5,\\
	&{\psi}^{1}_{1}(x)=\rho_0(x)
	\begin{bmatrix}
		1 & -1
	\end{bmatrix}
	^T,\quad &&{\psi}^{2}_{1}(x)=\rho_0(x)
	\begin{bmatrix}
		1 &1
	\end{bmatrix}
	^T.
\end{alignat*}
However, the question arises how to compute such a decomposition when only snapshot data are available.
In this case the POD is optimal in the sense that it minimizes the residual, i.\,e., it solves the optimization problem
\begin{equation}
\label{eq:PODoptimization}
\min_{{\psi},{\alpha}} \int\limits_0^T\int\limits_\Omega \left(q\left(x,t\right)-\sum_{{k}=1}^r{\alpha}_{k}(t){\psi}_{k}(x)\right)^2 \mathrm{d}x \mathrm{d}t\quad \mbox{s.\ t.} \left\langle{\psi}_i(x),{\psi}_j(x)\right\rangle_{L^2(\Omega)}=\delta_{ij}
\end{equation}
for $i,j=1,\ldots,r,$ where $\delta$ denotes the Kronecker delta. In this way the modes ${\psi}_j$
form an orthonormal basis with respect to the $L^2$ inner product in $\Omega$.

To extend this optimality of \eqref{eq:PODoptimization} to the more general decomposition \eqref{eq:shiftedSnapshotDecomposition}, we consider the optimization problem
\begin{equation}
\label{eq:sPODoptimization_fixedShifts}
\min_{{\psi},{\alpha}}\int\limits_0^T\int\limits_\Omega \left(q\left(x,t\right)-\sum\limits_{\ell=1}^{{N_\mathrm{s}}} {\mathcal{T}_{\mathrm{per}}}\left({\Delta}^{\ell}(t)\right) \sum\limits_{{k}=1}^{r^\ell} {\alpha}^{\ell}_{{k}}(t){\psi}^{\ell}_{{k}}(x)\right)^2 \mathrm{d}x \mathrm{d}t,
\end{equation}
where for the moment we assume that the shift frames ${\Delta}$ are available  or can be approximated before the optimization for the modes ${\psi}$ and their time amplitudes ${\alpha}$ is carried out.
Methods to estimate these shifts based on given snapshot data have been discussed in \cite{ReiSSM17_ppt}.

A crucial difference between \eqref{eq:sPODoptimization_fixedShifts} and \eqref{eq:PODoptimization} is that \eqref{eq:sPODoptimization_fixedShifts} is an unconstrained optimization problem without the orthonormality restriction for the modes ${\psi}_j$. The reason why we drop this orthonormality requirement is that in a decomposition of the form \eqref{eq:shiftedSnapshotDecomposition} even linearly dependent modes may lead to optimal approximations.

To illustrate the necessity to allow linearly dependent modes, consider again the linear wave equation but this time only the density, {i.\,e.}, take $q(x,t)=\rho(x,t)$. In this case a solution of the optimization problem \eqref{eq:sPODoptimization_fixedShifts} is obtained with
\begin{alignat*}{2}
	&{N_\mathrm{s}}=2,\quad r_1=r_2=1,\quad &&{\Delta}^{1}(t)=-{\Delta}^{2}(t)=t,\\
	&{\alpha}^{1}_{1}(t)={\alpha}^{2}_{1}(t)\equiv 0.5, \quad &&{\psi}^{1}_{1}(x)={\psi}^{2}_{1}(x)=\rho_0(x),
\end{alignat*}
which means that there exists an optimal approximation with linearly dependent modes ${\psi}^{1}_{1}={\psi}^{2}_{1}$.

For this reason, we prefer not to enforce orthogonality of the modes in \eqref{eq:sPODoptimization_fixedShifts} as in the POD optimization problem \eqref{eq:PODoptimization}, at least when there is more than one transport velocity.

\section{Residual Minimization\label{sec:optimization}}

In this section we discuss the optimization problem \eqref{eq:sPODoptimization_fixedShifts} with a general linear shift operator ${\mathcal{T}}$, {i.\,e.}, we consider
\begin{equation}
	\label{eq:generalTransformedModesApproximation}
	\min_{{\psi},{\alpha}}\int\limits_0^T\int\limits_\Omega \left(q\left(x,t\right)-\sum\limits_{\ell=1}^{{N_\mathrm{s}}} {\mathcal{T}}\left({\Delta}^{\ell}(t)\right) \sum\limits_{{k}=1}^{r^\ell}
	{\alpha}^{\ell}_{{k}}(t){\psi}^{\ell}_{{k}}(x)\right)^2 \mathrm{d}x \mathrm{d}t.
\end{equation}
The solution of the POD optimization problem \eqref{eq:PODoptimization} can be obtained by
solving an operator eigenvalue problem,  which in the discrete setting corresponds to computing an SVD.
Since in the setting of \eqref{eq:generalTransformedModesApproximation}, the modes may be linearly dependent, we have to solve a nonlinear optimization problem instead.
To this end, we apply numerical optimization techniques on the discrete level but, prior to that, we analyze some properties of \eqref{eq:generalTransformedModesApproximation}.

First, it should be noted that the solution is in general not unique. This can be seen by taking for instance the simple case where ${\mathcal{T}}={\mathcal{T}_{\mathrm{per}}}$ and $q(x,t)=q_1(x+t)+q_2(x-t)+\cos(t)q_3(x)$ with some arbitrary functions $q_i$ for $i=1,\ldots,3$.
Then, a solution of \eqref{eq:generalTransformedModesApproximation} is given by
\begin{alignat*}{2}
	&{N_\mathrm{s}}=3,\quad r_1=r_2=r_3=1,\quad &&{\Delta}^{1}(t)=-{\Delta}^{2}(t)=t,\quad {\Delta}^3(t)\equiv 0,\\
	&{\alpha}^{1}_{1}(t)={\alpha}^{2}_{1}(t)\equiv 1, \quad &&{\alpha}^{3}_{1}(t)=\cos(t),\quad {\psi}^{i}_{1}(x)=q_i(x),\quad\mbox{for }i=1,\ldots,3.
\end{alignat*}
On the other hand, by making use of the trigonometric identities $\sin(x\pm t)=\sin(x)\cos(t)\pm \cos(x)\sin(t)$, another solution is
\begin{align*}
	&{N_\mathrm{s}}=3,\quad r_1=r_2=r_3=1,\quad {\Delta}^{1}(t)=-{\Delta}^{2}(t)=t,\quad {\Delta}^3(t)\equiv 0\\
	&{\alpha}^{1}_{1}(t)={\alpha}^{2}_{1}(t)\equiv 1, \quad{\alpha}^{3}_{1}(t)= \cos(t),\\
	&{\psi}^{i}_{1}(x)=q_i(x)+\sin(x),\quad\mbox{for }i=1,2,\quad {\psi}^{3}_{1}(x) = q_3(x)-2\sin(x).
\end{align*}
Both these solutions are optimal, since the cost functional is zero.

As discussed in Sec.\ \ref{sec:intro}, many of the currently discussed model reduction approaches for transport-dominated phenomena consider the case of only one transport velocity (${N_\mathrm{s}}=1$) and periodic boundary conditions. In this special case the cost functional takes the form
\begin{equation}
	\label{eq:coMovingFrameApproximation}
	\min_{{\psi},{\alpha}}\int\limits_0^T\int\limits_\Omega \left(q\left(x,t\right)-{\mathcal{T}_{\mathrm{per}}}\left({\Delta}(t)\right) \sum\limits_{{k}=1}^{r}
	{\alpha}_{{k}}(t){\psi}_{{k}}(x)\right)^2 \mathrm{d}x \mathrm{d}t
\end{equation}
and one can enforce the modes to form an orthonormal basis, since orthogonality is preserved under the action of the periodic shift operator, {i.\,e.},
\begin{equation}
	\label{eq:orthogonalityConstraints}
	\left\langle {\mathcal{T}_{\mathrm{per}}}\left({\Delta}(t)\right){\psi}_i(x),{\mathcal{T}_{\mathrm{per}}}\left({\Delta}(t)\right){\psi}_j(x)\right
\rangle_{L^2(\Omega)}=\left\langle{\psi}_i(x),{\psi}_j(x)\right\rangle_{L^2(\Omega)}=\delta_{ij}
\end{equation}
for $i,j=1,\ldots,r$.
This follows, since ${\mathcal{T}_{\mathrm{per}}}(\cdot)$ is a unitary operator, cf. \cite{Coh13}.

