text:
\section{Introduction}\label{sec:intro}
It is a well known fact that for one-dimensional discrete data total variation (TV) regularization and TV flow are equivalent.
More specifically, denote by
	$$ J(u) =  \sum_{i=1}^{n-1} |u_i - u_{i+1}| $$
the total variation of $u \in \mathbb{R}^n$, and let $f\in \mathbb{R}^n$ and $\alpha>0$ be given.
Then, as was shown in \cite{Steidl1}, the minimizer $u_\alpha$ of the functional
	$$ \frac{1}{2} \|f - u\|^2_{2} + \alpha J(u)$$
coincides with the solution to the Cauchy problem
	\begin{align*}
		u'(t) &\in - \partial J(u(t)), \quad t>0,\\
		u(0)  &= f,
	\end{align*}
at time $t = \alpha$. That is, $u_\alpha = u(\alpha)$ for all $\alpha >0.$
On the other hand, it is known that $u_\alpha$ can also be obtained by means of the taut string algorithm (see \cite{Mammen}), which reads as follows.
\begin{enumerate}
	\item Identify the vector $f\in \mathbb{R}^n$ with a piecewise constant function on the unit interval and integrate it to obtain the linear spline $F$.
	\item Find the ``taut string" $U_\alpha$, that is, the element of minimal graph length in a tube of width $2\alpha$ around $F$ with fixed ends:
		$$ U_\alpha = \argmin \left\{ \int_0^1 \sqrt{1 + (U'(x))^2}\, dx: \|U - F\|_\infty \le \alpha, U(0)=F(0), U(1) = F(1) \right\} $$
	\item Differentiate $U_\alpha$ to obtain $u_\alpha.$
\end{enumerate}
Problems which essentially can be modelled and solved by the taut string algorithm appear in diverse applications. Examples include production planning, see for instance \cite{Modigliani1}, and energy and information transmission, e.g.\ \cite{Salehi1} and \cite{Yang1}.
Extensions of the taut string algorithm to more general data have been studied in \cite{Gra07,GraObe08,Hinterberger1}.
Further suggestions of generalizations of the taut string algorithm, in both discrete and continuous settings, can be found in \cite[Chap. 4.4]{Scherzer1}.

It turns out that the taut string does not only have minimal graph length, but actually minimizes \emph{every} functional of the form
\begin{equation*}
	U \mapsto \int^1_0 \varphi(U'(x))\, dx,
\end{equation*}
where $\varphi:\mathbb{R} \to \mathbb{R}$ is an arbitrary convex function and $U$ ranges over the $2\alpha$-tube around $F$. Recently, this intriguing situation was studied in greater generality in \cite{Kruglyak2,Kruglyak3}.
The authors coined the term \emph{invariant $\varphi$-minimal} for sets which, like the $2\alpha$-tube, have an element that simultaneously minimizes a large class of distances. In addition they characterized these sets in the discrete setting.

In this article we study relations between TV regularization, TV flow and taut strings in a setting that contains the one outlined above as a special case. More specifically, we consider data $f$ as given on the vertices of an oriented graph $G=(V,E)$ together with the total variation
\begin{equation} \label{eq:tv}
	J(f) = \sum_{v,w} |f(v) - f(w)|,
\end{equation}
where the sum runs over all adjacent pairs of vertices $v,w$.

Our first result concerns the subdifferential of $J$. In Theorem \ref{thm:subdifphimin} we prove that $\partial J(f)$ is an invariant $\varphi$-minimal set for every $f:V\to \mathbb{R}$. It is noteworthy that, as is shown in Remark \ref{rem:isorof}, this property is not shared by the discrete isotropic total variation, which for $f\in \mathbb{R}^{m \times n}$ reads\footnote{Here, $f\in \mathbb{R}^{m \times n}$ corresponds to $f$ being defined on the vertices of an $m\times n$ Cartesian graph as depicted in Figure \ref{F1}.}
\begin{equation}\label{eq:isotv}
	\sum_{i,j }\sqrt{(f_{i+1,j} - f_{i,j})^{2} + (f_{i,j+1}-f_{i,j})^{2}}
\end{equation}
and has been widely used in imaging applications, see \cite{AujAubBlaCha05,AujGilChaOsh06,Chambolle2} for instance.

Next we consider the Rudin-Osher-Fatemi (ROF) model \cite{Rudin1} on the graph
\begin{equation} \label{eq:rofgraph}
	\min_{u : V \to \mathbb{R}} \frac{1}{2} \sum_{v\in V}|f(v) - u(v)|^2 + \alpha J(u), \quad \alpha \ge 0.
\end{equation}
From its dual formulation and Theorem \ref{thm:subdifphimin} it follows that the solution $u_\alpha$ of problem \eqref{eq:rofgraph} has a characteristic feature resembling the universal minimality property of the taut string: It simultaneously minimizes
\begin{equation*}
	 \sum_{v\in V} \varphi (u(v))
\end{equation*}
over the set $f - \alpha \partial J(0)$ for every convex $\varphi,$ see Theorem \ref{T1}. We stress again that the minimizer of the isotropic ROF model, where $J(f)$ is given by \eqref{eq:isotv}, does not have this property.

Because of its anisotropy different variants of model \eqref{eq:rofgraph} have been used for imaging problems with an underlying rectilinear geometry \cite{BerBurDroNem06,choksi2011,SanOzkRomGok18,Setzer1}. Moreover, in contrast to \eqref{eq:isotv}, $J$ as given by \eqref{eq:tv} is submodular and for the minimization of submodular functions many efficient algorithms are available, for instance, graph cut algorithms \cite{Cha05,ChaDar09,DarSig06a,Hoc01}.

Finally, we examine the gradient flow for $J$ and how it relates to the ROF model. Such relations in higher dimensional settings have been the subject of recent investigations. In \cite{Burger1} discrete variational methods and gradient flows for convex one-homogeneous functionals are investigated and sufficient conditions for their equivalence are provided. A sufficient condition for the equivalence of TV regularization and TV flow with $\ell^{1}$-anisotropy in the continuous two-dimensional setting is given in \cite{Lasica1}. Considering the continuous setting with isotropic TV, it is shown in \cite{Jalalzai1} that TV regularization and TV flow coincide for radial data but in general are non-equivalent.

Our results in this direction are the following. First and foremost TV regularization and TV flow are not equivalent for general graphs and data $f$, see Theorem \ref{prop:noneq}. This result is based on a constructed example for which we are able to explicitly track the evolution of the two solutions $u_\alpha$ and $u(t)$ as $\alpha$ and $t$ range over an interval $[0,L].$ The example also shows that, in contrast to the one-dimensional setting, the jump sets do not necessarily evolve in a monotone way. Moreover, we investigate conditions for equality of $u_\alpha$ and $u(t=\alpha)$ and discuss situations in which they apply.

To summarize, let $\psi:\mathbb{R}\to\mathbb{R}$ by a strictly convex function, for the sake of analogy pick $\psi(x) = \sqrt{1 + x^2}.$ Then the problem
\begin{equation*}
	\min_{u \in f - \alpha \partial J(0)} \sum_{v\in V} \psi (u(v))
\end{equation*}
may be seen as a generalization of the taut string algorithm to oriented graphs for the following reasons.
\begin{itemize}
	\item The set $f - \alpha\partial J(0)$ reduces to the set of derivatives of the elements in the $2\alpha$-tube around $F$ in case the underlying graph is a path, that is, it models the one-dimensional situation described in the first paragraph of this introduction.
	\item The solution $u_{\alpha}$ in fact minimizes $\sum_{v\in V}\varphi(u(v))$ for any convex function $\varphi$.
	\item $u_{\alpha}$ minimizes the corresponding ROF model \eqref{eq:rofgraph}.
	\item Further, if $\alpha$ is either sufficiently small or sufficiently large, then $u_{\alpha}$ equals the TV flow solution at time $t=\alpha$.
\end{itemize}

This article is organized as follows. In Section \ref{S1} we introduce the graph setting and collect some properties of the total variation $J$. In particular we discuss the concept of invariant $\varphi$-minimal sets in Section \ref{sec:phiminimal}, while establishing a connection to base polyhedra in Section \ref{sec:submodular}. Sections \ref{sec:rof} and \ref{sec:tvflow} are dedicated to the two main problems considered in this paper, that is, total variation regularization and total variation flow, respectively. In Section \ref{sec:comparison} we compare the flow and ROF solutions. The detailed calculations underlying several results of Section \ref{sec:comparison} are collected in the appendix.

\section{Total variation on graphs} \label{S1}
Throughout this article, following the terminology of \cite{ChaLesZha10}, we consider oriented connected graphs $G=(V,E)$. That is, $V=\{v_1,\ldots,v_n\}$ and $E \subset V \times V$ with the additional conditions that, first, $(v_i,v_j) \in E$ implies $(v_j,v_i) \notin E$ and, second, there is a path between every pair of vertices (ignoring edge orientations). Whenever we simply write ``graph" below, we implicitly mean a graph of this type.
For $v,w \in V$ the edge $(v,w)\in E$ is interpreted as directed from $v$ to $w$.
Let $\mathbb{R}^{V}$ and $\mathbb{R}^{E}$ be the space of real-valued functions defined on the vertices and edges, respectively. We consider the usual $\ell^{p}$-norms on $\mathbb{R}^{V}$
\begin{align*}
	\|u\|^p_{p}		&=	\sum_{v\in V}|u(v)|^{p}, \quad 1\leq p<\infty, \\
	\|u\|_{\infty}	&=	\max_{v\in V}|u(v)|.
\end{align*}
Analogous $\ell^{p}$-norms will be considered on $\mathbb{R}^{E}$. In particular, denote the closed $\ell^{\infty}$-ball of radius $\alpha \ge 0$ in $\mathbb{R}^{E}$ by
\begin{align*}
 \mathcal{B}_{\alpha}=\{H\in \mathbb{R}^{E}:\lVert H\rVert_{\infty}\leq\alpha\}.
\end{align*}
Given $H\in \mathbb{R}^{E}$, define the \emph{divergence operator} $\divergence:\mathbb{R}^{E}\rightarrow \mathbb{R}^{V}$ according to
\begin{align*}
 (\divergence H)(v)=\sum_{w\in V:(w,v)\in E}H((w,v))-\sum_{w\in V:(v,w)\in E}H((v,w)).
\end{align*}
The divergence at the vertex $v$ can be thought of as the sum of the flows on the incoming edges minus the sum of the flows on the outgoing edges. We will frequently apply $\divergence$ to the unit ball $\mathcal{B}_{1}\in\mathbb{R}^{E}$ and its subset $\mathcal{B}_{1,u}$ defined, for given $u\in\mathbb{R}^{V}$, by
\begin{align*}
\mathcal{B}_{1,u}=\left\{H\in \mathbb{R}^{E}:
 H((v_{i},v_{j}))\in
 \left\{
\begin{array}{ll}
\{1\}, & u(v_{i})<u(v_{j}),\\
\left[-1,1\right], & u(v_{i})=u(v_{j}),\\
\{-1\}, & u(v_{i})>u(v_{j})
\end{array}
\right.
\right\}.
\end{align*}
Introduce further the natural scalar product on $\mathbb{R}^{V}$ according to
\begin{align*}
 \langle u,h\rangle_{\mathbb{R}^{V}}=\sum_{v\in V}u(v)h(v).
\end{align*}
For a closed and convex set $A\subset \mathbb{R}^V$ the \emph{support function} $\sigma_A : \mathbb{R}^V \to \mathbb{R}$ is given by
\begin{align*}
	\sigma_A (u) = \sup_{h\in A}\langle u , h \rangle_{\mathbb{R}^{V}}.
\end{align*}
\begin{definition}\label{def:tv}
The \emph{total variation} on $\mathbb{R}^{V}$ is defined as the support function of the set $\divergence \mathcal{B}_1$,
\begin{align}
 J(u)	&=	\underset{h\in\divergence \mathcal{B}_{1}}{\sup}\langle u,h\rangle_{\mathbb{R}^{V}}. \notag
\intertext{Since $J(u) = \langle u, \divergence H \rangle_{\mathbb{R}^V}$ for every $H \in \mathcal{B}_{1,u}$ we can rearrange the inner product to obtain}
 J(u)	&=	\sum_{(v_{i},v_{j})\in E}|u(v_{j})-u(v_{i})|. \label{TVG2}
\end{align}
\end{definition}

\begin{remark}
Equation \eqref{TVG2} shows that $J$ is independent of the orientation of edges, even though the divergence is not. All subsequent results remain true regardless of edge orientation, and also apply to simple undirected graphs once each edge has been oriented arbitrarily.
\end{remark}

\begin{definition}\label{dfn:subdif}
For every $u\in \mathbb{R}^V$ the \emph{subdifferential} $\partial J(u)$ is defined as the set of all elements $u^{*}\in \mathbb{R}^{V}$ such that
\begin{align*}
\langle h-u,u^{*}\rangle_{\mathbb{R}^{V}}+J(u)\leq J(h) \quad \text{for all } h\in \mathbb{R}^{V}.
\end{align*}
\end{definition}
Since $\partial J(u)$ is a closed, convex and non-empty subset of $\mathbb{R}^{V}$, we can highlight one particular element.
\begin{definition}
 The element of minimal $\ell^2$-norm in $\partial J(u)$ will be referred to as the \emph{minimal section} of $\partial J(u)$. It is denoted by $\partial^\circ J(u)$, that is,
 \begin{align*}
  \partial^\circ J(u) = \underset{u^{*}\in\partial J(u)}{\argmin}\lVert u^{*}\rVert_{{2}}.
 \end{align*}
\end{definition}
\par The following lemma collects some results for the subdifferential $\partial J$ which will be used in the sequel.
\begin{lemma} \label{L1}
\leavevmode
\begin{enumerate}
\item $\partial J(0)=\divergence\mathcal{B}_{1}$.
\item $\partial J(u)=\{u^{*}\in\partial J(0):\langle u,u^{*}\rangle_{\mathbb{R}^{V}}=J(u)\} \quad \text{for all } u\in \mathbb{R}^{V}$.
\item $\partial J(u)=\divergence\mathcal{B}_{1,u} \quad \text{for all } u\in \mathbb{R}^{V}$.
\end{enumerate}
\end{lemma}
\begin{proof}
The functional $J$ is the support function of the closed and convex set $\divergence\mathcal{B}_{1}$ and therefore $\partial J(0)=\divergence\mathcal{B}_{1}$.

