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text:
Space and Physics
NASA Image Of "Unusual" Hole On Mars Thought To Be An Underground Cavern
Mar 4 2020, 22:03 UTC
The hole is an opening to a deeper cavern on the western slopes of Pavonis Mons volcano on Mars. NASA, JPL, U. Arizona
Every day we seem to learn something new about Mars' surface thanks to the multiple missions either on or circling the Red Planet, snapping away. Like the fact that it looks delicious or has huge holes, which would make it a golf course fit for a giant.
We're pretty sure that's not what this hole is. According to NASA, who shared the image as part of its Astronomy Image of the Day series, the hole is likely an opening to an underground cavern. Still, excitingly mysterious.
The image of the hole, approximately 35 meters (115 feet) across, was discovered by accident in 2011. At the time, the HiRISE instrument aboard NASA's Mars Reconnaissance Orbiter, which is currently orbiting Mars, was snapping photos of the dusty slopes of Pavonis Mons volcano.
Analysis of the image, and follow up photos, revealed that it appears to be an opening to an underlying cavern. The shadow of the hole's entrance reveals its depth, thought to extend around 20 meters (65 feet) deep, though it could be much larger.
Gamma-Ray Eclipses Have Been Spotted For The First Time
The Moon Will Eclipse Mars On Monday Night
Astronomers Discover 25 New Fast Radio Bursts Unlike What We Have Seen Before
The most likely scenario is that it's the opening to a lava tube. The hole is on the western slopes of Pavonis Mons, which has many well-documented lava tubes. As the lava rolls down the slopes, it can solidify on the surface while still moving underneath. When the lava drains away, it can leave hollow tubes, and if a section of the roof collapses, it creates a skylight, or hole, much like those that occur on Earth. Why there is a circular raised crater around the hole, however, is still a matter for debate.
What isn't up for debate is how important these holes are for researchers to explore. Because their interior caves are relatively protected from Mars' harsh surface – with a thin atmosphere, the planet is more subject to the Sun's heat and radiation – such holes are considered good candidates for supporting or containing evidence of life on Mars. Probably not the alien cowering in a hole kind, though.
The deeper you go on Mars, the warmer it gets, which means at some point conditions may have been right for liquid water to be stable. Subterranean structures may have trapped water vapor at one point, protecting it from the surface onslaught. This makes them important targets for future missions looking for life. Plus it's a lot easier to drill down to look for minerals or gases when you're already 20 meters (65 feet) below the surface. As NASA's InSight mission is demonstrating, this is not as easy as hoped.
They could also make convenient potential habitats for future astronauts. It looks like the kind of place Mark Watney might have holed up in during The Martian's epic dust storm if he'd known about it (and, those kinds of dust storms could actually exist on Mars).
Katy is Managing Editor at IFLScience where she oversees editorial content from News articles to Features, and even occasionally writes some.
More Space and Physics Stories