Since the adjoint operator of ${\mathcal{T}_{\mathrm{per}}}({\Delta})$ is given by ${\mathcal{T}_{\mathrm{per}}}^\ast({\Delta})={\mathcal{T}_{\mathrm{per}}}(-{\Delta})$, the optimization problem \eqref{eq:coMovingFrameApproximation} associated with the constraints \eqref{eq:orthogonalityConstraints} is equivalent to
\begin{equation*}
	\min_{{\psi},{\alpha}}\int\limits_0^T\int\limits_\Omega \left({\mathcal{T}_{\mathrm{per}}}\left(-{\Delta}(t)\right)q\left(x,t\right)-\sum\limits_{{k}=1}^{r}
	{\alpha}_{{k}}(t){\psi}_{{k}}(x)\right)^2 \mathrm{d}x \mathrm{d}t,\quad \mbox{s.\,t.}\,\mbox{\eqref{eq:orthogonalityConstraints}}.
\end{equation*}
Thus in this special case, the optimization problem leads to a POD of the transformed function ${\mathcal{T}_{\mathrm{per}}}\left(-{\Delta}(t)\right)q\left(x,t\right)$, which has been used, {e.\,g.}, in \cite{CagMS16_ppt}.

In the general case of more than one transport velocity (${N_\mathrm{s}}>1$), we have to solve the optimization problem \eqref{eq:generalTransformedModesApproximation} numerically.
Carrying out a discretization, we have to solve the optimization problem
\begin{equation}
	\label{eq:generalTransformedModesApproximation_discrete}
	\min_{{w},{a}}\underbrace{\sum_{j=1}^n\left\lVert \left[X\right]_j-\sum\limits_{\ell=1}^{{N_\mathrm{s}}} {T}\left({d}^{\ell}_{j}\right) \sum\limits_{{k}=1}^{r^\ell}
	{a}^{\ell}_{{k},j}{w}^{\ell}_{{k}}\right\rVert^2}_{=\vcentcolon {J}}.
\end{equation}
where $n$ is the number of snapshots.
Introducing the notation
\begin{align*}
\nonumber
	 &{a}_j\vcentcolon=[{a}^{1}_{1,j}\quad\ldots\quad{a}^{1}_{r_1,j}\quad
{a}^{2}_{1,j}\quad\ldots\quad{a}^{{N_\mathrm{s}}}_{r_{N_\mathrm{s}},j}]^T,\\
	&K_j\vcentcolon=
	\begin{bmatrix}
		T\left({d}^{1}_{j}\right){w}^{1}_{1} & \ldots &{T}\left({d}^{1}_{j}\right){w}^{1}_{r_1} & {T}\left({d}^{2}_{j}\right){w}^{2}_{1} & \ldots & {T}\left({d}^{{N_\mathrm{s}}}_{j}\right){w}^{{N_\mathrm{s}}}_{r_{N_\mathrm{s}}}
	\end{bmatrix}
	,
\end{align*}
the cost functional in \eqref{eq:generalTransformedModesApproximation_discrete} can be expressed as the least squares problem
\begin{equation}
\label{eq:shortCostFunctional}
{J} = \sum\limits_{j=1}^n\left\lVert \left[X\right]_{j}-K_j{a}_j\right\rVert_2^2.
\end{equation}
Considering the dependency of $J$ with respect to the amplitudes ${a}_j$ for fixed modes ${w}$, the necessary optimality condition is given by
\[
	\nabla_{{a}_j}{J} = -2K_j^T\left(\left[X\right]_{j}-K_j{a}_j\right)=0,
\]
or equivalently
\begin{equation}
	\label{eq:alphaCondition}
K_j^TK_j{a}_j = K_j^T\left[X\right]_{j}
\end{equation}
for $j=1,\ldots,n$.
The general solution of \eqref{eq:alphaCondition} is given by
\begin{equation}
	\label{eq:solutionOfLS}
	{a}_j = V_{j,1}\Sigma_{j,1}^{-1}U_{j,1}^T\left[X\right]_{j}+V_{j,2}\beta_j,
\end{equation}
where $\beta_j$ is an arbitrary vector of suitable dimension, and the matrices $V_{j,1}$, $\Sigma_{j,1}$, $U_{j,1}$, and $V_{j,2}$ are defined via the SVD of $K_j$
\begin{equation*}
	K_j =
	\begin{bmatrix}
		U_{j,1} & U_{j,2}
	\end{bmatrix}
	\begin{bmatrix}
		\Sigma_{j,1} & 0\\
		0 & 0
	\end{bmatrix}
	\begin{bmatrix}
		V_{j,1}^T \\ V_{j,2}^T
	\end{bmatrix},
\end{equation*}
where $\Sigma_{j,1}$ contains the non-zero singular values of $K_j$ \cite{GanGK14}.
If the shifted modes are linearly independent at a time point $t$, then $V_{j,2}$ is void and the solution \eqref{eq:solutionOfLS} is unique, otherwise
\eqref{eq:alphaCondition} has infinitely many solutions.

Substituting \eqref{eq:solutionOfLS} into \eqref{eq:shortCostFunctional}, the cost functional takes the form
\begin{equation*}
	J = \sum\limits_{j=1}^n\left\lVert \left[X\right]_{j}-U_{j,1}U_{j,1}^T\left[X\right]_{j}\right\rVert_2^2,
\end{equation*}
which only depends on the modes ${w}$ hidden in the matrices $U_{j,1}$.
Simple calculations show that minimizing ${J}$ is equivalent to the minimization problem
\begin{equation}
	\label{eq:finalOptimizationProblem}
	\min_{{w}}
\tilde {J} = -\sum\limits_{j=1}^n\left\lVert U_{j,1}^T\left[X\right]_{j}\right\rVert_2^2.
\end{equation}
The gradient of $\tilde J$ with respect to a mode ${w}^{\ell}_{k}$ is given by
\begin{equation*}
	\label{eq:gradientOfJ}
	\nabla_{{w}^{\ell}_{k}}\tilde {J} = \sum\limits_{j=1}^n {a}^{\ell}_{k,j} {T}\left({d}^{\ell}_{j}\right)^T\left(I_{m}-U_{j,1}U_{j,1}^T\right)\left[X\right]_{j}.
\end{equation*}
An algorithm to compute an optimal solution is presented in Sec.\ \ref{sec:algorithm}.