Item 2 follows from Definition \ref{dfn:subdif} and the absolute $1$-homogeneity of $J$, that is, $J(tu) = |t|J(u)$ for all $t\in \mathbb{R}$ and $u \in \mathbb{R}^V$.

Regarding item 3, note that $J(u)= \langle u,\divergence H \rangle_{\mathbb{R}^V}$ for $H\in\mathcal{B}_{1}$ if and only if $H\in\mathcal{B}_{1,u}$. In view of item 2, it is then clear that $\partial J(u)=\divergence\mathcal{B}_{1,u}$.
\end{proof}
\begin{remark}\label{rem:subdifequal1}
\leavevmode
\begin{enumerate}
\item Since, according to item 3 in Lemma \ref{L1}, the set $\mathcal{B}_{1,u}$ only depends on $\sgn(u(v_i) - u(v_j))$ for every edge $(v_{i},v_{j})\in E$, we have
\begin{align*}
 \partial J(u)=\partial J(h),
\end{align*}
if and only if
\begin{align*}
 \sgn(u(v_{i})-u(v_{j}))=\sgn(h(v_{i})-h(v_{j}))
\end{align*}
for each $(v_{i},v_{j})\in E$.
\item It now follows immediately that, if the subdifferentials of $J$ at $u$ and $h$ coincide, then they also coincide for every convex combination of $u$ and $h$. That is, $\partial J(u) = \partial J(h)$ implies $\partial J(\lambda u + (1-\lambda)h) = \partial J(u)$ for every $\lambda \in (0,1)$.
\item Lemma \ref{L1} also implies that the number of different subdifferentials of $J$ is finite. In particular,
 \begin{align*}
  \left|\left\{\partial J(u):u\in \mathbb{R}^{V}\right\}\right|\leq 3^{\left|E\right|}.
 \end{align*}
 This must not be confused with the fact that for any given $u\in \mathbb{R}^{V}$ the subdifferential $\partial J(u)$ might have infinitely many elements.
\end{enumerate}
\end{remark}

\subsection{Connections to invariant $\varphi$-minimal sets}\label{sec:phiminimal}
In this subsection we recall the notion of invariant $\varphi$-minimal sets introduced in \cite{Kruglyak2} and show that the subdifferential $\partial J(u)$ is an example of such a set.
\begin{definition}\label{def:invphimin}
 A set $\Omega\subset\mathbb{R}^{n}$ is called \emph{invariant $\varphi$-minimal} if for every $a\in\mathbb{R}^{n}$ there exists an element $x_{a}\in\Omega$ such that
 \begin{align} \label{IP}
  \sum^{n}_{i=1}\varphi(x_{a,i}-a_{i})\leq\sum^{n}_{i=1}\varphi(x_{i}-a_{i})
 \end{align}
holds for all $x\in\Omega$ and all convex functions $\varphi:\mathbb{R}\rightarrow\mathbb{R}$.
\end{definition}
An interesting property of invariant $\varphi$-minimal sets is the following. By considering the particular convex function $\varphi(x)=\left|x\right|^{p}$, $1\leq p<\infty$, in \eqref{IP} we obtain
\begin{align*}
\sum^{n}_{i=1}\left|x_{a,i}-a_i\right|^{p} \leq \sum^{n}_{i=1} \left|x_i-a_i\right|^{p}
\end{align*}
for all $x\in\Omega$. Taking the $p$-th root and including the case $p=\infty$, which follows by limiting arguments, shows that the element $x_{a}$ satisfies
\begin{align*}
 \lVert x_{a}-a\rVert_{{p}}\leq\lVert x-a\rVert_{{p}}
\end{align*}
for all $x\in\Omega$ and $1\leq p\leq\infty$. That is, $x_{a}$ is an element of best approximation of $a$ in $\Omega$ with respect to all $\ell^{p}$-norms, $1\leq p\leq\infty$.

Before we can restate two characterizations of invariant $\varphi$-minimal sets from \cite{Kruglyak2} we have to introduce several notions about convex subsets of $\mathbb{R}^n$.

A hyperplane $H$ \emph{supports} a set $M\subset\mathbb{R}^{n}$ if $M$ is contained in one of the two closed halfspaces with boundary $H$ and at least one boundary point of $M$ is in $H$. Assume that $M\subset\mathbb{R}^{n}$ is convex. Following the terminology of \cite{Grunbaum2} a set $F\subset M$ is called a \emph{face} of $M$ if $F=\emptyset$, $F=M$ or if $F=M\cap H$ where $H$ is a supporting hyperplane of $M$. A \emph{convex polytope} $P$ in $\mathbb{R}^{n}$ is a bounded set which is the intersection of finitely many closed halfspaces. Note that a face of a convex polytope is itself a convex polytope.

 Let $\Omega\subset\mathbb{R}^{n}$ be closed and convex and denote by $\left\{e_{i}\right\}^{n}_{i=1}$ the standard basis of $\mathbb{R}^{n}$. For $x\in\Omega$, consider all vectors $y=e_{i}-e_{j}$ such that $x+\beta y\in\Omega$ for some $\beta>0$. Let $S_{x}$ denote the set of all such vectors at $x$. Further, let $K_{x}=\{z:z=\sum_{y\in S_{x}}\lambda_{y}y,\lambda_{y}\geq 0\}$ be the convex cone generated by the vectors in $S_{x}$. We say that $\Omega$ has the \emph{special cone property} if $\Omega\subset x+K_{x}$ for each $x\in\Omega$.
\begin{remark}
In \cite{Kruglyak2} vectors of the type $e_{i}$ and $e_{i}+e_{j}$ are considered in addition to $e_i-e_j$ in the definition of the special cone property. Including these vectors leads to a characterization of the related notion of invariant $K$-minimal sets.
\end{remark}

\begin{theorem}\label{thm:phimin}
Let $\Omega\subset\mathbb{R}^{n}$ be a bounded, closed and convex set. Then the following statements are equivalent.
\begin{enumerate}
	\item $\Omega$ is invariant $\varphi$-minimal.
	\item $\Omega$ has the special cone property.
	\item $\Omega$ is a convex polytope where the affine hull of any of its faces is a shifted subspace of $\mathbb{R}^{n}$ spanned by vectors of the type $e_{i}-e_{j}$.
\end{enumerate}
\end{theorem}
\begin{proof}
Equivalence of statements 1 and 2 follows from combining Thms.\ 3.2 and 4.2 in \cite{Kruglyak2}. Equivalence of statements 1 and 3 is precisely Thm.\ 4.3 in \cite{Kruglyak2}.
\end{proof}

An example of an invariant $\varphi$-minimal set in the plane is depicted in Figure \ref{fig:invphimin}, left panel.
\begin{figure}
\centering
\begin{tikzpicture}[scale=1]
    \draw[->] (-2.1,0)  -- (2.5,0) node(xline)[right] {$x_1$};
    \draw[->] (0,-2.1) -- (0,2.5) node(yline)[above] {$x_2$};
    \draw[-,thick] (-1,1) -- (1,-1);
    \draw[-] (1,0) -- (1,.1);
    \draw[-] (0,1) -- (.1,1);
    \draw[-] (-1,0) -- (-1,.1);
    \draw[-] (0,-1) -- (.1,-1);
    \node at (.875,-.2)  {$\phantom{-}1$};
    \node at (-.35,1)  {$\phantom{-}1$};
    \node at (-1.125,-.2)  {$-1$};
    \node at (-.35,-1)  {$-1$};
\end{tikzpicture}
\qquad
\begin{tikzpicture}[scale=1]
    \fill[gray!20] (-2,1) -- (0.5,1) -- (1.7,-0.2) -- (1.7,-2.1) --(-2.1,-2.1) -- (-2.1,1);
    \draw[->] (-2.1,0)  -- (2.5,0) node(xline)[right] {$x_{1}$};
    \draw[->] (0,-2.1) -- (0,2.5) node(yline)[above] {$x_{2}$};
    \draw[-,thick] (0.5,1) -- (1.7,-0.2);
    \node at (-1,-1)  {$P(g)$};
    \node at (1.4,0.75)  {$B(g)$};
\end{tikzpicture}
\caption{Left: The slanted line segment is an example of an invariant $\varphi$-minimal set as characterized by Theorem \ref{thm:phimin}. In fact, identifying $x_i$ with $u^*(v_i)$, it is the subdifferential $\partial J(0)$ for $J$ being defined on the graph with $V=\{v_1,v_2\}$ and $E=\{(v_1,v_2)\}$. All other invariant $\varphi$-minimal sets in $\mathbb{R}^2$ are translations and rescalings of $\partial J(0)$. Right: Submodular polyhedron (in grey) and base polyhedron (slanted line segment) in the plane.}
\label{fig:invphimin}
\end{figure}
We are now ready to show
\begin{theorem} \label{thm:subdifphimin}
 The subdifferential $\partial J(u)$ is an invariant $\varphi$-minimal set.
\end{theorem}
\begin{proof}
Consider first $\partial J(0)$. In \cite[Thm.\ 2.4, Rem.\ 2.5]{Kruglyak3} it is established that the bounded, closed and convex set $\divergence\mathcal{B}_{\alpha}\subset\mathbb{R}^{V}$ is invariant $\varphi$-minimal by showing that it has the special cone property. It follows that $\partial J(0)=\divergence\mathcal{B}_{1}$ is an invariant $\varphi$-minimal set.
\par Take next a general $u\in \mathbb{R}^{V}$. We have $\partial J(u)=H\cap\partial J(0)$ where $H=\{u^{*}\in\mathbb{R}^{V}:\langle u^{*},u\rangle_{\mathbb{R}^{V}}=J(u)\}$, recall Lemma \ref{L1}. Consider the halfspace $\widehat{H}=\{u^{*}\in\mathbb{R}^{V}:\langle u^{*},u\rangle_{\mathbb{R}^{V}}\leq J(u)\}$ with boundary $H$. Note that (i) $\partial J(0)\subset\widehat{H}$, (ii) $H\cap\partial J(0)\neq\emptyset$ and (iii) $\partial J(0)$ is a convex polytope. So, $H$ is a supporting hyperplane of $\partial J(0)$ and $\partial J(u)$ is a face of $\partial J(0)$ and itself a convex polytope. Further, every face of $\partial J(u)$ is a face of $\partial J(0)$. This follows from a general result on faces of convex polytopes, see e.g. \cite[Chap.\ 3.1, Thm.\ 5]{Grunbaum2}. Therefore $\partial J(u)$ satisfies statement 3 in Theorem \ref{thm:phimin}.
\end{proof}
\begin{remark}
 As $\partial J(u)$ is an invariant $\varphi$-minimal set, it follows that the minimal section $\partial^\circ J(u)$ not only has minimal $\ell^2$-norm in $\partial J(u)$, but satisfies
 \begin{align*}
  \sum_{v\in V}\varphi(\partial^\circ J(u)(v))=\underset{u^{*}\in\partial J(u)}{\min}\sum_{v\in V}\varphi(u^{*}(v))
 \end{align*}
for every convex function $\varphi:\mathbb{R}\rightarrow\mathbb{R}$.
\end{remark}

\subsection{Invariant $\varphi$-minimal sets and submodular functions} \label{sec:submodular}
To conclude this section, we present an interesting connection between submodular functions and invariant $\varphi$-minimal sets. Submodular functions play an important role in combinatorial optimization, similar to that of convex functions in continuous optimization. See \cite{Bach13,Fujishige1} for more details.
\par Let $S=\left\{1,...,n\right\}$. A set function $g:2^{S}\rightarrow\mathbb{R}$ is \emph{submodular} if
 \begin{align*}
  g(A)+g(B)\geq g(A\cup B)+g(A\cap B)
 \end{align*}
for all sets $A,B\subset S$. Given a submodular function $g$, assuming $g(\emptyset)=0$, the associated \emph{submodular polyhedron} $P(g)$ and \emph{base polyhedron} $B(g)$ are defined by
\begin{align*}
 P(g)=\left\{x\in\mathbb{R}^{n}:\forall A\subset S,\sum_{i\in A}x_{i}\leq g(A)\right\},
\end{align*}
\begin{align*}
 B(g)=\left\{x\in P(g):\sum_{i\in S}x_{i}=g(S)\right\}.
\end{align*}
Note that $B(g)$ is a bounded set and therefore a convex polytope. In the plane we can easily visualize submodular and base polyhedra, see Figure \ref{fig:invphimin}, right panel, for an example.

\par Define the \emph{tangent cone} $T_{P}(x)$ of a convex polytope $P\subset\mathbb{R}^{n}$ at $x\in P$ by
\begin{align*}
 T_{P}(x)=\left\{\lambda z:\lambda\geq 0,x+z\in P\right\}.
\end{align*}
N. Tomizawa characterized, see \cite[Thm. 17.1]{Fujishige1}, base polyhedra according to
\begin{theorem} \label{thm:basepoly}
 A convex polytope $P\subset\mathbb{R}^{n}$ is a base polyhedron if and only if for all $x\in P$, the tangent cone $T_{P}(x)$ is generated by vectors of the type $e_{i}-e_{j}$, $i\neq j$.
\end{theorem}
With this characterization at hand, the connection between invariant $\varphi$-minimal sets and submodular functions can be revealed.
\begin{proposition} \label{thm:equivbasephi}
 A bounded, closed and convex set $\Omega\subset\mathbb{R}^{n}$ is invariant $\varphi$-minimal if and only if it is a base polyhedron associated to a submodular function $g:2^{S}\rightarrow\mathbb{R}$.
\end{proposition}
\begin{proof}
 Recall from Theorem \ref{thm:phimin} that $\Omega$ is invariant $\varphi$-minimal if and only if it has the special cone property. Next, it is straightforward to derive that $\Omega$ has the special cone property if and only if the tangent cone $T_{\Omega}(x)$, for every $x\in\Omega$, is generated by vectors of the type $e_{i}-e_{j}$, $i\neq j$. This is precisely the characterization of a base polyhedron as given by Theorem \ref{thm:basepoly}.
\end{proof}
\begin{remark}
Figure \ref{fig:invphimin} illustrates the equivalence of invariant $\varphi$-minimal sets and base polyhedra. Note that the subdifferential $\partial J(0)$ is the base polyhedron $B(g)$ associated to the cut function $g$ on the graph, see \cite[Sec.\ 6.2]{Bach13}.
\end{remark}

\section{The ROF model on the graph}\label{sec:rof}
With the graph setting introduced, we now turn to an analogue of the ROF image denoising model on $\mathbb{R}^{V}$. Given $f\in \mathbb{R}^{V}$ and $\alpha \ge 0$ we consider the following minimization problem:
\begin{align} \label{ROFG}
 \underset{u\in \mathbb{R}^{V}}{\min} \frac{1}{2}\|f-u\|^{2}_{{2}}+\alpha J(u).
\end{align}
Throughout this article the unique solution to \eqref{ROFG} will be denoted by $u_\alpha$.