redpajama_set_name: RedPajamaCommonCrawl


text:
Now a days Piracy is a very big problem. There is still a mistake that the air that is based on the complex of the gaming world, a universe of abstract, that makes no difference. They are video games, not have clothes or food. Sure, but before video games, are work . Works in which artists, writers, programmers, and many other people with debility and cold-blooded use of the time to entertain and amaze people. So be paid , because they are the works. People pay a chair, a sandwich, a shirt, a carpet, even the device or the monitor on which you are reading this article must be paid: why are jobs and the world turns thanks to them (and to the market, of course).The question, however, is unknown and it is quite clear in the minds of many people, but perhaps they are not very clear repercussions on the market caused by piracy .
Take for example Gentlemen! , title of Lucky Frame recently landed on Android. One of the developers, Yann Seznec , gave an interview on Indie Statik where he speaks of the results obtained from the game after landing on the platform of the green robot. "In terms of players, big time! Currently, we have about 3000 gamer " The developers of Lucky Frame joke about it, but the whole thing is incredible. On 3000 only 20 players bought the game regularly . Just to make a comparison, Gentlemen! iOS has sold about 154 copies in three days . But where must look for the reasons for these poor sales on Android and the consequent massive piracy.
In short, Lucky Frame looks ahead and tries to collect the positive elements from this unfortunate event.After this long and incredible story, let's see 'What is this Gentlemen!. Basically it is a game where two players battle head to head in a duel using knives, bombs and many other weapons. It is also possible to reverse gravity to dodge the blows of the enemy. Gentlemen! is available on the Play Store at a cost of 4.49 € (perhaps another factor that has fueled the high amount of pirated copies?) exclusively for the tablet . We leave you with the badge of purchase, along with the trailer and screenshots to the gallery: let us know what you think of the story is that of the game.
Gentlemen game is like 2 Players, One Screen: Gentlemen!
A Victorian era head-to-head arcade battle for tablets – 7 inches and up!
Take on your friends in this frantically paced duel game. Knives, bombs, homing pigeons – choose your weapon wisely, and master the art of the gravity flip. An honorable duel, a dapper delight – GENTLEMEN!You'll need to hone a delicate balance of technique, knowledge, cunning, and blind panic (just like a real duel!). You'll also need a friend – two players, one screen. If you don't have a friend, go and get one.
Gentlemen Game Android Gentlemen Game android review Gentlemen Game video review
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Next: Rush Song a Running Game with Music : Android


There are six images in this sequence.
After more than an hour of fighting the warriors are already tired and thirsty.
King Harald waits until the two sides are locked together before making his move.
First, he commands a blast to be sounded which tells his army to advance. So all along the line the Norse warriors advance into the ditch and towards the English line.
Next, when the armies are locked together, Harold takes his banner down to the river bank and leads a devastating charge. This will be the turning point in the battle.
King Harald's style was to swing his axe while others protected him with their spears and shields and exploit the breach in the shield wall, or so we hear in the Saga.
King Harald was a giant and coming fresh to the fight, with his chosen band, he would have cut a way where the riverside-path crossed the Beck. From the moment when Harald broke cover until he reached the English line and Earl Edwin's small Mercian force, was probably less than a minute. The shock and momentum allowed King Harald to force his way over the Beck.
While some of Harald's Vikings pressed Earl Edwin's flank force back towards York, other Vikings now advanced along the defender's side of Germany Beck.


text:
\section{Introduction}
Consider the challenge of identifying a sparse source in an elliptic PDE from Dirichlet boundary data:
    \begin{equation}\label{eq1}
        \min_{(f,u) \in F_h \times H^1(\Omega)} \left\{ \frac{1}{2}\|u - d\|_{L^2(\partial\Omega)}^2 + \alpha \sum_i w_i |(f, \phi_i)_{L^2(\Omega)}| \right\}
    \end{equation}
    subject to
    \begin{equation}\label{eq2}
    \begin{split}
        -\Delta u + \epsilon u &= f \quad \mbox{in } \Omega, \\
        \frac{\partial u}{\partial {\mathbf{n}}} &= 0  \quad \mbox{on } \partial \Omega,
    \end{split}
    \end{equation}
where we employ weighted $\ell^1$-regularization in \eqref{eq1}. Here, $\{ \phi_1, \phi_2, \ldots, \phi_n \}$ is an $L^2$-orthonormal basis for $F_h$, $\{ w_i \}$ are positive weights, $\alpha > 0$ is a regularization parameter, \textcolor{black}{$d$ represents the Dirichlet boundary data}, $\epsilon$ is a parameter, ${\mathbf{n}}$ denotes the outwards pointing unit normal vector of the boundary $\partial \Omega$ of the bounded domain $\Omega$, and $f$ is the unknown source.

Variants of this inverse source problem appear in many applications, such as in crack determination, in the inverse ECG problem and in electroencephalography (EEG). Consequently, it has received much attention from researchers, see, e.g.,  \cite{babda09,cheng15,Han11,Het96,hinze19,BIsa05,kun94,ring95,song12}. However, in these studies the authors did not use $\ell^1$-regularization. A more detailed description of previous investigations is presented in \cite{Elv20}.