\section{Algorithm and Implementation\label{sec:algorithm}}

Since it is a priori unclear how many modes are necessary to achieve a certain error tolerance, we propose to solve the optimization problem \eqref{eq:finalOptimizationProblem} starting with a small number of modes and iteratively adding modes in a greedy fashion, cf. Algorithm \ref{alg:sPOD}.
\begin{algorithm}[htb]
	\caption{sPOD algorithm based on residual minimization}
	\label{alg:sPOD}
	\begin{algorithmic}[1]
		\Statex \textbf{Input:} snapshot matrix $X$; initial mode numbers ${\boldsymbol{r}}^0$; shifts ${d}^{\ell}_{j}$ for $j=1,\ldots,n$ and $\ell=1,\ldots,{N_\mathrm{s}}$; routine for the calculation of ${T}(\cdot)$; error
		tolerance \textit{tol}; maximum iteration number ${p}_{\mathrm{max}}$
		\Statex \textbf{Output:} modes ${w}^{\ell}_{{k}}$; amplitudes ${a}^{\ell}_{{k},j}$ for $j=1,\ldots,n$, $\ell=1,\ldots,{N_\mathrm{s}}$, and $k=1,\ldots,r^{\ell}$
		\State Solve \eqref{eq:finalOptimizationProblem} with mode numbers ${\boldsymbol{r}}^0$ for the modes ${w}$ \label{line:firstOptimization}
		\State Compute the amplitudes ${a}$ from \eqref{eq:solutionOfLS} \label{line:alpha}
		\State Reconstruct $\tilde X$ as in \eqref{eq:shiftedSnapshotDecomposition_discrete} and compute the error as in \eqref{eq:errorMeasure} \label{line:error}
		\State ${p}=0$
		\While{$(\text{error}>\text{tol})\ \mbox{\rm and } ({p}<{p}_{\mathrm{max}})$}\label{line:while}
			\State ${p} \gets {p}+1$
			\For{$i=1:{N_\mathrm{s}}$} \label{line:for}
				\State Solve \eqref{eq:finalOptimizationProblem} with mode numbers ${\boldsymbol{r}}^{{p}-1}+e_i$ for the modes ${w}$ \label{line:optimizationInForLoop}
				\State Compute the amplitudes ${a}$, $\tilde X$, and the error as in lines \ref{line:alpha} and \ref{line:error}
				\State $\text{tempError}(i)\gets\text{error}$
			\EndFor \label{line:endFor}
			\State Find the index $q$ for which $\text{tempError}$ is minimal
			\State $\text{error}\gets\text{tempError}(q) $
			\State ${\boldsymbol{r}}^{p}\gets {\boldsymbol{r}}^{{p}-1}+e_q$
		\EndWhile
	\end{algorithmic}
\end{algorithm}
To initiate the algorithm we choose a vector ${\boldsymbol{r}}^0\in\mathbb{N}^{{N_\mathrm{s}}}$ containing the initial mode numbers for each velocity frame and prescribed shifts ${d}^{\ell}_{j}$ for each velocity frame and discrete time step.
The algorithm starts with computing a mode decomposition with mode numbers ${\boldsymbol{r}}^0$.
For this, the optimization problem \eqref{eq:finalOptimizationProblem} is solved with a nonlinear optimization solver of choice, {e.\,g.}, Newton's method or quasi-Newton methods, see {e.\,g.}~\cite{NocW06}.
Since the optimization problem scales with the full dimension,
we recommend an inexact Newton method or a limited-memory quasi-Newton method which is more efficient \cite{NocW06}.
Motivated by the case with one velocity frame and a periodic shift operator discussed in Sec.\ \ref{sec:optimization}, we choose the first $[{\boldsymbol{r}}^0]_\ell$ singular vectors of the transformed snapshot matrix
\begin{equation*}
	\begin{bmatrix}
		{T}\left(-{d}^{\ell}_{1}\right)[X]_1 & \cdots & {T}\left(-{d}^{\ell}_{n}\right)[X]_n
	\end{bmatrix}
\end{equation*}
as starting values for the modes of the $\ell$th velocity frame.
Following this, in line \ref{line:while} the relative error is compared with the tolerance and if the tolerance is not achieved, then the algorithm continues by adding modes in a greedy manner.
More precisely, in the \textbf{for} loop in lines \ref{line:for}-\ref{line:endFor}, we add one mode to each frame at a time, solve the optimization problem \eqref{eq:finalOptimizationProblem}, construct $\tilde X$, and compute the error.
Subsequently, the errors corresponding to the different mode number vectors ${\boldsymbol{r}}^{{p}-1}+e_i$ are compared, where $e_i\in\mathbb{R}^{N_\mathrm{s}}$ denotes the $i$th unit vector, and only that mode is kept which results in the smallest error.
This \textbf{while} loop continues until the error is below the tolerance or the maximum iteration number is reached.

The major computational cost of Algorithm~\ref{alg:sPOD} arises from the solution of the optimization problems in lines \ref{line:firstOptimization} and \ref{line:optimizationInForLoop} and depends on the chosen solver.
The computation time can be decreased significantly by performing the \textbf{for} loop in lines \ref{line:for}-\ref{line:endFor} in parallel.
Another opportunity for a speedup is to use multigrid methods for the optimization, see {e.\,g.}{} \cite{Nas00}.

Most parts of Sec.\ \ref{sec:optimization}, as well as Algorithm \ref{alg:sPOD} are valid for general matrix functions ${T}$ which do not necessarily have to be associated with a shift operation. Thus, the use of matrix functions which simulate other effects like rotation or dilation is possible, however, this topic is not within the scope of this paper.

Instead, in Sec.\ \ref{sec:DDT} we use a shift operator with constant extrapolation, {i.\,e.},
\begin{equation*}
	{\mathcal{T}_{\mathrm{c}}}\left({\Delta}(t)\right)f\left(x,t\right)\vcentcolon=\left\{
	\begin{array}{lll}
		f\left(x-\Delta(t),t\right) & \text{for } 0\leq x-\Delta(t) \leq L,\\
		f\left(0\right) & \text{for } x-\Delta(t)<0,\\
		f\left(L\right) & \text{for } x-\Delta(t)>L.
	\end{array}
    \right.
\end{equation*}
For the discrete analogue ${T_{\mathrm{c}}}$ on a uniform grid with mesh width $h$, we distinguish between two cases: If the shift is a multiple of $h$, then ${T_{\mathrm{c}}}(\cdot)$ is defined as
\begin{equation}
	\label{eq:shiftOperatorWithConstantExtrapolation}
	{T_{\mathrm{c}}}\left(kh\right) = \left[
	\begin{array}{cccc|c}
		1 & 0 & \ldots & 0 & 0 \\
		\hline
		 & & & & 0\\
		 & & & & \vdots\\
		 \multicolumn{4}{c|}{\text{\smash{\raisebox{1.5\normalbaselineskip}{$I_{m-1}$}}}} & 0
	\end{array}
	\right]^k,\quad
	{T_{\mathrm{c}}}\left(-kh\right) = \left[
	\begin{array}{c|cccc}
		0 & & & & \\
		\vdots & & & & \\
		0 & \multicolumn{4}{|c}{\text{\smash{\raisebox{1.5\normalbaselineskip}{$I_{m-1}$}}}}\\\hline
		0 & 0 & \cdots & 0 & 1
	\end{array}
	\right]^k
\end{equation}
with $k\in\mathbb{N}$.
If the shift is not a multiple of $h$ we use an interpolation scheme, {i.\,e.}, for instance, a linear interpolation like ${T_{\mathrm{c}}}(0.5h)=0.5({T_{\mathrm{c}}}(0)+{T_{\mathrm{c}}}(h))$.
Similarly, a shift matrix function for the periodic case has been introduced in \cite{RimML17_ppt}.

\section{Test Case: Pulsed Detonation Combuster\label{sec:DDT}}
As a realistic test example, we consider density, velocity, pressure, and effective species snapshot data of a \emph{Pulsed Detonation Combuster (PDC)} with a shock-focusing geometry.
The data is based on a simulation of the reactive, compressible Navier-Stokes equations where physical parameters have been adjusted by a data assimilation, see \cite{GraLRPSM17}.
The density and species snapshots are depicted in Fig.\ \ref{fig:FOM_snapshots}.
\begin{figure}[htb]
	\centering
	\begin{tikzpicture}
		\node[inner sep=0pt] (rho_FOM) at (-3,0){\includegraphics[width=0.49\linewidth]{rho_FOM}};
		\node[inner sep=0pt] (Y_FOM) at (3,0){\includegraphics[width=0.49\linewidth]{Y_FOM}};
		\node[inner sep=0pt] (DDT) at (-3.8,0.1) {\scriptsize{DDT}};
		\node[inner sep=0pt] (LS) at (-3.5,-0.7) {\scriptsize{leading shock}};
		\node[inner sep=0pt] (RF) at (-3.4,-0.3) {\scriptsize{reaction front}};
		\node[inner sep=0pt] (RW) at (-3.8,0.9) {\scriptsize{reflected wave}};
		\node[inner sep=0pt] (RRW) at (-3.8,2.1) {\scriptsize{re-reflected wave}};
		\node[inner sep=0pt] (DW) at (-2.5,0.5) {\scriptsize{detonation wave}};
		\draw (-4.55,1.7) -- (-4.3,2);
		\draw (-4.72,0.45) -- (-4,0.8);
		\draw (-2.4,1.3) -- (-2.4,0.6);
		\draw (-4.51,-0.1) -- (-4.25,-0.3);
		\draw (-4.53,-0.32) -- (-4.35,-0.6);
	\end{tikzpicture}
	\caption{PDC: snapshots of the full-order solution for density (left) and species (right).}
	\label{fig:FOM_snapshots}
\end{figure}
In the snapshots of the species we observe a reaction front propagating through the domain.
The density snapshots show initially two transports, the reaction front and a leading shock, slightly diverging before they converge again and interact.
This \emph{deflagration to detonation transition (DDT)} is caused by a nozzle at around $x=0.2$, cf. \cite{GraLRPSM17}.
Following this, the reaction front and the leading shock continue as a detonation wave moving to the right.
At the same time, a reflected wave is moving to the left before being reflected at the boundary.
When it reaches the nozzle again, another partial reflection is visible.
The velocity and pressure snapshots look similar.

Before we apply Algorithm \ref{alg:sPOD} we need to find good candidates for the shifts corresponding to the transports of the system.
Here, we focus on the four most dominant transports: the reaction front, the leading shock, the reflected wave, and the partial reflection at the nozzle which is referred to as \textit{re-reflected wave} in the following.
We track these transports based on the snapshot data without any a priori knowledge of their velocities.
The reaction front is the easiest to detect, since it is clearly visible in the species snapshots as a large gradient.
To track it, we determine the location of the maximum in each column of the difference matrix whose $j$th column is defined as the difference between the $j+1$st and $j$th column of the species snapshot matrix.
The resulting tracked shift is depicted in Fig.\ \ref{fig:cs}, solid line.
\begin{figure}[b]
	\centering
	\includegraphics [width=0.325\linewidth]{cs}
	\caption{PDC: tracked shifts for the different transports.}
	\label{fig:cs}
\end{figure}
Here, negative shift values occur since the reaction front is shifted such that it is centered in the middle of the computational domain.