\subsection{Dual formulation and an invariance property of the ROF minimizer} \label{S31}
The next proposition remains true, if $J$ is replaced by the support function of an arbitrary closed and convex subset of $\mathbb{R}^V.$
\begin{proposition}\label{thm:rofconstrained}
For every $f\in \mathbb{R}^{V}$ and $\alpha \ge 0$ problem \eqref{ROFG} is equivalent to
\begin{align} \label{206}
\underset{u\in f - \alpha \partial J(0)}{\min}\|u\|_{{2}}.
\end{align}
\end{proposition}
\begin{proof}
The corresponding dual problem of \eqref{ROFG} can be expressed as
\begin{align} \label{ROFGD}
 \underset{u^*\in \mathbb{R}^{V}}{\min} \frac{1}{2}\|f-u^*\|^{2}_{{2}}+(\alpha J)^{*}(u^*),
\end{align}
where $(\alpha J)^{*}$ denotes the convex conjugate of $\alpha J$. For general results underlying the derivation of \eqref{ROFGD} and the optimality conditions \eqref{OC1} below, see \cite[Chap. III, Prop. 4.1, Rem. 4.2]{Ekeland2}. Let $u_{\alpha}$ and $u^*_{\alpha}$ denote solutions to the primal problem \eqref{ROFG} and the dual problem \eqref{ROFGD} respectively. The optimality conditions are
\begin{equation}\label{OC1}
	\begin{aligned}
		u^*_{\alpha}	&\in\partial(\alpha J)(u_{\alpha})=\alpha\partial J(u_{\alpha}) \\
		u_{\alpha}	&=f-u^*_{\alpha}.
	\end{aligned}
\end{equation}
As $\alpha J(u)$ is the support function of $\divergence\mathcal{B}_{\alpha} = \alpha \partial J(0)$, its convex conjugate $(\alpha J)^{*}$ is given by
\begin{equation*}
\begin{aligned}
(\alpha J)^{*}(u^*)=
\left\{
\begin{array}{lr}
0,			& u^* \in \alpha \partial J(0),\\
+\infty,	& u^* \notin \alpha \partial J(0).
\end{array}
\right.
\end{aligned}
\end{equation*}
Taking into account the characterization of $(\alpha J)^{*}$ in the dual formulation \eqref{ROFGD} yields
\begin{align*}
 u^*_{\alpha} = \underset{u^*\in\alpha \partial J(0)}{\argmin}\|f-u^*\|_{2}.
\end{align*}
That is, $u^*_{\alpha}$ is the orthogonal projection of $f$ onto the closed and convex set $\divergence\mathcal{B}_{\alpha}$. For $u_{\alpha}$ we now obtain using \eqref{OC1} that
\begin{align*}
	\left\|u_{\alpha}\right\|_{{2}}=\|f-u^*_{\alpha}\|_{{2}}
		= \underset{u^*\in\alpha \partial J(0)}{\min}\left\|f-u^*\right\|_{{2}}
		= \underset{u\in f-\alpha \partial J(0)}{\min}\left\|u\right\|_{{2}}.
\end{align*}
\end{proof}
\begin{theorem} \label{T1}
The ROF minimizer $u_{\alpha}$ satisfies
 \begin{align} \label{TVRP}
 \sum_{v\in V}\varphi(u_{\alpha}(v))=\underset{u\in f-\alpha\partial J(0)}{\min}\sum_{v\in V}\varphi(u(v))
\end{align}
for every convex function $\varphi:\mathbb{R}\rightarrow\mathbb{R}$.
\end{theorem}
\begin{proof}
According to Theorem \ref{thm:subdifphimin} the set $\alpha\partial J(0)$ is invariant $\varphi$-minimal, recall Definition \ref{def:invphimin}. From the above derivation of the dual formulation, we know that $u_{\alpha}$ is the $\ell^{2}$-minimizer in the set $f-\alpha\partial J(0)$. Taken together, this gives \eqref{TVRP}.
\end{proof}
While Proposition \ref{thm:rofconstrained} is valid for every support function of a closed and convex set, Theorem \ref{T1} fails in this more general case. The following remark discusses this failure for the so-called discrete isotropic total variation.
\begin{remark}\label{rem:isorof}
Let $G=(V,E)$ be an $M\times N$ Cartesian graph, as illustrated in Figure \ref{F1}. On such graphs the following variant of $J$ has been a popular choice, in particular for image processing applications
\begin{equation*}
\begin{aligned}
J_{\mathrm{iso}}(u)=\sum^{N-1}_{j=1}\sum^{M-1}_{i=1}\sqrt{\left|u(v_{i+1,j})-u(v_{i,j})\right|^{2}+\left|u(v_{i,j+1})-u(v_{i,j})\right|^{2}}+\\
\sum^{M-1}_{i=1}\left|u(v_{i+1,N})-u(v_{i,N})\right|+\sum^{N-1}_{j=1}\left|u(v_{M,j+1})-u(v_{M,j})\right|,
\end{aligned}
\end{equation*}
see, for instance, \cite{AujAubBlaCha05,AujGilChaOsh06,Chambolle2}. It can be shown that $J_{\mathrm{iso}}$ is the support function of $\divergence \mathcal{B}^{\text{iso}}_{1}$, where
\begin{align*}
	\mathcal{B}^{\text{iso}}_{1} = \left\{H\in \mathbb{R}^{E}: \max_{i,j} C_{ij}(H) \leq 1 \right\},
\end{align*}
and $C_{ij}(H)$ is given by
\begin{equation*}
	C_{ij}(H)=
 	\begin{cases}
 		\sqrt{H((v_{i+1,j},v_{i,j}))^{2}+H((v_{i,j+1},v_{i,j}))^{2}},	& i \le M-1 ,\; j \le N-1,\\
 		|H((v_{i+1,N},v_{i,N}))|,	& i \le M-1,\; j=N, \\
		|H((v_{M,j+1},v_{M,j}))|,	& i=M,\; j \le N-1,\\
		0, & i=M,\; j=N.
 	\end{cases}
\end{equation*}
Let $M,N>1$. From the construction of $\mathcal{B}^{\text{iso}}_{1}$ it follows that $\partial J_{\text{iso}} (0) = \divergence \mathcal{B}^{\text{iso}}_{1}$ is not a polytope and therefore, by Theorem \ref{thm:phimin}, it cannot be invariant $\varphi$-minimal. Consequently, the minimizer of the isotropic ROF model, which can be characterized as
	$$ \argmin_{u\in f-\alpha\partial J_{\text{iso}}(0)} \|u\|_2,$$
in general does not have property \eqref{TVRP}.
\begin{figure}[h]
\centering
 \begin{tikzpicture}[scale=0.75]
\tikzstyle{every node}=[draw, circle, thick, fill=gray!20, minimum width={width("fjhere")},]

	\node       	 (a) at (0,0)  {$v_{1,2}$};
	\node            (b) at (2,0)  {$v_{2,2}$};
	\node            (c) at (4,0)  {$v_{3,2}$};
	\node            (d) at (2,2)  {$v_{2,3}$};
	\node            (e) at (2,-2) 	{$v_{2,1}$};
	\node            (f) at (0,2)  {$v_{1,3}$};
	\node            (g) at (0,-2) 	{$v_{1,1}$};
	\node            (h) at (4,-2) 	{$v_{3,1}$};
	\node            (i) at (4,2)  {$v_{3,3}$};
	\draw[<-,thick] (a) edge (b);
	\draw[<-,thick] (b) edge (c);
	\draw[<-,thick] (b) edge (d);
	\draw[<-,thick] (e) edge (b);
	\draw[<-,thick] (a) edge (f);
	\draw[<-,thick] (g) edge (a);
	\draw[<-,thick] (f) edge (d);
	\draw[<-,thick] (g) edge (e);
	\draw[<-,thick] (d) edge (i);
	\draw[<-,thick] (c) edge (i);
	\draw[<-,thick] (h) edge (c);
	\draw[<-,thick] (e) edge (h);
\end{tikzpicture}
\caption{A $3 \times 3$ Cartesian graph.}
\label{F1}
\end{figure}
\end{remark}
\begin{remark}
In the continuous setting it is known that an analogue of Theorem \ref{T1} holds for isotropic total variation, see \cite[Thm.\ 4.46]{Scherzer1}.
\end{remark}

\subsection{Further properties of the ROF minimizer} \label{S3}
In this subsection we study further properties of the ROF minimizer $u_{\alpha}$. We first give an auxiliary result.
\begin{lemma} \label{thm:convexcombination}
 Let $0\leq\beta_{1}<\beta_{2}$. If $\partial J(u_{\beta_{1}})=\partial J(u_{\beta_{2}})$, then for every $\alpha \in (\beta_1,\beta_2)$ the ROF minimizer $u_\alpha$ is a convex combination of $u_{\beta_1}$ and $u_{\beta_2}$. That is,
 \begin{align} \label{u(alpha)}
   u_{\alpha} = \frac{\beta_{2}-\alpha}{\beta_{2}-\beta_{1}}u_{\beta_{1}} + \frac{\alpha-\beta_{1}}{\beta_{2}-\beta_{1}}u_{\beta_{2}},\quad \beta_{1} < \alpha < \beta_{2}.
 \end{align}
\end{lemma}
\begin{proof}
Denote the convex combination in \eqref{u(alpha)} by $c(\alpha)$. It suffices to verify that $c(\alpha)$ satisfies the optimality conditions \eqref{OC1}, that is, $f - c(\alpha) \in \alpha \partial J(c(\alpha))$. First, note that by item 2 in Remark \ref{rem:subdifequal1} we have $\partial J(c(\alpha)) = \partial J(u_{\beta_1}).$ Next, let $u^*_{\beta_{i}} = f - u_{\beta_i}$, $i=1,2$.

If $\beta_1>0$, we compute
\begin{align*}
	\frac{f-c(\alpha)}{\alpha}
		&= \frac{1}{\alpha} \left[ \frac{\beta_{2}-\alpha}{\beta_{2}-\beta_{1}}u^*_{\beta_{1}}+\frac{\alpha-\beta_{1}}{\beta_{2}-\beta_{1}}u^*_{\beta_{2}} \right] \\
		&= \frac{\beta_1}{\alpha} \frac{\beta_{2}-\alpha}{\beta_{2}-\beta_{1}} \frac{u^*_{\beta_{1}}}{\beta_1} +
		   \frac{\beta_2}{\alpha} \frac{\alpha-\beta_{1}}{\beta_{2}-\beta_{1}} \frac{u^*_{\beta_{2}}}{\beta_2}.
\end{align*}
It is straightforward to check that the last expression is a convex combination of $u^*_{\beta_{1}}/\beta_1$ and $u^*_{\beta_{2}}/\beta_2$. By optimality of $u_{\beta_i}$ and the assumption that $\partial J(u_{\beta_{1}})=\partial J(u_{\beta_{2}})$, both $u^*_{\beta_{i}}/\beta_i$ lie in the same convex set $\partial J(u_{\beta_1}).$ Therefore $(f-c(\alpha))/\alpha$ is in this set, too. We conclude that $c(\alpha)$ must be the ROF minimizer $u_\alpha$.

If $\beta_1=0$, then $u^*_{\beta_1} = 0$ and $(f-c(\alpha))/\alpha = u^*_{\beta_2}/\beta_2 \in \partial J(c(\alpha)).$
\end{proof}
We can now show the following properties of the ROF minimizer.
\begin{proposition} \label{P7}
\leavevmode
\begin{enumerate}
 \item Problem \eqref{ROFG} is mean-preserving, that is
 $$ \sum_{v\in V} u_{\alpha}(v) = \sum_{v\in V} f(v) \quad \text{for all }\alpha\ge 0.$$
 \item The function $\alpha \mapsto	\| u_{\alpha} \|_{2}$ is nonincreasing on $\left[0,\infty\right)$.
\item The solution $u_{\alpha}$ is a continuous piecewise affine function with respect to $\alpha$. Its piecewise constant derivative $d  u_{\alpha}/d \alpha$ exists everywhere except for a finite number of values of $0<\alpha_{1}<...<\alpha_{N}<\infty$. In particular,
 \begin{align}\label{eq:stationaryrof}
  u_{\alpha}(v)=\frac{1}{\left|V\right|}\sum_{w\in V}f(w),\quad \text{for all }\alpha\geq\alpha_{N}\text{ and }v\in V.
 \end{align}
\end{enumerate}
\end{proposition}
\begin{proof}
\leavevmode
\begin{enumerate}
\item According to Proposition \ref{thm:rofconstrained} we have $u_{\alpha}=f-\divergence H$ for an $H\in\mathbb{R}^E$. Summing this equation over all $v\in V$ and using the fact that $\sum_{v\in V} \divergence H(v)$ vanishes for every $H\in \mathbb{R}^E$ gives $\sum_{v\in V} u_{\alpha}(v) = \sum_{v\in V} f(v)$ for all $\alpha\geq 0$.
\item From the dual formulation of the ROF model, we know that $u_{\alpha}$ is the $\ell^{2}$-minimizer in the set $f-\divergence\mathcal{B}_{\alpha}$. Since $f-\divergence\mathcal{B}_{\beta_{1}}\subset f-\divergence\mathcal{B}_{\beta_{2}}$, $\beta_{1}\leq\beta_{2}$, it then follows that $\alpha\mapsto\| u_{\alpha} \|_{2}$ is nonincreasing.
\item We first prove that the map $\alpha \mapsto u_{\alpha}$ is continuous.
Consider a convergent sequence of regularization parameters $\alpha_n \to \alpha$. According to the optimality condition \eqref{OC1} the corresponding minimizers $u_n \coloneqq u_{\alpha_n}$ and $u \coloneqq u_{\alpha}$ can be expressed as
\begin{align*}
	u_{n}	&=	f - \alpha_n u_n^*, \\
	u	&=	f - \alpha u^*,
\end{align*}
for certain $u^*_n \in \partial J(u_{n})$ and $u^* \in \partial J(u)$. We compute
\begin{align*}
	\| u_{n} - u \|_2^2
		&= \langle u_{n} - u, u_{n} - u \rangle_{\mathbb{R}^V} \\
		&= \langle u_{n} - u, \alpha u^* - \alpha_n u^*_n \rangle_{\mathbb{R}^V} \\
		&= \alpha \langle u_{n}, u^*\rangle_{\mathbb{R}^V} -  \alpha_n \langle u_{n} , u^*_n \rangle_{\mathbb{R}^V} - \alpha \langle u, u^* \rangle_{\mathbb{R}^V} + \alpha_n \langle u, u^*_n \rangle_{\mathbb{R}^V}.
	\intertext{Using the fact that $\langle u,u^* \rangle = J(u)$ and $\langle u_n,u_n^* \rangle = J(u_n)$ while $\langle u_n,u^* \rangle \le J(u_n)$ and $\langle u,u_n^* \rangle \le J(u)$ according to Lemma \ref{L1}, we obtain}
	\| u_{n} - u \|_2^2
		&\le \alpha J(u_{n}) - \alpha_n J(u_{n}) - \alpha J(u) + \alpha_n J(u) \\
		&\le |\alpha_n - \alpha| |J(u) + J(f)|,
\end{align*}
and therefore $u_n \to u$.