Regularization of inverse problems with sparsity promoting methods has increased in popularity in
recent years \cite{daubechies04, grasmair08, Jin_2012, Jin_2017,lorenz08, Lu_2019, Flemming16}. In this context, the notion of sparsity, with respect to a given basis $\{\phi_i\}$ for the control space, means that the inverse solution $f^*$ can be represented using very few of the basis functions. That is, if we expand the inverse solution $f^*$ as $f^* = \sum_i x_i^*\phi_i$, then $x_i^* = (f^*,\phi_i)_{L^2(\Omega)} \neq 0$ only for very few of the basis functions. If we have $s$ such non-zero components, we say that the solution $f^*$ is $s$-sparse.

In \textit{compressed sensing} the study of $\ell^1$-regularization for the exact recovery of a sparse source from noise free data has been studied in detail \cite{candes06,candes05,donoho03}. To elaborate the main findings in \cite{candes05}, let us assume that ${\mathsf{A}}$ is a matrix with a significant null space. If ${\mathbf{b}}^\dagger$ is generated from a sparse source ${\mathbf{x}}^\dagger$, i.e., ${\mathbf{b}}^\dagger = {\mathsf{A}}{\mathbf{x}}^\dagger$, then the minimizer of
\begin{equation*}
    \min_{{\mathbf{x}}\in{\mathbb{R}^n}} \|{\mathbf{x}}\|_1 \quad \textnormal{subject to} \quad {\mathsf{A}}{\mathbf{x}} = {\mathbf{b}}^\dagger,
\end{equation*}
is the true sparse solution ${\mathbf{x}}^\dagger$ when a certain assumption, known as the \textit{restricted isometry property}, is fulfilled.

In \cite{grasmair10} the authors unified this result with the theory developed by the inverse problem community. In particular, they showed that the commonly used range condition, combined with an additional restricted injectivity condition on the forward operator, are weaker assumptions than the previously mentioned \textit{restriced isometry property}, and they proved that the former conditions are the weakest which admit linear convergence rates for the regularized problem
\begin{equation*}
    \min_{{\mathbf{x}}\in{\mathbb{R}^n}} \left\{ \frac{1}{2} \|{\mathsf{A}}{\mathbf{x}} - {\mathbf{b}}\|_2^2 + \alpha \|{\mathbf{x}}\|_1 \right\}.
\end{equation*}
\begin{comment}
That is, for every $C > 0$, there exists $c > 0$ such that one of the solutions ${\mathbf{x}}_\alpha$ of the problem above satisfies
\begin{equation*}
    \|{\mathbf{x}}_\alpha - {\mathbf{x}}^\dagger\| \leq c\zeta
\end{equation*}
when the noise level is such that $\|{\mathbf{b}} - {\mathbf{b}}^\dagger\| = \|{\mathbf{b}} - {\mathsf{A}}{\mathbf{x}}^\dagger \| \leq \zeta$ and $\alpha = C\zeta$.
\end{comment}

\textcolor{black}{Numerical experiments indicate that the identification criteria, mentioned in the previous paragraph, are not fulfilled by the forward/transfer matrix ${\mathsf{A}}$ associated with \eqref{eq1}-\eqref{eq2},
see Figure \ref{fig:ex0} (and appendix \ref{app:discretization} for details about the matrix ${\mathsf{A}}$). This is the main motive for the present study. Observe also that the inverse solution in panel (b) in Figure \ref{fig:ex0} only is nonzero close to where observations are made, i.e., close to the boundary $\partial \Omega$ of $\Omega$. In view of Proposition 2.3 in \cite{casas2012approximation}, this is not surprising.}

\begin{figure}[h]
    \centering
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex0/true.png}
        \caption{True source.}

    \end{subfigure}
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex0/inverse.png}
        \caption{Inverse solution.}

    \end{subfigure}\par
    \caption{Panel (b) shows the outcome of attempting to use \eqref{eq1}-\eqref{eq2}, with $w_i=1$ for $i=1,2,\ldots,n$ and $\alpha = 10^{-4}$, to recover the true interior source depicted in panel (a). That is, the Dirichlet boundary data $d$ in \eqref{eq1} was generated by the true source (a).}

    \label{fig:ex0}
\end{figure}

Many researchers have explored the use of sparsity regularization in connection with PDEs, see, e.g., \cite{casas2012approximation,casas2013parabolic,Golmohammadi2018,herman2020randomization,Xiang_2020,li2019sparse,Rudy2009,Wang2012}. However, as far as the authors know, the use of weighted $\ell^1$-regularization to solve inverse source problems for PDEs, employing only boundary measurements, has not previously been attempted. In \cite{mehr_2015} the authors apply an iterative reweighted $\ell^1$-regularization technique \cite{candes2008enhancing} to recover sources, but their approach differs significantly from ours and involve data recorded at equidistant locations inside the domain.