The tracking of the other transports works similarly, but is a little more elaborate since we need to distinguish them from each other.
To this end, we restrain the region of the computational domain where the location of the maximum slope is computed. This subregion depends on both the considered transport and time interval.
In our tests, this decomposition in subregions has been done manually based on the velocity snapshots. The corresponding tracked shifts are depicted in Fig.\ \ref{fig:cs}.
In addition, we also add a frame with zero velocity to account for the structures that we cannot capture well by the other velocity frames.

We apply Algorithm \ref{alg:sPOD} with a shift operator with constant extrapolation as in \eqref{eq:shiftOperatorWithConstantExtrapolation} with Lagrange polynomials of degree three for the interpolation.
In addition, we specify $\mathrm{tol}=0.01$, ${p}_{\mathrm{max}}=n$, and ${\boldsymbol{r}}^0=[1\;1\;1\;1\;0]$, {i.\,e.}, one mode for each of the non-zero velocity frames.
The nonlinear optimization problem is solved using the MATLAB package HANSO which is based on a limited-memory BFGS \cite{LewO13}.
Moreover, to avoid parasitic structures in the approximation of the species, we force those parts of the modes which correspond to the species and to other transports than the reaction front, to be zero.

In this test case we  have to deal with data of physical variables with highly different scales.
To avoid that the approximation of the physical variable with the highest scale becomes dominant we scale the snapshots such that the snapshot matrices of the different physical variables have the same Frobenius norm.
We build the snapshot matrix $X$ for Algorithm \ref{alg:sPOD} by concatenating the scaled snapshot matrices of the different physical variables.

Algorithm \ref{alg:sPOD} terminates after $3$ iterations in the {\bf while} loop with an error of $0.71\%$ and ${\boldsymbol{r}}^3=[3\;1\;1\;1\;1]$, {i.\,e.}, two modes have been added to the reaction front and one mode to the zero velocity frame. This means that we meet the error tolerance with $7$ modes in total. The sPOD approximation for the density and the species is depicted in Fig.\ \ref{fig:FOMvsSPODvsPOD}, middle column.
\begin{figure}[t]
	\centering
	\includegraphics [width=0.325\linewidth]{rho_FOM}
	\includegraphics [width=0.325\linewidth]{rho_sPOD}
	\includegraphics [width=0.325\linewidth]{rho_POD}
	\includegraphics [width=0.325\linewidth]{Y_FOM}
	\includegraphics [width=0.325\linewidth]{Y_sPOD}
	\includegraphics [width=0.325\linewidth]{Y_POD}
	\caption{PDC: comparison between full-order solution (left column), sPOD approximation with $7$ modes (middle column), and POD approximation with $7$ modes (right column). The top
	row shows the results for the density, the bottom row for the species.}
	\label{fig:FOMvsSPODvsPOD}
\end{figure}
Although some deviations to the full-order solution are visible, the sPOD captures the dynamics well and the dominant transports are clearly distinct.
This becomes even more striking when comparing it to the POD with the same number of modes which is plotted in Fig.\ \ref{fig:FOMvsSPODvsPOD}, right.
As is common in the POD literature, we first subtracted the mean value of each row of the snapshot matrix to center the data around the origin, cf. \cite{Cha00}.
The POD approximation of the density features a high peak in the region of the DDT while the other structures are hardly recognizable.
For the species, the reaction front is at least indicated, but blurred, and further distortions are visible especially near the DDT.
To obtain a POD approximation of the same accuracy as the sPOD with $7$ modes, $73$ POD modes are needed for this example.

Another advantage of the sPOD becomes clear when looking at the POD and sPOD modes.
In Fig.\ \ref{fig:Ymodes} the first sPOD mode for the species in the reaction front frame is depicted and compared to the first POD mode.
\begin{figure}[t]
	\centering
	\includegraphics [width=0.325\linewidth]{Y_modes}
	\caption{PDC: comparison of first POD mode and first sPOD mode for the species.}
	\label{fig:Ymodes}
\end{figure}
While the sPOD mode clearly reveals the reaction front as a jump in the middle, the POD is rather smooth and does not show any structure resembling a reaction front.

In Fig.\ \ref{fig:rhoModes} the first sPOD mode for the density is depicted for the reaction front, the leading shock, and the reflected wave and compared to the first three POD modes.
\begin{figure}
	\centering
	\includegraphics [width=0.325\linewidth]{rho_sPODmode_reactionFront}
	\includegraphics [width=0.325\linewidth]{rho_sPODmode_leadingShock}
	\includegraphics [width=0.325\linewidth]{rho_sPODmode_reflectedWave}
	\includegraphics [width=0.325\linewidth]{rho_1stPODmode}
	\includegraphics [width=0.325\linewidth]{rho_2ndPODmode}
	\includegraphics [width=0.325\linewidth]{rho_3rdPODmode}
	\caption{PDC: comparison of first sPOD modes (top row) for the reaction front, the leading shock, and the reflected wave (from left to right) and first three POD
	modes (bottom row, from left to right) for the density.}
	\label{fig:rhoModes}
\end{figure}
The latter ones mainly focus on the DDT which agrees with Fig.\ \ref{fig:FOMvsSPODvsPOD}, top right, while the shapes of the moving fronts are not captured.
The sPOD modes are not as clear as in Fig.\ \ref{fig:Ymodes} but still each of them features a clear front profile in the middle (marked by dashed lines) which reflects the physics properly.
However, especially at the left boundary they differ strongly: The mode for the reflected wave, top right in Fig.\ \ref{fig:rhoModes}, reveals a flat profile at the left boundary.
This is due to the fact that this part of the mode is not used in the sPOD approximation, since the corresponding shift, depicted in Fig.\ \ref{fig:cs}, does not attain values greater than $-0.16$.
The modes for the reaction front and the leading shock reveal some oscillations at the left boundary.
A possible reason for this is the use of the shift operator with constant extrapolation which provides the values at the boundaries of the mode with a disproportionate weight.

\section{Summary and Outlook\label{sec:conclusion}}

We have presented a new algorithm for computing a shifted proper orthogonal decomposition (sPOD) based on a residual minimization applied to snapshot data.
We have applied the algorithm to snapshots  determined from a pulsed detonation combuster (PDC) and compared the results with the standard proper orthogonal decomposition (POD).
The sPOD yields a reasonable approximation of the snapshots with only very few modes.
In contrast, the POD approximation with the same number of modes is blurred and the dynamics is not captured well.
Moreover, the sPOD modes clearly reveal the front profiles of the different transports, whereas the POD is not suitable for identifying structures in this test case.
In comparison to the heuristic sPOD algorithm proposed in \cite{ReiSSM17_ppt}, the new algorithm is based on a residual minimization and hence at least locally optimal.
A drawback of the new algorithm is that it is more expensive than the  POD
and the original sPOD approach of \cite{ReiSSM17_ppt}.
The reason is that a large-scale nonlinear optimization problem has to be solved.

The results of the new sPOD algorithm look promising in terms of both the number of required modes and the physical structures identified by the sPOD modes.
The use of the sPOD modes to obtain a reduced-order model via projection is currently under investigation. Further interesting research directions are a rigorous treatment of non-periodic boundary conditions and an optimization of the shifts together with the modes and the amplitudes.\\

\noindent\textbf{Acknowledgements.} The authors gratefully acknowledge the support by the Deut\-sche Forschungsgemeinschaft (DFG) as part of the collaborative research center SFB 1029 \textit{Substantial efficiency increase in gas turbines through direct use of coupled unsteady combustion and flow dynamics}, project A02 \textit{Development of reduced-order models for pulsed combustion}.