The piecewise affine structure of $u_{\alpha}$ has been shown in \cite[Thm.~4.6]{Burger1}. However, since our proof relies on different arguments, we choose to include it.

From Lemma \ref{thm:convexcombination} as well as Remark \ref{rem:subdifequal1}, items 2 and 3, we can derive two important facts. These two facts, combined with continuity of the map $\alpha \mapsto u_\alpha$, show that it must be piecewise affine on $[0,\infty)$. First, the subdifferential $\partial J(u_\alpha)$ can only change a finite number of times. Second, in intervals where it does not change, the minimizer $u_\alpha$ is an affine function of $\alpha$.
\par Finally, consider $u_{\alpha}$ for $\alpha\geq\alpha_{N}$, where $\alpha_{N}$ is the last time $\partial J (u_\alpha)$ changes. Let $\bar{f}$ denote the averaged initial image $f$, i.e.
 \begin{align}\label{eq:faverage}
  \bar{f}(v)=\frac{1}{\left|V\right|}\sum_{w\in V}f(w), \quad \text{for all }v\in V.
 \end{align}
For $\alpha\geq C$, where $C>0$ is chosen large enough, it follows that $\bar{f}\in f-\divergence\mathcal{B}_{\alpha}$. Clearly, $\bar{f}$ is the $\ell^{2}$-minimizer in $f-\divergence\mathcal{B}_{\alpha}$. Combined with the piecewise affine structure of $u_{\alpha}$, we conclude that $u_{\alpha}=\bar{f}$ for $\alpha\geq\alpha_{N}$.
\end{enumerate}
\end{proof}
\begin{remark}
Recall that in Section \ref{S1} we have assumed the graph to be connected. If this assumption is dropped, then \eqref{eq:stationaryrof} does not hold in general, since $\bar f$ might not be a minimizer for any $\alpha.$ If the graph is disconnected, however, the ROF problem decouples into mutually independent subproblems, one for each connected component of the graph. Statement \eqref{eq:stationaryrof} then applies to each subproblem. An analogous remark can be made about property \eqref{eq:stationaryflow} of the TV flow.
\end{remark}

\section{The TV flow on the graph}\label{sec:tvflow}
In this section we consider the gradient flow associated to $J$. That is, given an initial datum $f:V\to \mathbb{R}$ we want to find a function $u:[0,\infty) \to \mathbb{R}^V$ that solves the Cauchy problem
\begin{equation} \label{TVFG}
	\begin{aligned}
		u'(t)	&\in 	-\partial J(u(t)) \quad \text{for a.e. } t > 0,\\
		u(0)	&=		f.
	\end{aligned}
\end{equation}
The statements in the next theorem follow from general results on nonlinear evolution equations and semigroup theory. See \cite[Chap.\ 4]{Barbu1} for a detailed treatment and \cite[Sec.\ 2.1]{Santambrogio1} for a brief introduction to the finite-dimensional setting.
\begin{theorem}\label{thm:flowwellposed}
Solutions to problem \eqref{TVFG} have the following properties.
\begin{enumerate}
	\item For every $f\in \mathbb{R}^V$ there is a unique solution and this solution depends continuously on $f$. In particular, if $u_1$ and $u_2$ are two solutions corresponding to initial conditions $f_1$ and $f_2$, respectively, then
		$$ \| u_1(t) - u_2(t)\|_{2} \le \| u_1(s) - u_2(s) \|_{2} \quad \text{for all } t \ge s \ge 0.$$
	\item The solution $u$ lies in $C([0,\infty),\mathbb{R}^V) \cap W^{1,\infty}([0,\infty),\mathbb{R}^V)$ and satisfies
		$$ \|u'(t)\|_{2} \le \| \partial^\circ J(f) \|_{2} \quad \text{for a.e. } t \ge 0.$$
	\item The solution is right differentiable everywhere. Its right derivative is right continuous, it satisfies
		\begin{equation}\label{eq:rightderivative}
			\frac{d^+}{dt} u(t)	= -\partial^\circ J(u(t)), \quad \text{for all } t \ge 0,
		\end{equation}
		and the map
		$$ t \mapsto \Big\| \frac{d^+}{dt} u(t) \Big\|_{2} $$
		is nonincreasing.
	\item Define $S_t(f) = u(t)$. Then, for every $f\in \mathbb{R}^V$, we have
		$$ S_t (S_s (f)) = S_{t+s}(f) \quad \text{for all } t,s \ge 0.$$
	\item The function $u(t) \in  \mathbb{R}^V$ converges to a minimizer of $J$ as $t\to \infty$.
\end{enumerate}
\end{theorem}
Equation \eqref{eq:rightderivative} is a strengthening of the inclusion in \eqref{TVFG}. It implies, for instance, that whenever $u'$ exists, it equals $-\partial^\circ J (u)$. Note that Theorem \ref{thm:flowwellposed} actually holds true for any convex real-valued functional, which admits a minimizer on $\mathbb{R}^V$, in place of $J$. For $J$ being the total variation, however, we have in addition the following analogue of Proposition \ref{P7}.
\begin{proposition}\label{thm:flowproperties}
\leavevmode
\begin{enumerate}
	\item Problem \eqref{TVFG} is mean-preserving, that is,
		$$ \sum_{v\in V} u(t)(v) = \sum_{v\in V} f(v) \quad \text{for all } t \ge 0.$$
	\item The function $t \mapsto	\| u(t) \|_{2}$	is nonincreasing on $[0,\infty)$.
	\item The solution $u$ is piecewise affine with respect to $t$. More specifically, the derivative $u'(t)$ does not exist for only a finite number of times $0<t_1 < \cdots < t_M$ and it is constant in between. It follows that a stationary solution is reached in finite time:
	\begin{equation}\label{eq:stationaryflow}
		u(t)(v) = \frac{1}{|V|}\sum_{w\in V} f(w)  \quad \text{for all } t \ge t_M \text{ and } v\in V.
	\end{equation}
\end{enumerate}
\end{proposition}
\begin{proof}
\leavevmode
\begin{enumerate}
	\item Since the subdifferential of $J$ consists entirely of divergences of edge functions, for a.e.\ $t\ge 0$ there is an $H(t) \in \mathbb{R}^E$ such that
		\begin{equation*}
			u'(t) = -\divergence H(t).
		\end{equation*}
		Summing this equation over all $v\in V$ and using the fact that $\sum_{v\in V} \divergence H(v)$ vanishes for every $H\in \mathbb{R}^E$ gives
		$$ \frac{d}{dt} \sum_{v\in V} u(t)(v) = 0 \quad \text{for a.e. } t\ge 0.$$
		Since $u\in W^{1,\infty}([0,\infty),\mathbb{R}^V)$, the assertion follows.
	\item From $-u'(t) \in \partial J(u(t))$ and the characterization of the subdifferential in Lemma \ref{L1}, it follows that
		$\langle u(t), -u'(t) \rangle_{\mathbb{R}^V} = J(u(t)).$ Therefore
		$$-J(u(t)) = \langle u(t), u'(t) \rangle_{\mathbb{R}^V} = \frac{1}{2} \frac{d}{dt} \|u(t)\|^2_{2}$$
		for a.e.~$t>0$, which shows that $t \mapsto \| u(t) \|_{2}$ is nonincreasing.
	\item As for the ROF minimizer the piecewise affine behaviour has been shown in \cite[Thm.~4.6]{Burger1}. Our proof uses different arguments. According to item 3 in Remark \ref{rem:subdifequal1} the number of different values the right derivative of $u$ can take is finite. Since $d^+u/dt$ is also right continuous, there must be an $\epsilon>0$ for every $t_0 \ge 0$ such that
		$$ \frac{d^+}{dt} u(t) = -\partial^\circ J(u(t_0)) \quad \text{for all } t \in [t_0,t_0+\epsilon)$$
with $d^+u/dt = u'$ on $(t_0,t_0+\epsilon)$. This proves that $t \mapsto u(t)$ is piecewise affine on $[0,\infty).$

That $d^+u/dt$ only changes a finite number of times follows from the fact that, if it changes, then its norm becomes strictly smaller. To see this let $\hat t>0$ and assume that $d^+u(t)/dt \equiv c$ is constant on $(\hat t -\epsilon,\hat t)$ for some $\epsilon>0$ and that $d^+u(\hat t)/dt \neq c$. We now have
\begin{align*}
	J(u(\hat t)) =	\lim_{t\to \hat{t}^-} J(u(t)) =	\lim_{t\to \hat{t}^-} \langle u(t), -c \rangle  = \langle u(\hat t), -c \rangle,
\end{align*}
and therefore $-c \in \partial J(u(\hat t))$. However, since $-c = \partial^\circ J(u(t))$ for $t\in (\hat t - \epsilon, \hat{t})$ and the minimal section is the unique element of minimal norm in the subdifferential, we must have $\|d^+u(t)/dt\|_2 > \|d^+u(\hat t)/dt\|_2$. This combined with the fact that $d^+u/dt$ can take only a finite number of values, implies that it can change only a finite number of times.

Thus $t\mapsto u(t)$ is a continuous piecewise affine function with a finite number of slope changes. Since, by item 5 in Theorem \ref{thm:flowwellposed}, $u(t)$ is convergent, it must reach its limit in finite time. Due to mean preservation, this limit has to be the averaged initial datum.
\end{enumerate}
\end{proof}

\section{Comparison of TV regularization and TV flow} \label{sec:comparison}
In this section we first provide and analyze various conditions for the equivalence of TV regularization and TV flow on graphs. We then show that they are non-equivalent methods by constructing a counterexample.

\subsection{Conditions for equivalence of TV regularization and TV flow}
Proposition \ref{thm:norm} below relates the norms of the solutions of the TV regularization and the TV flow to each other. Recall that $\bar{f}$ denotes the averaged datum $f$, see \eqref{eq:faverage}.
\begin{proposition} \label{thm:norm}
For every $\alpha>0$ let $u_\alpha$ and $u(\alpha)$ be the ROF and TV flow solutions, respectively, both corresponding to the same datum $f\in \mathbb{R}^V$. They satisfy
$$ \|\bar f \|_{2} \le \|u_\alpha\|_{2} \le \| u(\alpha) \|_{2} \le \| f \|_{2}, \quad \text{for all } \alpha>0.$$
It follows that in general $u_\alpha$ reaches $\bar f$ before $u(t)$, that is, $\alpha_N \le t_M$, see Propositions \ref{P7} and \ref{thm:flowproperties}.
\end{proposition}
\begin{proof}
Both $\|u_\alpha\|_{2}$ and $\|u(\alpha)\|_{2}$ are nonincreasing functions of $\alpha$, recall property 2 in Propositions \ref{P7} and \ref{thm:flowproperties}, and therefore bounded from above by $\| f \|_{2}$. On the other hand, due to mean preservation, recall property 1 in Propositions \ref{P7} and \ref{thm:flowproperties}, they are bounded from below by $\|\bar f \|_{2}$. It remains to show that $\|u_\alpha\|_{2} \le \| u(\alpha) \|_{2}$. To see this, observe that both $u_\alpha$ and $u(\alpha)$ lie in $f-\divergence\mathcal{B}_{\alpha}$ with $u_\alpha$ being the element of minimal norm in this set according to \eqref{206}.
\end{proof}
The next proposition collects several conditions for equality of ROF and TV flow solutions. The second condition is an adaptation of \cite[Thm.\ 10]{Lasica1} to the graph setting.
\begin{proposition}\label{thm:eqcond}
Let $u_\alpha$ and $u(t)$ be the ROF and TV flow solutions for a given common datum $f\in \mathbb{R}^V$.
\begin{enumerate}
	\item Let $\alpha> 0.$ We have $u_\alpha = u(\alpha)$ if and only if
		 \begin{align} \label{eq:necsuffcond}
 			 -\frac{1}{\alpha}\int^{\alpha}_{0}u^{\prime}(t)dt\in\partial J(u(\alpha)).
 		\end{align}
 	\item Let $\alpha> 0.$ If
 	\begin{align}\label{eq:suffcond}
		-\langle u'(t), u(\alpha) \rangle_{\mathbb{R}^V} = J(u(\alpha)) \quad \text{for a.e. } t \in (0,\alpha)
	\end{align}
		then $u(\alpha) = u_{\alpha}$. Moreover, condition \eqref{eq:suffcond} is always satisfied for $\alpha=t_1$, where $t_1$ is the first time $u'(t)$ does not exist.
	\item Define $T_\alpha(f) = u_\alpha$. We have
		$$u_\alpha = u(\alpha) \quad \text{for all} \quad \alpha \ge 0$$
		if and only if
 	\begin{align} \label{eq:semigrouprof}
  		T_{t}(T_{s}(f))=T_{t+s}(f) \quad \text{for all} \quad  t,s\geq 0.
 	\end{align}
\end{enumerate}
\end{proposition}
\begin{proof}
\leavevmode
\begin{enumerate}
	\item We can express $u(\alpha)=f+\int^{\alpha}_{0}u^{\prime}(t)dt$. Recalling the optimality conditions \eqref{OC1} for the ROF minimizer $u_{\alpha}$, it follows that $u(\alpha)=u_{\alpha}$ if and only if $-\frac{1}{\alpha}\int^{\alpha}_{0}u^{\prime}(t)dt\in\partial J(u(\alpha))$.
	\item The proof is analogous to the one of \cite[Thm.\ 10]{Lasica1}. We include it for the sake of completeness.