\bibliographystyle{abbrv}


There are two kinds of people in the Philippines: those who take "Local" transportation, and those who go "First Class," as they'll call them. I've found that Local transport usually entails a jeepney (the converted U.S. army jeeps with dozens of passengers crammed in the back that litter the streets by the tens of thousands), shared vans, no-frills ferry rides, trikes (the converted motorcycles with sidecars) and buses. First Class, however, is reserved for those people who can take a metered taxi, a Grab (their version of Uber or Lyft), or are lucky enough to have a car.
The number of people who go "Local" versus "First Class" aren't always easily segregated by homegrown Filipinos versus foreigners, either. Although there may be a correlation, a lot of foreign travelers are broke backpackers who are brave enough to sweat and push it out like a native son or daughter. Conversely, while most travelers do opt for planes rides whenever possible and hire car services or take Grab cars, etc. for long trips, a surprising number of middle-class Filipinos have cars.
For me, I've been a complete travel snob whenever possible, because the comfort, safety, and convenience is worth it. Plus, in a country where plane tickets routinely cost $40-$50 to some beautiful island, why bother with the ferry that takes 12 hours just to save $30?
However, I'm living in Manila now, which I'm finding is NOT a cheap city. In fact, my relatively-humble budget made me a baller in Dumaguete where I lived the last year, but barely allows me to get by in Manila, where a whole lot of locals have the means comparable with the average person in the U.S.
That being said, I can't spend frivolously anymore, so I'll save a buck whenever it REALLY makes sense. Such is the case this weekend when I had to get from my McKinley Hill neighborhood in Manila up to Angeles City, the Las Vegas-like smaller city in the province of Pampanga. This was also a necessity because I left several suitcases with friends up there while I was in the States, so I needed to finally pick them up.
Hiring a car costs about $60, but of course, it's the fastest and most comfortable method. The Fly The Bus option isn't bad if you don't mind being crammed in a van, but it is safe and has AC, etc. That only runs about $10 if I remember right – huge savings, but you have to pick it up at the Swagman Hotel in Malate, which is way too far from my neighborhood to make sense.
And then the bus, which I opted for.
The Victory Liner bus in the Philippines – could be a lot worse!
My mini adventure started with a lot of Googling and asking friends where to go and how to do it, because it's not as simple as you may think. First, I had to take a taxi to the Pasay area to the Victory Liner bus terminal there. That taxi ride ran me 170 Pesos ($3) and took 20 minutes, although that could easily be an hour Monday through Friday because I departed on a Sunday (by plan). From there, I found the outdoor queue for tickets and was told by one lady to stand in a certain line. After a fifteen minute wait, I got to the front, only to be told that I was in the wrong line, so I had to get wayyyy in the back of the line next door. It's more confusing in the Philippines!
But no worries, because just when I settled in for a 20 or 30-minute wait in line, one of the bus driver's assistants called over to me and asked where I was going. When I told him that I was heading to Dau (the small town/bus station adjacent to Angeles City), he ushered me on and told me that I could easily pay on the bus without a ticket – which ran 163 Pesos – less than the taxi to get there!
I did double check that we were heading towards Angeles City because you never know if my pronunciation of "Dau" is off and I actually end up in the far end of "Mau" twelve hours away! Haha.
I found my way to the back and found an empty pair of seats, putting my backpack on one to try to discourage someone sitting next to me. Trust me when I tell you that this is important, because the seats are not quite U.S. sizes, and I'm pretty wide in the shoulders and hate being jammed in next to someone.
Usually, on a REAL local bus in the province (or countryside, meaning anywhere outside of Manila!) they will not only pack people in 2 or 3 to a 2-person row of seats, but fill up the aisles with people standing, until you're so jammed against the window and other people that you literally feel claustrophobic and like you can't breathe! Those buses also stop just about every fifty feet, as anyone along the side of the road can flag them down and get on, bringing their chickens/dogs/puking babies/fish or anything else with them.
Luckily, the Victory Liner is much more of a first-rate option, and there were still some seats to spare, so I did have a seat next to me empty. The air conditioning was fine and my seat also had sufficient leg room, which makes it comfortable.
It took us about an hour just to get to the outskirts of Manila and on the open highway, but that can easily be 3-4 hours, believe it or not, in traffic. So MAKE SURE you go to the bathroom before you depart and don't drink anything along the way! There is a rest stop about half way to Dau, but the bus didn't stop because we were making such good time, although they may if the trip takes hours and hours.
In all, it only took us about 2 hours from the Victory Liner station in Pasay to the station in Dau, which is just a flyspeck of a suburb. While the trip was simple, comfy, and enjoyable, I would recommend you check for another thing when boarding – calculate which side of the bus will get direct sunlight and sit on the OTHER side. The sun is no joke beaming through the windows for hours, and even with the window curtains drawn, it really heats it up.
Once in Dau, I got my bag from the luggage storage compartment underneath (I would never let my bag out of sight on a more "local" bus because it would most likely be stolen, but the Victory Liner is a good operation) and found a trike driver easily to take me to my hotel in Angeles City, which only cost me 150 Pesos ($3) and less than fifteen minutes.
The grand total? About $9 and change and less than three hours door-to-door. Of course, the caveat it that it could easily be double that time or more on a normal day with traffic, but that would be true if you took a taxi or car, too.
I'll definitely take the Victory Liner again on certain trips!


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The Chief of Civil/Military Affairs, Army HQ, Brig. Gen. Nicholas Rogers says territories held by the insurgents have been liberated as the people are returning back to their communities gradually.
Rogers was the guest of Channels Television's Sunrise Daily on Thursday, December 10, where he gave an update on the progress made by the Nigerian military in its fight against terrorism in the Northeast.
"What is left are some villages between the boundaries of Nigeria, Chad and Cameroun particularly along the Lake Chad region," he said.
Brig. Gen. Rogers said that stability and calm have returned in Maiduguri, Damaturu, Gwoza, Bama, Baga, Monguno and other towns, adding that some buildings like schools, churches that were burnt down are being rebuilt by the military.
He, however, noted that there are some challenges along the line.
According to the Brig. Gen, in continuing the fight against the insurgents and protecting the towns that have been liberated, other security agencies are yet to get fully involved and this situation has been preventing the military from carrying out their full operations.
Brig. Gen. Rogers also disclosed that several meetings have been held and is still ongoing to discuss the challenges in the force but noted that the responses have been slow.


Read my interview on the above subject at the healthcentral website. Here is the link.
my limbs and can actually go for walks unassisted.
same kinds of tasks as they do, and work in the same mechanism like them.


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text:
Tudulinna (Estisch: Tudulinna vald) was tot in 2017 een gemeente in de Estische provincie Ida-Virumaa. De gemeente telde 423 inwoners op 1 januari 2017 en had een oppervlakte van 269,2 km². De hoofdplaats van de gemeente was Tudulinna.

Sinds oktober 2017 maakt de gemeente deel uit van de gemeente Alutaguse, net als de voormalige gemeenten Alajõe, Iisaku, Illuka en Mäetaguse.

Gemeente in Ida-Virumaa
Voormalige gemeente in Estland

redpajama_set_name: RedPajamaWikipedia


6 Lucky winners will each receive a huge package prize full of gift certificates, merchandise, lodging & more all donated by local merchants. This year's recipients are LOVE I.N.C. and the Newaygo Museum. Tickets are $5 each and can be purchased at LOVE, Sportsman's and many local Newaygo area merchants.
Newaygo Troutfest 2015 Prize packages are subject to change and be even better!!
- we are adding more prizes from sponsors as they come in!
Who were the raffle winners??


text:
kwachu123 4 years ago
Institution RL9
In this text I would like to give you a picture of how Lewandowski influenced the football environment in my country, Poland.
We all know that Robert Lewandowski is a phenomenon on a national scale. He carries the burden of the Polish National Team, as well as Bayern Munich. Hence the pressure it has to cope with. We love him for goals, which he scores, but when disasters come, a lot of criticism falls on him. Not everyone knows, however, that apart from scoring priceless goals, he does and has already done much more for Polish football.
It is a model that he missed
"I want to be like the Lewandowski"- that's what every dad of a young footballer has heard. The fact that Robert has contributed to the increase in the number of children in football schools is obvious. Young people are staring at our representative. When they see on the TV screen how he shoots a goal behind a goal, it works on their imagination. They imitate his behaviour, after scored goals they enjoy like him. The youngest have an idol from their country who grew up and trained in the same places. They do not have to look abroad, such as RL9. Having an idol from Poland makes us even more aware that we can achieve success. It gives hope to all those who wait for hours for an autograph and have posters with its image in their room. He is for all young players a personality, which was lacking in his life. As in one of his advertising campaigns, he is at the forefront of the RL9 generation. As we all know, Lewandowski is a 100% professional. He cannot imagine sitting at home at the end of his career and reaping the benefits of what he has earned over the years. He has many plans in his head. As he says himself, he is closer to creating football academies for children than to being a professional trainer. He intends to share the knowledge he has acquired over the years of training. After all, he knows everything that no trainer in Poland has had the opportunity to get to know. Again, he tries to give others what he has not experienced himself.
He cleared the routes
Thanks to his talent and hard work, he managed to get out of the Polish club ball and hit the West. It might seem that he had it easier than two other Poles in Dortmund: Lukasz Piszczek and Jakub Blaszczykowski. After all, he wasn't there first and he could move into the band much faster. However, the truth is different. Robert wanted to write his own story, show that he can do it himself. This is one of the reasons why he spent more time with players of German origin. However, he was not ashamed of the country he came from. On the contrary, he made sure that our players were treated in the same way as others. He was able to point out to the then BVB coach Juergen Klopp, even after one of the matches, that he has too many claims to the Poles. We were able to find out several times that he is not afraid to express his opinion. As in the situation when he was critical of Bayern Munich's transfer policy. The fact that he is the best earning footballer of the Bavarian club is also due to his obstinacy and tenacity. Now nobody will say that a foreigner will not earn the most. He can be self-confident because he has already done a lot of good for the Munich people.
He changed thinking
Robert's success had an impact on the awareness of part of football society. In many interviews he emphasizes the importance of physical development, not only from a football point of view. The fact that he became such a good striker is the result of his training in many other disciplines, such as judo. Already in childhood, thanks to practicing many sports, he acquired skills which he could later use in professional football. These words certainly reach the coaches in Poland and some of them are modernizing their training plans in order to provide the best possible conditions for the development of young players. Thanks to Lewandowski one more thing has changed. They started to pay more attention to the diet of the athletes. It would be hard to make it different if a star of such a large format emphasizes how much it owes to well-balanced meals. He said in one of his books that while playing in Borussi Dortmund, he ate a lot of sweets. He did not see the problem in this, because it was not back and was still well built. It was only when he changed his diet that he saw how much his training performance increased. Together with his wife, they make athletes more willing to reach for innovative nutritional solutions. They simply know that this will help them.
As you can see, Robert Lewandowski is not only a footballer who scores a lot of goals, but also a person who makes the public aware. It has a real impact on the development of Polish football and the changes taking place in it. Thanks to his attitude on the pitch, many young people started to play football. However, for those who have been training for years, he showed that when they go abroad, they cannot feel worse than others.
Soccerrobert lewandowskirl9bayern munichpoland

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10 Thoughts on Gun Control in the U.S.
HomePolitics10 Thoughts on Gun Control in the U.S.
Gun control is the most emotionally charged issue in America. Critics of health care legislation may talk about how thousands will die; tax reform opponents can warn of Armaggedon. But in the wake of a mass shooting such as the recent incident in Parkland, Fla., real victims and their families step forward and share their stories of terror and grief. But how can we prevent another Parkland, or Sandy Hook, or Columbine? More to the point, how can we prevent another mass shooting without affecting millions of law-abiding citizens exercising their Second-Amendment rights? It's not the first time we've had this debate; sadly, it will not be the last.
— Conservative columnist Ben Shapiro, saying anti-gun advocates need to outline specific proposals that would prevent mass shootings.
— President Donald Trump, in a tweet asking for bipartisan support. Among President Trump's other suggestions: a ban on bump stocks, an age limit on the purchase of AR-15s, and arming school teachers, all of which have been met with varying degrees of support.
— Sen. Marco Rubio (R-Fla.), during a CNN town hall event, on the concept of arming teachers. One of his concerns is that an armed school staff member could be mistaken for the shooter by responding law enforcement officers.
— Conservative radio host Kevin McCullough. This is a common mantra of those who want to maintain a status quo on gun laws. They note that many mass shooters violate multiple laws, and passing more laws will be ineffective. McCullough offered several other points he believe would help in mass school shootings, from holding active shooter drills to placing metal detectors in every school.
— New York Times writer Bret Stephens, renewing a call he made after last year's Las Vegas shooting to repeal the second amendment. Even in the aftermath of a great tragedy this proposal is on the fringe. It would mean a huge reversal of a constitutional right that would be virtually impossible to pass through Congress and even more difficult to enforce without enacting a police state. Just for the sake of argument though, to repeal a constitutional amendment requires the approval of a two-thirds majority in both the House and Senate — no easy feat on any issue these days. And keep in mind that 44 states have constitutional provisions specifying the right to keep and bear arms.
— Andrew Pollack, whose daughter Meadow Pollack was among the victims at Marjory Stoneman Douglas High School. Pollack's emotional plea came during a special session at the White House involving family members of school shootings at Parkland, Columbine, Sandy Hook and elsewhere.


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Warren Hegg
Warren Kevin Hegg
February 23, 1968, Whitefield, Manchester, Lancashire
54y 343d
Fielding Position
Wicketkeeper
England Lancashire
A consistent performer behind the stumps in a long career with Lancashire, with the ability to score useful lower-order runs, Warren Hegg was 30 before he achieved full international recognition. Selected to tour West Indies with England A in 1990-91, illness prevented him taking part, and a loss of confidence kept him from further international honours until an A tour to Australia in 1996-97. Chosen at last in the full England squad to tour Australia two years later, Hegg expected to carry out the tour as understudy to the captain, Alec Stewart. However, England's dismal performance in the Ashes series, and a growing perception that Stewart was being overburdened, prompted a re-think after the third Test. To free up the captain to open the batting, Hegg made his Test debut at Melbourne on Boxing Day. He held on to his place for the final Test at Sydney, but although he kept tidily he failed to make much impression with the bat. After Stewart reverted to all three roles in the World Cup the following summer, Hegg was overlooked at the start of the series against New Zealand, when the selectors instead chose the young Nottinghamshire keeper Chris Read. By the end of the summer Stewart had reverted to the role of wicket-keeper batsman, which he held for another two years. Stewart's unavailability for the 2001 tour of India prompted the selectors to consider Hegg again. After another admirable season with Lancashire in 2001 (in which he had averaged over 40 with the bat) he was included in the squad along with the young Essex wicketkeeper James Foster. However it was Foster who played throughout the two Test series in India and New Zealand. To Hegg's disappointment, the selectors reverted to Stewart when Foster had the misfortune to break an arm in the nets at the start of the domestic season. Hegg holds a notable record of 11 catches in a first-class match, set in 1989. His batting, like his keeping, is consistent and has steadily improved. Hegg was appointed captain of Lancashire in 2002. After finishing third and second in Division One of the County Championship in his first two years in charge, Lancashire were hotly-tipped to be champions, but when they were relegated after a season of internal unrest, he resigned. Although he was relatively injury-free throughout his career, an injury to his hand caused him to retire in September 2005, a matter of a couple of weeks before the end of the season.
Martin Williamson (September 2005)
Test 2 4 0 30 15 7.50 86 34.88 0 0 1 0 8 0
FC 348 504 99 11302 134 27.90 null null 7 55 null null 857 94
List A 409 245 79 3313 81 19.95 null null 0 5 null null 466 61
T20 20 13 4 104 45 11.55 107 97.19 0 0 7 1 10 8
Test 2 - - - - - - - - - - - -
FC 348 null 6 7 0 - - - 7.00 - 0 0 0
List A 409 null 0 0 0 - - - - - 0 0 0
T20 20 - - - - - - - - - - - -
Australia vs England at Melbourne - December 26 - 29, 1998
Australia vs England at Sydney - January 02 - 05, 1999
Essex vs Lancashire at Chelmsford - September 09 - 11, 2005
List A Matches
Essex vs Lancashire at Chelmsford - September 07, 2005
T20 Matches
Lancashire vs Notts at Nottingham - June 16, 2003
Lancashire vs Somerset at The Oval - July 30, 2005
PCA XI vs ACA Masters 0c/1s 25-Jul-2013 London OTHERT20
Akram's yorkers, Hegg's hits, and Chapple's 6 for 18
This week, we relive the glorious one-day triumphs of Lancashire in the 1990s
27-Apr-2020•Andrew McGlashan
Late bloomers: Joe Denly joins England's list of 30-plus debutants
From Alan Wells' disappointment at The Oval to another replacement for Keaton Jennings
30-Jan-2019•Andrew Miller
Teams prepare for Kingfisher Cup semis
Bamford Fieldhouse, York, Havant, and Wansted and Snaresbrook will contest the semi-finals of the Kingfisher Cup
24-Aug-2012•George Dobell