Integrating \eqref{eq:suffcond} from $t=0$ to $t=\alpha$ gives
	\begin{equation}\label{eq:suffcond1}
		\langle f - u(\alpha), u(\alpha) \rangle_{\mathbb{R}^V} = \alpha J(u(\alpha)).
	\end{equation}
On the other hand, since $-u'(t)$ lies in $\partial J (0)$ for almost every $t$, so does its average $-\frac{1}{\alpha}\int_0^\alpha u'(t)\,dt$. Therefore
	\begin{equation}\label{eq:suffcond2}
		f - u(\alpha) = -\int_0^\alpha u'(t)\,dt \in \alpha \partial J (0).
	\end{equation}
Combining \eqref{eq:suffcond1} and \eqref{eq:suffcond2} shows that $f - u(\alpha) \in \alpha \partial J(u(\alpha))$, recall Lemma \ref{L1}. But this is just the optimality condition \eqref{OC1} for the ROF model, hence $u(\alpha) = u_\alpha.$

Next, recall that the flow solution satisfies
	$$ -u'(t) = \partial^\circ J (f) \in \partial J (u(t)), \quad t \in [0,t_1).$$
This implies by Lemma \ref{L1} that
	$$ \langle \partial^\circ J (f),u(t) \rangle = J(u(t)), \quad t \in [0,t_1),$$
and since $u$ is continuous in $t$
	$$ \langle \partial^\circ J (f),u(t_1) \rangle = J(u(t_1)).$$
Therefore condition \eqref{eq:suffcond} is satisfied for every $\alpha\in\left[0,t_1\right]$.
	\item Let $u_{\alpha}=u(\alpha)$ for all $\alpha\geq 0$. It then follows from property 4 in Theorem \ref{thm:flowwellposed} that $T_{t}(T_{s}(f))=T_{t+s}(f)$ for all $t,s\geq 0$.
\par Start now with the assumption $T_{t}(T_{s}(f))=T_{t+s}(f)$ for all $t,s\geq 0$. As the TV flow has an analogous property and the solutions to TV regularization and TV flow always coincide for the interval $\left[0,t_{1}\right]$ according to item 2 it is then immediate that they coincide for all $\alpha\geq 0$.
\end{enumerate}
\end{proof}
\begin{remark}\label{rem:eqcond}
\leavevmode
\begin{enumerate}
	\item Proposition \ref{thm:eqcond}, item 1, gives that $u(\alpha)=u_{\alpha}$ if and only if the average time derivative $\frac{1}{\alpha}\int^{\alpha}_{0}u^{\prime}(t)dt$ is in $-\partial J(u(\alpha))$. Compare with the pointwise inclusion $u^{\prime}(t)\in-\partial J(u(t))$ which holds for a.e. $t>0$. Note further that condition \eqref{eq:necsuffcond} is strictly weaker than \eqref{eq:suffcond}.
	\item Condition \eqref{eq:suffcond} holds true, given any $\alpha>0$, for graphs of the type displayed in Figure \ref{Figure1D} corresponding to one-dimensional space-discrete signals. This follows directly from the inclusion
 \begin{align} \label{inc}
 \partial J(u(s))\subset\partial J(u(t)),\quad s \leq t,
 \end{align}
which applies in this setting. The derivation of \eqref{inc} can be done with the following arguments. Consider a pair of adjacent vertices $v_{i}$ and $v_{i+1}$. In \cite[Prop.\ 4.1]{Steidl1}, it is shown that if $u(s)(v_{i})=u(s)(v_{i+1})$ then $u(t)(v_{i})=u(t)(v_{i+1})$ for any $t\geq s$. Taking into account the continuity of $t \mapsto u(t)$ and the characterization of the subdifferential given by item 3 in Lemma \ref{L1}, \eqref{inc} then follows.
\item Another family of instances where $u_\alpha = u(\alpha)$, for all $\alpha \ge 0$, arises from the eigenvalue problem for the TV subdifferential. This problem seems to have originally been studied in the continuous setting, where it was realized to give rise to explicit solutions of both the TV flow and the ROF model. See, for instance, \cite{AndCasDiaMaz02,BelCasNov02}. In the discrete setting the situation is similar. Following \cite{Burger1,Gil14} we call $f\in \mathbb{R}^V$ an \emph{eigenfunction} of $J$, if it satisfies $\lambda f \in \partial J (f)$ for some $\lambda \ge 0$. If the datum of the ROF model has this property, then the optimality condition \eqref{OC1} directly implies that
\begin{equation*}
	u_\alpha =
	\begin{cases}
		(1-\alpha \lambda) f,	& \alpha\lambda < 1, \\
		0,						& \alpha\lambda \ge 1.
	\end{cases}
\end{equation*}
See also \cite[Thm.\ 5]{BenBur13}. In other words $T_\alpha(f)$ is a nonnegative multiple of $f$, hence again an eigenfunction. A brief calculation now shows that \eqref{eq:semigrouprof} is satisfied.
\end{enumerate}
\end{remark}
\begin{figure}
\centering
 \begin{tikzpicture}[scale=1]
 \node		(d) at (4,0) {};
 \node		    at (3.8,0) {...};
\tikzstyle{every node}=[draw, circle, thick, fill=gray!20, minimum width={width("fjhere")},]

	\node       	 (a) at (0,0)  {$v_{1}$};
	\node            (b) at (2,0)  {$v_{2}$};
	\node            (c) at (5.5,0)  {$v_{n}$};
	\draw[<-,thick] (a) edge (b);
	\draw[<-,thick] (b) edge (3.5,0);
 	\draw[<-,thick] (d) edge (c);
\end{tikzpicture}
\caption{Graph corresponding to a one-dimensional space-discrete signal with $n$ pixels.}
\label{Figure1D}
\end{figure}

\subsection{Negative results}\label{sec:example}
All results in this section are derived from the counterexample given by the graph and datum displayed in Figure \ref{Figure1}. While the corresponding solutions $u_\alpha$ and $u(t)$ are illustrated in Figures \ref{Figure2} and \ref{Figure3}, the underlying computations can be found in the appendix. Our main considerations in constructing this counterexample are explained below.

Proposition \ref{thm:rofconstrained} together with the fact that $\mathcal{B}_{\alpha} = -\mathcal{B}_{\alpha}$ implies that the ROF minimizer $u_{\alpha}$ can be written as
\begin{align} \label{reprof}
	u_{\alpha}=f+\divergence F_{\alpha},
\end{align}
for an $F_\alpha \in \mathcal{B}_{\alpha}$.
Regarding the TV flow, note that Lemma \ref{L1} and Proposition \ref{thm:flowproperties} guarantee the existence of a piecewise constant function $t \mapsto H(t) \in \mathcal{B}_{1,u(t)}$ with finitely many discontinuities satisfying
$$ u'(t) = - \divergence H(t)$$
for all but a finite number of times. Integrating and setting $F(t) = -\int_0^t H(s)ds$ we obtain the following representation
\begin{align}\label{repflow}
	u(t) = f + \divergence F(t).
\end{align}
Two properties concerning these representations are worth mentioning. First, the edge functions $F_\alpha$ and $F(t)$ are not uniquely determined in general. Second, $F(t)$ satisfies
$$\Big\|\frac{d^+}{dt} F(t) \Big\|_\infty \le 1$$
for all $t$, while the derivative of $F_\alpha$ in general is not bounded by one. The counterexample displayed in Figure \ref{Figure1} was constructed in such a way that $F_\alpha$ is uniquely determined and satisfies $\|dF_\alpha/d\alpha\|_\infty > 1$ for certain values of $\alpha.$ In fact, on the edge $e=(v_{32},v_{22})$ we have $dF_\alpha(e)/d\alpha = -3/2$ for $2/5 < \alpha < 2$, see Figure \ref{Figure2}.

\subsubsection{Nonequivalence of TV flow and TV regularization}
In spite of the similar qualitative properties of TV flow and TV regularization, recall Propositions \ref{P7} and \ref{thm:flowproperties}, the solutions $u(\alpha)$ and $u_{\alpha}$ do not coincide in general.
\begin{theorem} \label{prop:noneq}
\leavevmode
There exist graphs $G=(V,E)$ and data $f\in\mathbb{R}^{V}$ for which the TV regularization problem and the TV flow problem are non\-equivalent, i.e.
 \begin{align*}
  u_{\alpha}\neq u(\alpha),\text{ for some }\alpha>0.
 \end{align*}
\end{theorem}
\begin{proof}
 Consider the graph and the datum $f$ given in Figure \ref{Figure1}. For this example, the evolutions of $u_{\alpha}$ and $u(\alpha)$ on the interval $\left[0,4\right]$ are displayed in Figure \ref{Figure2} and Figure \ref{Figure3}, respectively. Note that $u_{\alpha}\neq u(\alpha)$ for $\alpha\in\left(2/5,4\right]$.
\end{proof}
\begin{figure}
\centering
 \begin{tikzpicture}[scale=1]
\tikzstyle{every node}=[draw, circle, thick, fill=gray!20, minimum width={width("fjhere")},]

	\node       	 (a) at (0,0)  {$v_{12}$};
	\node            (b) at (2,0)  {$v_{22}$};
	\node            (c) at (4,0)  {$v_{32}$};
	\node            (d) at (2,2)  {$v_{23}$};
	\node            (e) at (2,-2) 	{$v_{21}$};
	\node            (f) at (0,2)  {$v_{13}$};
	\node            (g) at (0,-2) 	{$v_{11}$};
	\node            (h) at (4,-2) 	{$v_{31}$};
	\node            (i) at (4,2)  {$v_{33}$};
	\draw[<-,thick] (a) edge (b);
	\draw[<-,thick] (b) edge (c);
	\draw[<-,thick] (b) edge (d);
	\draw[<-,thick] (e) edge (b);
	\draw[<-,thick] (a) edge (f);
	\draw[<-,thick] (g) edge (a);
	\draw[<-,thick] (f) edge (d);
	\draw[<-,thick] (g) edge (e);
	\draw[<-,thick] (d) edge (i);
	\draw[<-,thick] (c) edge (i);
	\draw[<-,thick] (h) edge (c);
	\draw[<-,thick] (e) edge (h);
\end{tikzpicture}
\qquad
 \begin{tikzpicture}[scale=1]
\tikzstyle{every node}=[draw, circle, thick, fill=gray!20, minimum width={width("fjhere")},]

	\node       	 (a) at (0,0)  {$100$};
	\node            (b) at (2,0)  {$18$};
	\node            (c) at (4,0)  {$20$};
	\node            (d) at (2,2)  {$100$};
	\node            (e) at (2,-2)  {$100$};
	\node            (f) at (0,2)  {$200$};
	\node            (g) at (0,-2)  {$200$};
	\node            (h) at (4,-2)  {$200$};
	\node            (i) at (4,2)  {$0$};
	\draw[<-,thick] (a) edge (b);
	\draw[<-,thick] (b) edge (c);
	\draw[<-,thick] (b) edge (d);
	\draw[<-,thick] (e) edge (b);
	\draw[<-,thick] (a) edge (f);
	\draw[<-,thick] (g) edge (a);
	\draw[<-,thick] (f) edge (d);
	\draw[<-,thick] (g) edge (e);
	\draw[<-,thick] (d) edge (i);
	\draw[<-,thick] (c) edge (i);
	\draw[<-,thick] (h) edge (c);
	\draw[<-,thick] (e) edge (h);
\end{tikzpicture}
\caption{Left: graph structure. Right: datum $f$.}
\label{Figure1}
\end{figure}