Claire Kreger-Boaz is a Senior Project Manager for the National Association of Music Merchants (NAMM) Public Affairs and Government Relations and The NAMM Foundation. Claire fights for music education and helps bring college students to The NAMM Show to open their eyes to all the opportunities for creating a career in the music industry.
She hopes The NAMM Show experience will help students consider new paths in their career they may not have considered or known about before attending the show.
Claire began her career journey at Montclair State University as an English literature major. She had a love for poetry, romanticism and lyricism and wanted to create things that gave people a similar feeling to what she experienced when studying the classics.
Although she was actively writing her own poetry and work, when she graduated she didn't know how this passion would translate into a career.
After going to bartending school and moving to Alaska—documenting her experience all the while—she found her way to San Diego where she worked as a Production Assistant for Greenhaven Press. In this position she felt she was able to integrate creativity into her career.
She worked her way up in that company until she left to work for a well-known literary agent. However, she realized the intense, cut-throat environment was not the experience or career path she wanted for herself.
Claire then returned to the Greenhaven Press family under Black Birch Press as a Content Editor. She moved on to work in higher education as an Associate Sponsoring Editor where she managed over a million dollar budget for producing learning products and e-textbooks. However, 2014 brought with it big cuts for the company and Claire found herself included in the wave. This unexpected turn left Claire feeling lost in her career journey.
Shocked and unclear of what her next career move would be, she reached out to friends, family and even casual acquaintances to ask for feedback on how they viewed her strengths.
The feedback she received helped her change her thought process in what skills she possessed and how they could transfer to other career paths.
This new language and mindset shift led her to apply with The NAMM Foundation where she was able to connect the dots of her content and management expertise with her role.
On the podcast, Claire explains she considers her work something she gets to do rather than has to do. At The 2019 NAMM Show, Claire lead the foundation in creating a one-of-a-kind experience in which students and music educators had the opportunity to sing along with iconic, Grammy Award winning composer Eric Whitacre.
On the podcast, Claire encourages you to reach out to those around you when you feel stuck or unsure of next steps as she did when seeking a new direction.
Be courageous enough to ask for this feedback from loved ones and people of influence in your life. Listen in and learn from Claire on how you can approach the career journey with curiosity and an open mind.
Download our podcast interview with Claire here on iTunes!
Once you're inspired by Claire's story we would love to read your review. Thanks so much!


What Level of Carbon Pricing Would Limit Climate Change?
Carbon pricing is a complicated topic, but one question in particular is usually overlooked in the flood of carbon pricing conversations. What level of carbon pricing would be required to limit climate change to 2oC? There is no simple answer, but this Doorway pulls together information and insight you can use to explore the question (and answer) for yourself.


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Infographic: these countries have had a female leader
Sirimavo Bandaranaike (Sri Lanka) was the first female prime minister ever elected in the world. Since this moment in 1960, more than 70 female prime ministers and presidents have been elected worldwide.
This infographic shows the years of service by the female heads of state. Since 2014 several more woman can be added, like Kolinda Grabar-Kitarovic who became Croatia's first female president in 2015.
Vigdís Finnbogadóttir was both Iceland's and Europe's first female president. To date, she is the longest-serving elected female head of state in the world. She was the president for 16 years.
100 female role models who belong in your history book
Why Rituals Are Good For Us: 3 Ways How We Can Benefit
Dry January: This Is What Happens To Your Body & Brain When You Give Up Alcohol
Caffeine And The Brain: The Good, The Bad And The Ugly
10 jobs that attract the most psychopaths – and why
Bizarre correlations between very unrelated things


text: Boston, Massachusetts native, Crosby Tatum, secured distinction as a television and motion picture producer, director, writer, and editor with works featured on Netflix, BET, Amazon Prime, and global video on demand platforms. Tatum's mantra is evident in his work: "Create opportunity where it seems as if there is no opportunity to be had at all." The concept motivates Tatum's loyalty to thorough work, the latest film and video technologies available, and to resilience in his profession. Tatum graduated from the New England Institute of Art with an associate degree in television broadcasting. His dedication and ability to excel during his time at the New England Institute of Art earned him the Most Outstanding Broadcaster Award in 2003. Two years following this achievement, Tatum founded Triceptus Studios, a motion picture production and post-production company. Triceptus Studios ensures excellence in every stage of the creative and technical processes, handling everything including the producing, directing, budgeting, screenwriting, post-production work and finishing services for clients. Triceptus Studios' film, television, and new media projects are shown worldwide, presented at film festivals, and complimented with various awards. As his company advances, Tatum continues to expand his personal expertise. Working with elite creative professionals, Tatum shares his talent, and in return learns from renowned colleagues he has worked with on some of his projects including Christina Keyes (Tyler Perry: A Madea Christmas), Jazsmin Lewis (Barbershop I and Barbershop II), Robin Givens, Keith Robinson, Tangi Miller, Christopher Nolen, Mickey Factz, and Kdita Otaku. Two feature films, Surprise, Surprise!!! and Confused…by Love, highlight Tatum's skill set in producing, directing, writing and distributing. An additional seven feature films specifically exemplify Tatum's capabilities as a producer. Five of these works earned placement on BET, Netflix, and Amazon Prime. Further information regarding these films can be found on Tatum's IMDB page at http://bit.ly/TatumIMDB . Tatum proudly supports his hometown sports teams when away from work, cheering on the New England Patriots, Boston Celtics, and the Boston Red Sox with pride. His love for sports also expands into professional wrestling, and Tatum enthusiastically participates in mixed martial arts, Brazilian Jiu-Jitsu and Muay Thai. When sports subside, Tatum loves to explore the culinary and physical worlds, experimenting with foods and traveling whenever possible.


Joseph likes books that make him feel sad, even though he doesn't necessarily feel like a sad person. He loves when he's reading a sad book and suddenly there's a moment that makes him laugh so hard that he drops the book. Most mornings, he writes, primarily short stories and poetry. These are sometimes funny, but mostly sad. He lives in Tempe with his wife, Harmony, and two cats.
She Would Be King is a brilliant and unexpected combination - an historical fiction infused with magical realism, and also, in part, a superhero story. We meet three characters, all with special powers and tragic origin stories, who come together to use their abilities to help others. Moore's ambitious vision of the founding of Liberia is devastating, but not without a current of hope. Her characters never give up on the future, on the possibility of a better tomorrow for themselves and their loved ones despite the trauma and abuse of their histories, or the violence and uncertainty of their present moment, and they implore us to do the same.
Any comic by Brian K. Vaughn goes on my radar, but even without his name on the cover, Paper Girls has the makings of a classic. It's a sweet coming-of-age story that crashes head-first into a temporal apocalypse, and then gets weirder from there. The first volume ends in a cliff hanger that totally changes the game. If you like Vaughn's other work, including Saga, but especially Runaways, you should check this out. If you like Stranger Things, Doctor Who, The Sandlot, or It, you should check this out.
Featuring the work of Elizabeth Acevedo, Morgan Parker, and many, many other amazing writers, this collection is full of power and pain, frustration and elation. The tremendous amount of variation in this collection underscores one of its unifying thematic elements, the paradox of home among the plurality of a diaspora. Pick this up and read "Ode to Fetty Wap." You will be surprised, filled with awe, and in love with this book by the end of it.
Ta-Nehisi Coates' Between the World and Me is essential contemporary reading, so just the fact that he's writing this series is enough to make it a must-read title. Add to that, Brian Stalfreeze's exceptional line art and a dynamic color palette, and you've got an instant classic run. Coates' story is thematically similar to the movie, exploring what it means to lead a powerful country, and by extension what it means to be a powerful country in a world of inequality. Welcome to T'Challa's world, to Coates', to ours.
As a big fan of short story collections and graphic novels, I was immediately intrigued by the idea of a book of graphic short stories. I recognized Adrian Tomine's art from his wonderful New Yorker covers, but here he demonstrates an ability to switch up his style depending on the needs of his story. Some are done in the style of newspaper serials, some in a more contemporary realism, but each one beats with an undeniable humanity, full of pain and beauty, humor and heartache. Visually and emotionally striking, Killing and Dying isn't like anything I've ever read.
Once I started this book, I found it very difficult to put down and wound up finishing it within 24 hours. It was an emotional day. The characters in Lot are reaching for epiphany, for absolution, and not all of them get there. In these stories, Washington has created, or maybe translated is a better word, a rough vision of Houston and the communities living there in the margins of society, of those grappling with poverty, homophobia and xenophobia, the aftermath of Hurricane Harvey, and impending gentrification. Recommended for fans of We the Animals, The Incendiaries, Friday Black, and Stuart Dybek.
At times brutal, at other times hilarious, this collection treads the intersections of race, class, and gender in its exploration of the ways in which we punish others and ourselves for our differences. While the subject matter here isn't anything to laugh about, Thompson-Spires manages to write stories that aren't just witty, but genuinely laugh-out-loud funny. Be aware, though, these stories mean business. Those moments of levity are bracketed by moments of heartache, frustration, and loss, and that variation of tone only adds another layer of complexity to the material. Definitely recommended for fans of Friday Black, Zadie Smith, and George Saunders.
Sayaka Murata's novel tells the story of a woman finding her place in the world and learning to navigate the pressures and expectations of society, but unlike most novels that tell this kind of story, this one is from the perspective of someone with little comprehension of how human emotions work. Murata uses her protagonist's unique perspective to hold a mirror up to ourselves, revealing the complicated relationship we have with our own rules and the people who do and don't follow them. She displays a tremendous amount of empathy for Keiko, a character utterly incapable of feeling any herself. You'll see yourself on both sides of Keiko's experience, which is just one of the novel's impressive feats. Definitely check it out. For fans of The Incendiaries or Heads of the Colored People.
R.O. Kwon lets you know where this is going from the opening of the novel, but the road there is utterly absorbing. The Incendiaries is an impressive and haunting debut novel, a story of loss, anger, regret, and the extremes to which we might go to soothe our own pain. Kwon makes certain choices in the presentation of this story that are at first curious, but eventually reveal themselves to be essential to her exploration of cult behavior, leadership, and persuasion. It's a fascinating puzzle to sort through, one that may or may not be solvable.
It's an oversimplification to call this a love triangle story. It is that, but it's also much more. Lucy Tan's debut novel is an illuminating exploration of the pressures put on us by familial and cultural expectations, and the ways in which those expectations are complicated by exposure to other ways of life. Tan presents a network of juxtapositions--contrasts of class, culture, and desire--that make this story of missed connection, of love potentially lost, one you'll be thinking about, and feeling, for a while after the last page.
Nguyen's poems are vibrant and mischievous, astoundingly witty and deeply intimate. To read this collection is to not only see the work of a keenly intelligent and clever mind, but to see that mind working as it examines the ways, and the costs, of carving out spaces for ourselves and our identities.
I urge you to read this book, but I'm telling you right now - you're not prepared for it. The worlds of Friday Black aren't exactly ours, but they are certainly, and disturbingly, familiar. One of the great accomplishments of these stories is their ability to contain together shocking brutality and enduring hope. At times darkly funny, at others deeply unsettling, Adjei-Brenyah has crafted a visceral collection of stories, each brimming with humanity that demands to be recognized and acknowledged. Please read.