\begin{figure}
\centering
\scalebox{0.575}{
\begin{tikzpicture}
\node at (2.5,3.75) {$0\leq\alpha\leq 2/5$:};
{\footnotesize
\node       	  at (1.25,0.2)  {$-\alpha$};
\node       	  at (3.75,0.2)  {$\alpha$};
\node       	  at (1.25,2.7)  {$-\alpha$};
\node       	  at (3.75,2.7)  {$-\alpha$};
\node       	  at (1.25,-2.3)  {$-\alpha$};
\node       	  at (3.75,-2.3)  {$\alpha$};
\node       	  at (-0.3,-1.25)  {$-\alpha$};
\node       	  at (-0.2,1.25)  {$\alpha$};
\node       	  at (2.2,-1.25)  {$-\alpha$};
\node       	  at (2.3,1.25)  {$\alpha$};
\node       	  at (4.7,-1.25)  {$-\alpha$};
\node       	  at (4.7,1.25)  {$-\alpha$};
\tikzstyle{every node}=[draw, circle, thick, fill=gray!20, minimum width={width("fjhererleheh")},]
	\node       	 (a) at (0,0)  {$100+\alpha$};
	\node            (b) at (2.5,0)  {$18+4\alpha$};
	\node            (c) at (5,0)  {$20-\alpha$};
	\node            (d) at (2.5,2.5)  {$100-\alpha$};
	\node            (e) at (2.5,-2.5)  {$100+\alpha$};
	\node            (f) at (0,2.5)  {$200-2\alpha$};
	\node            (g) at (0,-2.5)  {$200-2\alpha$};
	\node            (h) at (5,-2.5)  {$200-2\alpha$};
	\node            (i) at (5,2.5)  {$2\alpha$};
	\draw[<-,thick] (a) edge (b);
	\draw[<-,thick] (b) edge (c);
	\draw[<-,thick] (b) edge (d);
	\draw[<-,thick] (e) edge (b);
	\draw[<-,thick] (a) edge (f);
	\draw[<-,thick] (g) edge (a);
	\draw[<-,thick] (f) edge (d);
	\draw[<-,thick] (g) edge (e);
	\draw[<-,thick] (d) edge (i);
	\draw[<-,thick] (c) edge (i);
	\draw[<-,thick] (h) edge (c);
	\draw[<-,thick] (e) edge (h);
	}
\end{tikzpicture}
\qquad
\begin{tikzpicture}
\node at (2.5,3.75) {$2/5\leq\alpha\leq 2$:};
{\footnotesize
\node       	  at (1.25,0.2)  {$-\alpha$};
\node       	  at (3.79,0.35)  {$\frac{2-3\alpha}{2}$};
\node       	  at (1.25,2.7)  {$-\alpha$};
\node       	  at (3.75,2.7)  {$-\alpha$};
\node       	  at (1.25,-2.3)  {$-\alpha$};
\node       	  at (3.75,-2.3)  {$\alpha$};
\node       	  at (-0.3,-1.25)  {$-\alpha$};
\node       	  at (-0.2,1.25)  {$\alpha$};
\node       	  at (2.2,-1.25)  {$-\alpha$};
\node       	  at (2.3,1.25)  {$\alpha$};
\node       	  at (4.7,-1.25)  {$-\alpha$};
\node       	  at (4.7,1.25)  {$-\alpha$};
\tikzstyle{every node}=[draw, circle, thick, fill=gray!20, minimum width={width("fjhererleheh")},]
	\node       	 (a) at (0,0)  {$100+\alpha$};
	\node            (b) at (2.5,0)  {$19+\frac{3\alpha}{2}$};
	\node            (c) at (5,0)  {$19+\frac{3\alpha}{2}$};
	\node            (d) at (2.5,2.5)  {$100-\alpha$};
	\node            (e) at (2.5,-2.5)  {$100+\alpha$};
	\node            (f) at (0,2.5)  {$200-2\alpha$};
	\node            (g) at (0,-2.5)  {$200-2\alpha$};
	\node            (h) at (5,-2.5)  {$200-2\alpha$};
	\node            (i) at (5,2.5)  {$2\alpha$};
	\draw[<-,thick] (a) edge (b);
	\draw[<-,thick] (b) edge (c);
	\draw[<-,thick] (b) edge (d);
	\draw[<-,thick] (e) edge (b);
	\draw[<-,thick] (a) edge (f);
	\draw[<-,thick] (g) edge (a);
	\draw[<-,thick] (f) edge (d);
	\draw[<-,thick] (g) edge (e);
	\draw[<-,thick] (d) edge (i);
	\draw[<-,thick] (c) edge (i);
	\draw[<-,thick] (h) edge (c);
	\draw[<-,thick] (e) edge (h);
	}
\end{tikzpicture}
\qquad
\begin{tikzpicture}
\node at (2.5,3.75) {$2\leq\alpha\leq 4$:};
{\footnotesize
\node       	  at (1.25,0.2)  {$-\alpha$};
\node       	  at (3.79,0.35)  {$-\alpha$};
\node       	  at (1.25,2.7)  {$-\alpha$};
\node       	  at (3.75,2.7)  {$-\alpha$};
\node       	  at (1.25,-2.3)  {$-\alpha$};
\node       	  at (3.75,-2.3)  {$\alpha$};
\node       	  at (-0.3,-1.25)  {$-\alpha$};
\node       	  at (-0.2,1.25)  {$\alpha$};
\node       	  at (2.2,-1.25)  {$-\alpha$};
\node       	  at (2.3,1.25)  {$\alpha$};
\node       	  at (4.7,-1.25)  {$-\alpha$};
\node       	  at (4.7,1.25)  {$-\alpha$};
\tikzstyle{every node}=[draw, circle, thick, fill=gray!20, minimum width={width("fjhererleheh")},]
	\node       	 (a) at (0,0)  {$100+\alpha$};
	\node            (b) at (2.5,0)  {$18+2\alpha$};
	\node            (c) at (5,0)  {$20+\alpha$};
	\node            (d) at (2.5,2.5)  {$100-\alpha$};
	\node            (e) at (2.5,-2.5)  {$100+\alpha$};
	\node            (f) at (0,2.5)  {$200-2\alpha$};
	\node            (g) at (0,-2.5)  {$200-2\alpha$};
	\node            (h) at (5,-2.5)  {$200-2\alpha$};
	\node            (i) at (5,2.5)  {$2\alpha$};
	\draw[<-,thick] (a) edge (b);
	\draw[<-,thick] (b) edge (c);
	\draw[<-,thick] (b) edge (d);
	\draw[<-,thick] (e) edge (b);
	\draw[<-,thick] (a) edge (f);
	\draw[<-,thick] (g) edge (a);
	\draw[<-,thick] (f) edge (d);
	\draw[<-,thick] (g) edge (e);
	\draw[<-,thick] (d) edge (i);
	\draw[<-,thick] (c) edge (i);
	\draw[<-,thick] (h) edge (c);
	\draw[<-,thick] (e) edge (h);
	}
\end{tikzpicture}
}
\caption{The evolution of the ROF minimizer $u_{\alpha}$ (on the vertices) and the function $F_{\alpha}$ (on the edges) on the interval $0\leq\alpha\leq 4$. The underlying computations can be found in the appendix.}
\label{Figure2}
\end{figure}

\begin{figure}
\centering
\scalebox{0.8}{
\begin{tikzpicture}
\node at (2.5,3.75) {$0\leq t\leq 2/5$:};
{\footnotesize
\node       	  at (1.25,0.2)  {$-t$};
\node       	  at (3.75,0.2)  {$t$};
\node       	  at (1.25,2.7)  {$-t$};
\node       	  at (3.75,2.7)  {$-t$};
\node       	  at (1.25,-2.3)  {$-t$};
\node       	  at (3.75,-2.3)  {$t$};
\node       	  at (-0.3,-1.25)  {$-t$};
\node       	  at (-0.2,1.25)  {$t$};
\node       	  at (2.2,-1.25)  {$-t$};
\node       	  at (2.3,1.25)  {$t$};
\node       	  at (4.7,-1.25)  {$-t$};
\node       	  at (4.7,1.25)  {$-t$};
\tikzstyle{every node}=[draw, circle, thick, fill=gray!20, minimum width={width("fjhererlehe")},]
	\node       	 (a) at (0,0)  {$100+t$};
	\node            (b) at (2.5,0)  {$18+4t$};
	\node            (c) at (5,0)  {$20-t$};
	\node            (d) at (2.5,2.5)  {$100-t$};
	\node            (e) at (2.5,-2.5)  {$100+t$};
	\node            (f) at (0,2.5)  {$200-2t$};
	\node            (g) at (0,-2.5)  {$200-2t$};
	\node            (h) at (5,-2.5)  {$200-2t$};
	\node            (i) at (5,2.5)  {$2t$};
	\draw[<-,thick] (a) edge (b);
	\draw[<-,thick] (b) edge (c);
	\draw[<-,thick] (b) edge (d);
	\draw[<-,thick] (e) edge (b);
	\draw[<-,thick] (a) edge (f);
	\draw[<-,thick] (g) edge (a);
	\draw[<-,thick] (f) edge (d);
	\draw[<-,thick] (g) edge (e);
	\draw[<-,thick] (d) edge (i);
	\draw[<-,thick] (c) edge (i);
	\draw[<-,thick] (h) edge (c);
	\draw[<-,thick] (e) edge (h);
	}
\end{tikzpicture}
\qquad
\begin{tikzpicture}
\node at (2.5,3.75) {$2/5\leq t\leq 4$:};
{\footnotesize
\node       	  at (1.25,0.2)  {$-t$};
\node       	  at (3.79,0.35)  {$\frac{4}{5}-t$};
\node       	  at (1.25,2.7)  {$-t$};
\node       	  at (3.75,2.7)  {$-t$};
\node       	  at (1.25,-2.3)  {$-t$};
\node       	  at (3.75,-2.3)  {$t$};
\node       	  at (-0.3,-1.25)  {$-t$};
\node       	  at (-0.2,1.25)  {$t$};
\node       	  at (2.2,-1.25)  {$-t$};
\node       	  at (2.3,1.25)  {$t$};
\node       	  at (4.7,-1.25)  {$-t$};
\node       	  at (4.7,1.25)  {$-t$};
\tikzstyle{every node}=[draw, circle, thick, fill=gray!20, minimum width={width("fjhererlehe")},]
	\node       	 (a) at (0,0)  {$100+t$};
	\node            (b) at (2.5,0)  {$\frac{94}{5}+2t$};
	\node            (c) at (5,0)  {$\frac{96}{5}+t$};
	\node            (d) at (2.5,2.5)  {$100-t$};
	\node            (e) at (2.5,-2.5)  {$100+t$};
	\node            (f) at (0,2.5)  {$200-2t$};
	\node            (g) at (0,-2.5)  {$200-2t$};
	\node            (h) at (5,-2.5)  {$200-2t$};
	\node            (i) at (5,2.5)  {$2t$};
	\draw[<-,thick] (a) edge (b);
	\draw[<-,thick] (b) edge (c);
	\draw[<-,thick] (b) edge (d);
	\draw[<-,thick] (e) edge (b);
	\draw[<-,thick] (a) edge (f);
	\draw[<-,thick] (g) edge (a);
	\draw[<-,thick] (f) edge (d);
	\draw[<-,thick] (g) edge (e);
	\draw[<-,thick] (d) edge (i);
	\draw[<-,thick] (c) edge (i);
	\draw[<-,thick] (h) edge (c);
	\draw[<-,thick] (e) edge (h);
	}
\end{tikzpicture}
}

\caption{The evolution of the TV flow $u(t)$ (on the vertices) and the function $F(t)$ (on the edges) on the interval $0\leq t\leq 4$. The underlying computations can be found in the appendix.}
\label{Figure3}
\end{figure}
\begin{remark}
\leavevmode
\begin{enumerate}
	\item Proposition \ref{thm:eqcond}, item 3, combined with Theorem \ref{prop:noneq} gives that the ROF model in general does not possess the semigroup property \eqref{eq:semigrouprof}. This is in contrast to the situation for the TV flow, recall property 4 in Theorem \ref{thm:flowwellposed}.
	\item Recall Theorem \ref{thm:flowwellposed}, item 3, stating that $t \mapsto \|d^+u(t)/dt\|_2$ is nonincreasing. The ROF minimizer, in contrast, does not have an analogous property. Consider Figure \ref{Figure2}, from where it can be seen that $\lVert  du_{\alpha}/d\alpha \rVert_{{2}}$ increases from the interval $(2/5,2)$ to $(2,4)$.
	\item In \cite[Thm.\ 4.7]{Burger1} the authors give a sufficient condition for equivalence of the variational method and the gradient flow associated to a proper, convex, lower semicontinuous and absolutely one-homogeneous function $J$ on $\mathbb{R}^n$. This condition, called MINSUB, requires
	$$\langle \partial^\circ J(u) , \partial^\circ J(u) - u^* \rangle = 0$$
to hold for all $u \in \mathbb{R}^n$ and $u^* \in \partial J(u).$ Theorem \ref{prop:noneq} implies that the total variation as given in Definition \ref{def:tv} does not meet MINSUB on general graphs.
\end{enumerate}
\end{remark}

\subsubsection{Nonmonotone behaviour of jump sets}
For a given graph $G=(V,E)$ and datum $f\in \mathbb{R}^V$ we define the jump sets of the ROF and TV flow solutions in the following way
\begin{align*}
	\Gamma_\alpha	&= \left\{ (v,w) \in E :  u_\alpha(v) \neq u_\alpha(w) \right\}, \quad \alpha \ge 0, \\
	\Gamma(t)		&= \left\{ (v,w) \in E :  u(t)(v) \neq u(t)(w) \right\},\quad t \ge 0.
\end{align*}
Clearly, for $\alpha$ or $t$ large enough these two sets are empty. They do not, however, necessarily evolve in a monotone way.
\begin{proposition}\label{thm:jumpsets}
There are graphs $G=(V,E)$, data $f \in \mathbb{R}^V$ and numbers $\beta_2 > \beta_1 \ge 0$, $s_2 > s_1 \ge 0$, such that
\begin{align*}
	\Gamma_{\beta_1}	&\subsetneq \Gamma_{\beta_2}, \\
	\Gamma(s_1)			&\subsetneq \Gamma(s_2).
\end{align*}
\end{proposition}
\begin{proof}
Consider the graph and datum of Figure \ref{Figure1}.

For the TV regularization, Figure \ref{Figure2} shows that
\begin{align*}
\sgn(u_{\alpha}(v_{32})-u_{\alpha}(v_{22}))=
 \left\{
\begin{array}{rl}
1, & 0\leq\alpha< 2/5,\\
0, & 2/5\leq\alpha\leq 2,\\
-1, & 2<\alpha\leq 4.
\end{array}
\right.
\end{align*}
That is, the jump between $u_{\alpha}(v_{22})$ and $u_{\alpha}(v_{32})$ disappears for $2/5\leq\alpha\leq 2$ but appears again, with reversed sign, for $2<\alpha\leq 4$. For all other edges $(v,w)$ the quantity $\sgn(u_{\alpha}(v)-u_{\alpha}(w))$ is constant on $[0,4].$ This shows that $\Gamma_{\beta_1} \subsetneq \Gamma_{\beta_2}$ for every $\beta_1 \in [2/5,2]$ and $\beta_2 \in (2,4].$

For the TV flow, see Figure \ref{Figure3}, we have
\begin{align*}
\sgn(u(t)(v_{32})-u(t)(v_{22}))=
 \left\{
\begin{array}{rl}
1, & 0\leq t< 2/5,\\
0, & t=2/5,\\
-1, & 2/5<t\leq 4.
\end{array}
\right.
\end{align*}
Here the jump between $u(t)(v_{22})$ and $u(t)(v_{32})$ disappears at $t=2/5$ and then a jump with reversed sign appears for $2/5<t\leq 4$. Again, for all other edges $(v,w)$ the quantity $\sgn(u(t)(v)-u(t)(w))$ is constant on $[0,4].$ Thus, $\Gamma (2/5) \subsetneq \Gamma(s_2)$ for every $s_2\in (2/5,4].$
\end{proof}

\begin{remark}
For one-dimensional graphs, however, the jump sets are nonincreasing, see item 2 in Remark \ref{rem:eqcond}. On the other hand, in the continuous anisotropic setting it is known that jumps can be created in the solution, see \cite[Rem.~4]{Casalles1} and \cite[Ex.~1]{Lasica1}.
\end{remark}
\begin{remark}
We stress that $\beta_1$ and $s_1$ can be equal to zero in Proposition \ref{thm:jumpsets}. To see this consider the datum $\tilde{f}$ and solutions $u_\alpha=u(\alpha)$ given in Figure \ref{Figure4}.
\begin{figure}
\centering
\scalebox{0.8}{
 \begin{tikzpicture}
\tikzstyle{every node}=[draw, circle, thick, fill=gray!20, minimum width={width("fjhererlehe")},]
{\footnotesize
	\node       	 (a) at (0,0)  {$100$};
	\node            (b) at (2.5,0)  {$20$};
	\node            (c) at (5,0)  {$20$};
	\node            (d) at (2.5,2.5)  {$100$};
	\node            (e) at (2.5,-2.5)  {$100$};
	\node            (f) at (0,2.5) {$200$};
	\node            (g) at (0,-2.5)  {$200$};
	\node            (h) at (5,-2.5)  {$200$};
	\node            (i) at (5,2.5)  {$0$};
	\draw[<-,thick] (a) edge (b);
	\draw[<-,thick] (b) edge (c);
	\draw[<-,thick] (b) edge (d);
	\draw[<-,thick] (e) edge (b);
	\draw[<-,thick] (a) edge (f);
	\draw[<-,thick] (g) edge (a);
	\draw[<-,thick] (f) edge (d);
	\draw[<-,thick] (g) edge (e);
	\draw[<-,thick] (d) edge (i);
	\draw[<-,thick] (c) edge (i);
	\draw[<-,thick] (h) edge (c);
	\draw[<-,thick] (e) edge (h);
	}
\end{tikzpicture}
\qquad
 \begin{tikzpicture}
{\footnotesize
\node       	  at (1.25,0.2)  {$-\alpha$};
\node       	  at (3.75,0.2)  {$-\alpha$};
\node       	  at (1.25,2.7)  {$-\alpha$};
\node       	  at (3.75,2.7)  {$-\alpha$};
\node       	  at (1.25,-2.3)  {$-\alpha$};
\node       	  at (3.75,-2.3)  {$\alpha$};
\node       	  at (-0.3,-1.25)  {$-\alpha$};
\node       	  at (-0.2,1.25)  {$\alpha$};
\node       	  at (2.2,-1.25)  {$-\alpha$};
\node       	  at (2.3,1.25)  {$\alpha$};
\node       	  at (4.7,-1.25)  {$-\alpha$};
\node       	  at (4.7,1.25)  {$-\alpha$};
\tikzstyle{every node}=[draw, circle, thick, fill=gray!20, minimum width={width("fjhererlehe")},]
	\node       	 (a) at (0,0)  {$100+\alpha$};
	\node            (b) at (2.5,0)  {$20+2\alpha$};
	\node            (c) at (5,0)  {$20+\alpha$};
	\node            (d) at (2.5,2.5)  {$100-\alpha$};
	\node            (e) at (2.5,-2.5)  {$100+\alpha$};
	\node            (f) at (0,2.5)  {$200-2\alpha$};
	\node            (g) at (0,-2.5)  {$200-2\alpha$};
	\node            (h) at (5,-2.5)  {$200-2\alpha$};
	\node            (i) at (5,2.5)  {$2\alpha$};
	\draw[<-,thick] (a) edge (b);
	\draw[<-,thick] (b) edge (c);
	\draw[<-,thick] (b) edge (d);
	\draw[<-,thick] (e) edge (b);
	\draw[<-,thick] (a) edge (f);
	\draw[<-,thick] (g) edge (a);
	\draw[<-,thick] (f) edge (d);
	\draw[<-,thick] (g) edge (e);
	\draw[<-,thick] (d) edge (i);
	\draw[<-,thick] (c) edge (i);
	\draw[<-,thick] (h) edge (c);
	\draw[<-,thick] (e) edge (h);
	}
\end{tikzpicture}
}
\caption{Left: datum $\tilde{f}$. Right: $u_{\alpha}=u(\alpha)$ (on the vertices) and $F_{\alpha}=F(\alpha)$ (on the edges) for
$\alpha \in [0,4]$.}
\label{Figure4}
\end{figure}
Note that $\tilde{f}$ is equal to $f$ from Figure \ref{Figure1} except for $v_{22}$ where $\tilde{f}(v_{22})=20$. The underlying calculations are analogous to the ones for $f$ and are therefore omitted.
A jump between the vertices $v_{22}$ and $v_{32}$, which is not present in the datum $\tilde{f}$, is created in $u_{\alpha}=u(\alpha)$, $0<\alpha\leq 4$. Thus the jump set of an image resulting from TV regularization or TV flow can strictly contain the jump set of the datum.
\end{remark}

\section{Conclusion}

In this article we have studied and compared TV regularization and TV flow for functions defined on the vertices of an oriented connected graph. Our motivation was the discrete one-dimensional setting, where the two problems are known to be equivalent and their solution minimizes a large class of convex functionals in a certain neighbourhood of the data.

It turns out that in the graph setting this situation can only be recovered for $\alpha,t \in [0,t_1] \cup [t_M,\infty)$, the reason being that on the complement $(t_1,t_M)$ the ROF and flow solution are in general different. Here $t_1$ and $t_M$ are the first and last times, respectively, the time derivative of the flow solution changes.

In addition we have shown that for every $\alpha \ge 0$ the ROF minimizer $u_\alpha$ simultaneously minimizes all functionals of the form
\begin{equation}\label{eq:asf}
	\sum_{v\in V} \varphi (u(v))
\end{equation}
over the set $f - \alpha \partial J(0)$, where $\varphi:\mathbb{R}\to \mathbb{R}$ is convex but otherwise arbitrary. In doing so we have relied on the fact that $\partial J$ is invariant $\varphi$-minimal. Since invariant $\varphi$-minimal sets must be polyhedra, the subdifferential of discrete isotropic total variation cannot be such a set. Consequently, the minimizer of the isotropic ROF model in general does not have property \eqref{eq:asf}.

\subsection*{Acknowledgements}
We acknowledge support by the Austrian Science Fund (FWF) within the national research network S117 ``Geometry $+$ Simulation," subproject 4.
In addition, the work of OS is supported by project I 3661 ``Novel Error Measures and Source Conditions of Regularization Methods," jointly funded by FWF and Deutsche Forschungsgemeinschaft (DFG).
We are grateful to an anonymous referee of a previous version of this article, who pointed out the connection between invariant $\varphi$-minimal sets and submodular functions.

redpajama_set_name: RedPajamaArXiv


Nearby parks include Miramar Park and Rex L Meador Park.
Regatta Bay Apartments has a Walk Score of 33 out of 100. This location is a Car-Dependent neighborhood so most errands require a car.
Regatta Bay Apartments offers spacious one, two, and three-bedroom floor plans and were designed with your needs in mind. Each of our apartment homes features an all-electric kitchen, lofty 9-foot, vaulted ceilings, and wood-like flooring. Enjoy the views from your private balcony or patio. We are proud to say we are pet friendly! Call our leasing office today to learn about apartment rental availability. Choose Regatta Bay and Live Happy!
Explore how far you can travel by car, bus, bike and foot from Regatta Bay Apartments.


The Sludge Metal band as it exists today was formed in July 2011 in Geneva by members having previously taken part in various bands of the local metal scene.
It is not easy to apply COLOSSUS FALL's rich and expanded sound to any existing style, being both brutal and sophisticated at the same time, but rather it is clearly an infl uence of a number of quite diff erent genres. However, COLOSSUS FALL might be a kind of mix between Mastodon, Keelhaul, Coalesce, Converge and Isis. The constant evolution of the group is felt through each piece composed in diff erent ways. The themes of the songs are also very varied. The emotions of the moment direct our inspirations.


The WHOIS info of a domain name is a collection of various details which are publicly accessed through special lookup Internet sites or a command line. The protocol which makes this possible bears the same name and you could easily see the firm where an Internet domain has been registered, the creation, expiration and last update dates in addition to the names, postal and e-mail address of those listed as Registrant (owner), Administrative, Technical and Billing contacts for a particular domain name. All of this info ought to be legitimate and up-to-date constantly; otherwise the domain name registration can be challenged. The latter is a policy of ICANN (the Internet Corporation for Assigned Names and Numbers), so you must always ensure that the WHOIS details of your domain addresses are valid. Updating the WHOIS for numerous country-code TLDs is limited, so when you register a completely new domain, make sure you double-check the information that you're submitting.
When you have a website hosting plan through our company and you register or transfer a domain address, you will have full control of its WHOIS information. Using the Domain Manager tool inside our custom Hepsia hosting Control Panel, you will be able to see and change every single detail associated with your domain names and even change the info of several domain addresses simultaneously with just a few clicks. Our tool is rather user-friendly and you will save time and efforts when you manage the WHOIS details of your Internet domains. Any updates that you make will take effect right away. Needless to say, that is valid for the details which can be edited considering that some country-code TLDs have a number of restrictions in this matter, for instance not being able to edit the owner names once an Internet domain is already registered. We'll be able to aid you 24/7 if such a situation appears for any of your domains.
When you have a semi-dedicated server plan with our company, you're going to be able to check out and update the WHOIS information of any domain registered here using the same Hepsia Control Panel used to control the hosting space, so you'll not have to log in and out of different systems. By clicking on a given domain address, you will see its current details and all it will take to edit any of them will be to input the new info and save the changes. You can even select a few domain addresses and modify their WHOIS info at the same time, so even when you update ten or fifteen domain names, it will not take you more time than to update 1. Due to the fact that some country-code extensions support updates, yet not automatic ones via the CP, you can contact us and we'll aid you with the process until the necessary change takes effect.


Our goal is to develop new algorithms and analytics strategies to bring personalized digital engagement to all patients. At the heart of every data point in healthcare is a patient. Our mission is to harness and activate the utility of that data to empower patients on a personal level to reach their health goals. Using machine learning and AI technologies, we create personalized digital health interventions to support, educate, and sustain behavior change in patients.
We uncover specific learnings to tailor our digital engagement programs to serve the needs of the patient and our client partners.
We build interventions that learn over time using 2-way communication and react based on the individual's unique behavior while providing valuable insights (e.g. risk flagging) to all healthcare stakeholders. Human behavior is complex and highly unique. These methodologies allow us to describe and label unknowns by predicting what will happen for specific patient cases (e.g. disease trajectories). Our R&D team also creates new clinically-relevant metrics to quantify health behaviors and outcomes (e.g. our 'Balanced Adherence Metric' (BAM) initiative).


A lot of people have experienced injuries that are caused by other people who are careless or reckless in Sacramento. Injuries resulting from such cases can be filed against those people who have caused them. You should turn to a lawyer before you sue them in Sacramento law courts. A personal injury will be the one you should look for in such cases. Cases that are caused by the negligence of another person or business are handled by these personal injury lawyers. Different types of personal injury claims are filed in Sacramento law courts by people. Medical malpractice, workplace injuries, slip and falls and car accidents are some of those cases. These days, people are even taking businesses and companies to courts in Sacramento to seek justice. The businesses are being accused because they are supplying defective products that cause injury.
People in Sacramento would like to be compensated from the injuries they have received, and that's why they seek legal justice. You will be compensated but the amount that you receive will depend on the extent of injury, loss of work or loss of wages. In Sacramento, those lawyers who specialize in personal injury cases are the ones you should look for if you need one. The reason for this is because there are many lawyers who specialize in other fields like in car accident or bankruptcy. Personal injury lawyers also are different, and because of this, you should find the one who specializes in that specific type of injury you have. The lawyer also should be experienced and have enough knowledge so that you may win your compensation claim.
The personal injury lawyer who has several medical experts at his disposal is the one you should hire in Sacramento. Your case will be strengthened by those medical experts of your lawyer. The personal injury lawyer that you will choose should have knowledge to the cases that are similar to yours. When you decide to file a personal injury case in Sacramento, you will have to sacrifice most of your time to prepare the case. The personal injury lawyer who can relive your stress by filing motions and gathering witness statements when needed is the one you should hire.
In Sacramento, every lawyer should have a specialty so that he may be able to handle personal injury claims of all kinds. You will have to look for a different lawyer if you would like to file a claim against a company for a defective product. In Sacramento, you will come across many different cases some even from injuries that prevent them from working. You will even find people who need lifelong medical care due to injuries. The victims should hire personal injury lawyers who deal with those types of cases if such cases happen.


Things got ugly on Valentine's Day when Nancy Pelosi (D-CA) tried to speak without a script.
The House speaker once again found herself stumbling and bumbling through a public appearance on Thursday, when she managed to forget which holiday she was celebrating and mix up the amount of money Democrats plan to spend on rebuilding America's infrastructure.
Pelosi's first slip-up occurred during a public bill signing held by the California congresswoman on Thursday.
As is customary, she began with a few remarks, commemorating the day.
No attempt to correct the mistake was made by Pelosi, who seemed to be oblivious to the error.
But that wasn't Pelosi's only major snafu of the day.
When discussing the allocation of federal funds, Pelosi said "1 billion — uh, 17 billion dollars" would be set aside to "rebuild America's infrastructure" before mixing up "$9 million" and "$9 billion" on another topic.
Later, when speaking about six appropriation bills that Democrats want to pass, Pelosi held up only five fingers before adding her other thumb to correct her mistake — but that just led to another struggle to find the words she wanted to say.
Suffice it to say, it was a difficult Valentine's Day for the House speaker.


Another tragedy is being reported in India, this time over a DJ request that turned deadly.
Last week we reported that an 18 year old in Boulder, CO pulled a gun on the DJ because he wouldn't take a song request. Now the tables have turned and a 22 year old man in Gurgaon, India was beaten to death because he made a song request that the DJ and his friends did not respect.
According to the initial report, Rohit Bhardwaj was at a party at a vacant plot of land when he went up to the DJ and asked to change the music. The DJ did not take kindly to this request and he began to get angry with Rohit. Soon a brawl erupted and people at the party, who were allegedly drunk, began to beat Rohit with hockey sticks. Rohit's father was on his way to find his son when he arrived at the party and was pushed into a wall where he hit his head and was knocked unconscious.
"We sent a team to the spot after receiving a PCR call at around 12.30am. They took Rohit to Civil hospital where doctors declared him brought dead. We have registered a murder case and forensic teams have visited the spot," said Inspector Babulal of the City police station.
Police have arrested three suspects, Raju, Dhuru and Amit Kumar who are all local DJs. Authorities also confiscated laptops, sticks and hockey sticks from the scene of the crime.


text:
L2, L-2, L.II ou L02 est un sigle qui peut signifier :

Domaine militaire
 , un porte-avions de la Royal Australian Navy ;
 , un sous-marin de la Royal Navy ;
 PZL Ł.2, un avion de liaison polonais ;
 Taylorcraft L-2, un avion de liaison et d'observation américain.

Formation
 L2, la deuxième année du cycle universitaire aboutissant à l'obtention de la licence.

Informatique et télécommunications

Informatique
 L2, le deuxième niveau de mémoire cache (entre L1 et L3) d'un microprocesseur moderne ; la L2 est intermédiaire entre la L1 (plus rapide mais généralement de plus petite taille) et la L3 (plus lente et de plus grande taille).

Téléphonie
 Motorola L2, un téléphone mobile de la marque Motorola.