I'm sharing 30+ of my very best Holiday Side Dishes recipes to get you ready for the holiday season. You'll find recipes for both traditional and unique side dishes!
Since Thanksgiving is right around the corner, I felt that I should share 30+ of my very best holiday side dishes that I've created. I'm sharing some recipes that are way back in the archives (with not so pretty pictures–but I assure you, that the recipes taste great) and then some recipes that are fairly new. Whether your looking for traditional holiday side dishes like mashed potatoes, stuffing, and sweet potato casserole or ready to spice it up with something new like ginger butternut squash mashed potatoes or maybe a broccoli and cheese casserole with jalapeño jack potato topping this list is for you! I hope this this list of 30+ holiday side dishes will officially put you at ease as we approach into the holiday season!


Description: This brand new 3 bedroom home offers an open living/dining layout, offering an indoor utility/mud room and outdoor covered patio. The kitchen features stainless steel appliances with gas range, microwave, quartz countertops, tile backsplash, and breakfast bar. The home boasts a HERS score of 64, guaranteeing you low heating and cooling costs all year long. Complete with fencing and landscaping, this home is now move-in ready!


The events for DPGs, MIGs and Affiliates; the Expo and the Foundation that enriched the FNCE® experience.
The conference and expo also offered events for both before and during the event.
FNCE® 2019 DPG, MIG and Affiliation information will be shared closer to the event.
FNCE® 2019 Expo Hall events will be shared closer to the event.
This new program offers six high-level discussions on hot topics.
Learn about the excursions and workshops available before the conference.


A video showing our Chamberlain Wireless Pedestrian & Vehicle Alert System with 2-Way Voice Communication and Remote Control .
This is a wireless intercom system that is pretty hard to find where you can use the intercom in 2-way.
We have it on the front and back of the house and it makes different chime sounds for each.
We are thinking about adding two more for a total of four.


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Fresh Music Friday
For Laughs
Celeb Chatter
Fashion Friday
Saturday Shade
ThisIsMoscatoLifeTV
Contact Us/Inquires
This Is Moscato Life
A Lifestyle Companion For The 9 to 5 Urbanite
Edmonia Lewis: First Female sculptor
Posted on February 1, 2017 by TIML Staff Posted in art, Feel Good News, History, News, UncategorizedTagged art, Edmonia Lewis, History, Lifestyle, News, Women
(TIML HISTORY) She was the first female African American sculptor to achieve international acclaim at a time when slavery was legal.
Mary Edmonia Lewis was a trailblazer who shattered racial barriers as the first professional African American sculptor in the mid-1800s, becoming famous for her 1,408kg marble sculpture, The Death of Cleopatra.
In honouring Lewis , America paid tribute to her artistic legacy and her effort to forge a path "for women and artists of color".
"Today, we celebrate her and what she stands for – self-expression through art, even in the face of [adversity],"
Born around July 1844, in New York, her father was of African Haitian origin and her mother of African American and Native American descent.
Lewis and her older half-brother, Samuel, were orphaned at a young age and raised by their maternal aunts, who lived near Niagara Falls, in northern New York state.
According to a Montana newspaper, Samuel started working as a barber at age 12, allowing him to support his sister. He would later become a respected barbershop entrepreneur in Bozeman, Montana.
At the age of 15, Lewis enrolled at Oberlin College, a private liberal arts school in the US state of Ohio. The year was 1859, and Oberlin was one of the very few institutions to admit women and people of color at a time when slavery was still legal. It was at Oberlin where she began her arts studies.
In 1862, she was accused of poisoning two classmates. Lewis was subsequently badly beaten by Whites and arrested. She was later acquitted, but the incident prevented her from completing her degree.
Shortly after, she left Ohio and headed east, arriving in Boston in 1864 to pursue a career as a sculptor.
Working in a field that was at the time dominated by white men, she was repeatedly rejected by instructors, until she met Edward A Brackett, a sculptor whose clients included some well-known advocates for the abolition of slavery.
Lewis earned her name in Boston with her many works paying homage to abolitionists and heroes of the Civil War. Abolitionist publications also noticed her work, featuring her in magazines and newspapers.
Her local success and popularity in Boston made possible her decision to move to Rome, and it was in the Italian capital that she became a highly respected artist. There, she focused on naturalism and themes relating to African American and Native American issues, achieving both financial and critical acclaim.
But it was her Cleopatra sculpture for which she became most well-known. It was featured at the 1876 Centennial Expo in the US city of Philadelphia and is now part of the collection of the Smithsonian American Art Museum.
She was also commissioned in 1877 to work on a portrait of US President Ulysses S Grant.
Among her other works is the 1867 Forever Free marble sculpture commemorating the ratification of the 13th Amendment. Ratified by sufficient state legislatures in December of that year, the amendment abolished slavery in the United States.
The National Freedom Day is the forerunner of Black History Month (BHM), which was conceived by a Harvard-trained historian in 1925.
In 1976, President Gerald R Ford urged Americans to "seize the opportunity to honour the too-often neglected accomplishments of black Americans in every area of endeavor throughout our history". Since then, every American president has officially proclaimed February as BHM. FYI everyday is Black History.
Despite achieving phenomenal success, she remained rooted to her Native culture.
"There is nothing so beautiful as the free forest. To catch a fish when you are hungry, cut the boughs of a tree, make a fire to roast it, and eat it in the open air, is the greatest of all luxuries. I would not stay a week pent up in cities, if it were not for my passion for art," Edmonia Lewis, quoted in "Letter From L Maria Child," National Anti-Slavery Standard, February 27, 1864.
Did You Know: The nephew of one of Emmett Till's killers is the governor of Mississippi
Beyoncé. Is Pregnant With Twins.
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'Pearl' Necklaces: The People Getting Jewelry Made From Semen (LOL) Seriously
Mike Tyson: 'They Stole My Life Story. To Hulu Executives I'm Just A N****r They Can Sell In The Auction Block.'
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Man with 9 wives has 'sex schedule' to 'fulfill them sexually at any cost'
Brooklyn subway shooting suspect's gun jammed during carnage: police sources
Yale Medical School Employee Admits To Stealing $40M From University, Buying Luxury Items
IHOP and Pepsi Made a Maple Syrup Cola — Will It Pair Well with Pancakes?


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