Jeux vidéo
 Lineage II, un jeu vidéo MMORPG ;
 L2, une touche sur les manettes de PlayStation, PlayStation 2, PlayStation 3 et PlayStation 4.

Photographie
 Nikon Coolpix L2, un appareil photographique numérique de Nikon.

Sciences

Anatomie
 L2, la deuxième vertèbre lombaire.

Astronomie
 L, le deuxième point de Lagrange, dans le système solaire astronomique ;
 L Puppis, étoile variable de la constellation de la Poupe.

Linguistique
 L2, la langue seconde.

Mathématiques
 Espace L des fonctions de carré intégrable ou l'espace des suites .

Médecine
 Abcès cutané, furoncle et anthrax (L02), d'après la Classification internationale des maladies ;
 Classe ATC L02, sous-groupe thérapeutique de la classification ATC.

Métallurgie
 L2, un acier dont l'énergie de rupture est garantie à  à .

Sport
 Ligue 2 de football, voir Championnat de France de football de Ligue 2.

Transport

Aviation/Aéronautique
 Arado L.II, un avion de tourisme allemand ;
 .

Métro
 Ligne L2 du métro de Barcelone.

Route
 L2, la rocade périphérique de Marseille (en construction).

Train
 Ligne L2 du Léman Express à Genève.

Voir aussi
 Sigles d'une lettre suivie de deux chiffres

redpajama_set_name: RedPajamaWikipedia


Looking for discounts or deals on Canadian Gas Mask today? Get more information about Canadian Gas Mask on our site. Did you know you can save money and time by purchasing Canadian Gas Mask on Ebay. Browse the large listings of products below and click for more detailed info.


text:
Q: No adapter for handler in spring I'm trying to change url-patter in dispather servler mapping:
dispatcher-servlet.xml:
<bean id="urlMapping" class="org.springframework.web.servlet.handler.SimpleUrlHandlerMapping">
    <property name="mappings">
        <props>
            <prop key="index.htm">indexController</prop>
        </props>
    </property>
</bean>

<context:component-scan base-package="Controllers"/>
<mvc:annotation-driven/>

<bean
    class="org.springframework.web.servlet.mvc.SimpleControllerHandlerAdapter">
</bean>
<bean class="org.springframework.web.servlet.mvc.annotation.DefaultAnnotationHandlerMapping"/>

<bean id="viewResolver"
      class="org.springframework.web.servlet.view.InternalResourceViewResolver"
      p:prefix="/WEB-INF/jsp/"
      p:suffix=".jsp" />

<bean name="indexController"
      class="org.springframework.web.servlet.mvc.ParameterizableViewController"
      p:viewName="index" />

Chunk of web.xml:
<servlet>
    <servlet-name>dispatcher</servlet-name>
    <servlet-class>org.springframework.web.servlet.DispatcherServlet</servlet-class>
    <load-on-startup>2</load-on-startup>
</servlet>
<servlet-mapping>
    <servlet-name>dispatcher</servlet-name>
    <url-pattern>/</url-pattern>
</servlet-mapping>

and Controller:
@Controller
public class SaleController {
    @RequestMapping(value = "/sales/info", produces = "application/json", headers = "")
    public @ResponseBody Product getProductJson(@RequestParam String id) throws SQLException
    {
        ProductDAO mapping = new ProductDAOImpl();
        Product product = mapping.getProductById(Integer.parseInt(id));
        return product;
    }
}

But when I'm trying to get http://localhost:8080/appk/sales/info I have exception:
javax.servlet.ServletException: No adapter for handler [Controllers.SaleController@15370df]: The DispatcherServlet configuration needs to include a HandlerAdapter that supports this handler
    org.springframework.web.servlet.DispatcherServlet.getHandlerAdapter(DispatcherServlet.java:1144)
    org.springframework.web.servlet.DispatcherServlet.doDispatch(DispatcherServlet.java:923)
    org.springframework.web.servlet.DispatcherServlet.doService(DispatcherServlet.java:876)
    org.springframework.web.servlet.FrameworkServlet.processRequest(FrameworkServlet.java:961)
    org.springframework.web.servlet.FrameworkServlet.doGet(FrameworkServlet.java:852)
    javax.servlet.http.HttpServlet.service(HttpServlet.java:620)
    org.springframework.web.servlet.FrameworkServlet.service(FrameworkServlet.java:837)
    javax.servlet.http.HttpServlet.service(HttpServlet.java:727)
    org.apache.tomcat.websocket.server.WsFilter.doFilter(WsFilter.java:52

What exactly handler I need to configure and how it can be done?

redpajama_set_name: RedPajamaStackExchange


This is an exciting yet challenging time in the world, with many changes happening now or on the horizon. "Change" seems to be happening everywhere, and many people are seeing the need for transformation in their personal and professional lives. These changes can be facilitated with the help of a skillful coach.
Clearly define your goals and dreams.
Overcome obstacles and limitations that stand between you and your success.
Get you into action to start to create the results you are seeking.
Coaching is the perfect partnership between coach and client. We believe that there is no magic in the achievement of goals. Instead, the intention with which we set goals sets the stage for the fulfillment of those goals.
We combine ideas from the fields of finance, psychology, business management, systems theory, management consulting, change management and philosophy, with some creative ideas and exercises mixed in.
Because each person is different, after an initial assessment, we tailor the program and working relationship to the needs of the individual, designing specific tools to facilitate goal achievement for the client. Designed to promote your personal and professional achievement, coaching facilitates self-awareness, the development of success-producing actions, and proactive living.
Accelerate your personal and professional development.
Identify and integrate your core operating values.
Establish short- and long-term professional goals.
Evaluate your skills for personal and organizational success.
Generate strategies that support success in your current environment.
Build self-awareness and your ability to influence and lead effectively.
Increase your capacity to cope, learn, and manage change.
Achieve and sustain extraordinary performance.
Establish balance between work and home.
Maintain a career portfolio of marketable skills, abilities, and knowledge.
Develop and implement a step-by-step action plan to achieve your professional goals.
A Friend – Friends are great, but they have different agendas and skills than professional coaches. As your coach, we don't have to worry about offending you by challenging you to the fullest.
A Therapist – Therapists are great, but a professional coach does not deal with relationship issues, analyze your past, or deal with mental health issues such as depression.
A Crutch – We emphasize a time-limited relationship and do not support coaching as a form of ongoing and indefinite dependency.
Listens and skillfully helps you identify and refine your goals.
Helps you set goals that are SMART (specific, measurable, attainable, realistic and timely).
Helps you identify and define priorities and values.
Helps you identify and remove personal obstacles.
Helps you set both short- and long-term goals.
Introduces you to new ways of thinking.
Helps you identify your strengths and teaches you how to capitalize on them.
Helps you identify your weaknesses and teaches you how to minimize them.
Helps you identify resources with which to explore your ideas.
Provides you with valuable feedback.
Helps you evaluate and celebrate your successes.


For the moment I have given Garden Dad charge of the garden. I have managed to line a couple hanging baskets and fill them with some compost and I have done some idle weeding but really Garden Dad is enjoying the time with Garden Girl and Garden Boy in the garden while I feed a hungry Garden Lass and try to get some rest.
Garden Boy's enthusiasm for planting and sowing seeds remains and he has helped me transplant courgettes to bigger pots and helped Garden Dad sow swede and turnip seeds. Garden Girl has been less interested in the garden since the arrival of Garden Lass, preferring instead to remain close by the side of her new little sister, She intermittently runs out to assist Garden Dad but she quickly returns to check on Garden Lass. Garden Boy meanwhile has been enoying some father/son time with Garden Dad, helping him dig and weed and playing with his new wheelbarrow. Today between the three of them they managed to plant our pumpkins out, build a couple of wigwams for the runner beans and plant them and prepare the bed for planting.And Garden Lass? Well she has done a lot of sleeping and feeding, enjoying the garden from the cosy comfort of her moses basket or relaxing on my knee taking in all the new sights and sounds of our garden.
The garden is in good hands and Garden Dad has been working hard to keep things going out there and I'm sure it won't be long before I get the opportunity to do more but I'm not complaining. I've got the best job of all, looking after Garden Lass.


text:
CONCERTS AT THE CLOISTER
Ten night concerts in the magnificent cloister of San Nicolò with the artists Paulo Szot, Mattew Aucoin, Keir GoGwilt.
Tuesday July 2 at 21.00
piano Matthew Aucoin
violin Keir GoGwilt
Janacek Sonata per violino e pianoforte
Bach Sonata per violino e tastiera #4
Mozart Sonata per violino e pianoforte K. 454
Wednesday July 3 at 21.00
Berg "Die Nachtigall" (arrangiamento di Keir GoGwilt)
Bach Ciaccona
Schumann Sonata per violino e pianoforte #1
Gershwin Selezione di arrangiamenti di Jascha Heifetz per violino e pianoforte
Thursday July 4 at 21.00
Aucoin Scherzo (dei due mondi)
Rogerson Once
Messiaen "Louange à l'immortalitede Jesus" dal Quatuor pour le fin du temps
Friday July 5 at 22.00
Bartok Ciaccona dalla Sonata per violino solo
Aucoin Whitman Suite
Sarasate Zigeunerweisen
Saturday 6 and Sunday July 7 at 22.00
An Evening With Paulo Szot
baritone Paulo Szot
arrangiaments and piano Matthew Aucoin
double-bass David Finck
drums David Meade
director Joe Langworth
Wednesday July 10 at 21.00
Thursday July 11 at 21.00
pianoforte Matthew Aucoin
violino Keir GoGwilt
Friday July 12 at 22.00
thanks to Angelo Fabbrini Pianoforti - Pescara and to the Conservatorio Francesco Morlacchi of Perugia for the invaluable support
Matthew Aucoin is a composer, conductor, pianist, and poet. His next opera has been commissioned by the American Repertory Theater, where it will be directed by Diane Paulus in 2015. Aucoin is currently an Assistant Conductor at the Metropolitan Opera, and he will make his debut with the Orchestra del Teatro dell'Opera di Roma in the fall; he has guest-conducted orchestras including the Longwood Symphony and the Brattle Street Chamber Players. Aucoin is currently music-directing Paulo Szot's new one-man show, which will arrive in New York in August, and composing new orchestral works for the Encounters Ensemble orchestra and the Fire Island Opera Festival. His past operas have been performed at The New York Composers Collaborative and the Loeb Drama Center. A 2012 graduate of Harvard College (summa cum laude), Aucoin has published poems and essays in publications including The Boston Globe, The Harvard Advocate, and The Colorado Review.
KEIR GOGWILT
Keir GoGwilt was born in Edinburgh, Scotland and grew up in New York City. Recent performances include the Beethoven Violin Concerto with the Bowdoin International Music Festival Orchestra, Mozart Violin Concerto #3 with the Greenwich Village Orchestra, Vivaldi Four Seasons with the Bach Society Orchestra at Harvard, the Berg Violin Concerto and Aucoin's This Same Light with the Encounters Ensemble. Keir has performed recitals at the Century Club, Miller Theater, the Canadian Opera Company, and in collaboration with Robert Levin on the world premiere of Levin's completion of a Mozart piano trio at the Sarasota Opera House. Tzadik records will release Keir's recordings of Tobias Picker's violin and chamber music, in collaboration with the Tobias Picker Ensemble and the Brentano Quartet. A 2013 graduate from Harvard in the literature department, Keir was awarded the Louis Sudler Prize in the Arts. His senior thesis, "Performing Responsibility," considers the implications of scientific studies of musicians' technique for literary theory and philosophy. Keir has studied with Lewis Kaplan, Christian Tetzlaff, Ute Hasenauer, Jorie Graham, and John Hamilton among others.
PAULO SZOT
In his return to Spoleto, after his debut as Donato in Maria Golovin, Paulo Szot recently starred in the broadway revival of South Pacific at Lincoln Center Theater and received several broadway awards in 2008: TONY Awards, Drama Desk, Outer Critics Circle and Theater World Award as best actor in a musical for his portrayal of Emile de Becque.
The Brazilian baritone made his acclaimed Metropolitan Opera debut as "Kovalyov" in a new production of Shostakovich's The Nose in 2010. Career highlights include the role of Don Giovanni with Opéra de Bordeaux, Michigan Opera, and the Dallas Opera; the title role of Eugene Onegin, Guglielmo (Così fan tutte), Aeneas (Dido and Aeneas) in Marseille; Count Almaviva (Le Nozze di Figaro) at New York City Opera, Boston Lyric Opera, De Vlaamse Opera and Aix-En-Provence; Escamillo with the Metropolitan Opera, Canadian Opera Company, Teatro Municipal de Santiago, Sao Paulo and Rio de Janeiro; Manon at the Metropolitan Opera.
Most recently made his debut at Opera de Paris/Palais Garnier in Cosi fan Tutte, Opera di Roma in The Nose and Alla Scala di Milano in A Dog's Heart.
He also performed at Carnegie Hall with the New York Pops, Avery Fischer Hall with the New York Philharmonic where he presented his acclaimed by critics and audience show An Evening with Paulo Szot, also presented as part of the American Songbook Series at Jazz at Lincoln Center and at the traditional Cafe Carlyle in New York.
His upcoming engagements includes: The Nose at The Metropolitan Opera, Die Fledermaus at the Metropolitan Opera, Le Nozze di Figaro in Bahrein among others.
San Nicolò Chiostro
...................................................
one seat € 20,00

redpajama_set_name: RedPajamaCommonCrawl


This Texas Boot Suncatcher is proudly made in the USA. This Texas Home Decor art is hand painted on 3/16" thick beveled glass using brilliantly colored crystalline, frosted and translucent lacquers by talented artisans in the USA.
This oval-shaped Texas suncatcher measures 9" high by 6" wide and includes a cord for hanging.
Let this beautiful Texas sun catcher welcome guests to your home. This fun Texas Gift will be treasured for years!


220 people volunteered at the Woodridge Park District's annual Community Clean Up the Parks Day on Saturday, April 13. Volunteers cleaned 16 park sites, collected 40 55-gallon bags of trash and three 35-gallon bags of recyclables.
Construction at Triangle Park began in January in order to improve overall storm water management functions and preservation of the park land.
Summer camps, sport camps, leagues, special events and more are packed into the 2019 Summer Activity Guide!


These day Angularjs gained much focus from developer community. If you an Angular js developer and willing to know what versions of AngularJs are available in Market. Then you can read it from here.
Angular 1.x, Angular 2, Angular 4, Angular 5, Angular 6, Angular 7 and Angular 8.