We will start with a brief motivation for developing weighted sparsity promoting regularization for ECG and EEG applications in Section \ref{sec:motivation}. In Section \ref{sec:analysis} we prove that our methodology can recover, without any errors or blurring, a single local source, regardless of whether it is interior or at the boundary. We also analyze a problem for which it is impossible to recover two separate sources. This implies that we can not guarantee, in general, that our approach can recover multiple local sources. However, in Section \ref{sec:numerical_experiments} many numerical experiments are presented, for both single and multiple sources, and we observe that several sources often can be successfully recovered.

We would like to emphasize that, even though our motivation originates from inverse source problems for elliptic PDEs, the general theory presented in this paper is applicable to any linear finite-dimensional inverse problem where the forward operator has a non-trivial null space.

\section{Motivation}
\label{sec:motivation}
The purpose of EEG is to recover electrical activity in the brain from voltage recordings on the scalp. If one suspects that the true signal is spatially local, e.g., for focal epileptic seizures, it is natural to search for sparse solutions.

Similarly, in the inverse problem of ECG, the aim can be to locate an ischemic\footnote{Ischemia is a precursor of a heart infarct.} region of the heart. This area will have an electrical potential which is different from the voltage in healthy tissue. The difference in the potential can be interpreted as the source in an elliptic PDE. If we assume that the ischemic region is small, with a sharp transition between ischemic and healthy tissue, it is reasonable to search for a sparse inverse solution.

\begin{comment}
The optimization problem which must be solved in connection with EEG and ECG recordings differs slightly from \eqref{eq1}-\eqref{eq2}:
\begin{equation}\label{EEG-ECG1}
        \min_{(f,u) \in F_h \times H^1(\Omega)} \left\{ \frac{1}{2}\sum_{i=1}^m (u_i - d_i)^2 + \alpha\|{\mathsf{W}} f\|_{L^1(\Omega)} \right\}
    \end{equation}
    subject to
    \begin{equation}\label{EEG-ECG2}
    \begin{split}
        -\nabla \cdot (\sigma \nabla u) &= \underbrace{f}_{= \nabla \cdot \mathbf{j}^p} \quad \mbox{in } \Omega, \\
        \frac{\partial u}{\partial {\mathbf{n}}} &= 0  \quad \mbox{on } \partial \Omega,
    \end{split}
    \end{equation}
where we, as above, employ weighted $L^1$-regularization. Here, $u_i$ and $d_i$ denote the simulated and recorded electrical potentials at the position of electrode number $i$, respectively, $\sigma=\sigma(x)$ is a conductivity tensor \cite{PMID:16364662, MacLachlan}, and $\mathbf{j}^p$ denotes the primary or impressed current. See, e.g., \cite{Wan13,Nie13a,Sarvas_1987,Hallez2007ReviewOS} for further details. The lack of the zero order term $\epsilon u$ in \eqref{EEG-ECG2}, compared to \eqref{eq2}, implies that the solution of \eqref{EEG-ECG2} is not unique. It is only determined up to a constant, but this matter can be handled by the techniques described in the appendix of \cite{Elv20a2}. (We also mention that \eqref{EEG-ECG1}-\eqref{EEG-ECG2} differs significantly from the inverse ECG problem which seeks to estimate the electrical potential at the heart surface from body surface measurements \cite{Rudy2009,Wang2012}. The latter problem typically does not have a significant null space because it is closely related to the Cauchy problem for Laplace's equation.)

After discretization, both \eqref{eq1}-\eqref{eq2} and \eqref{EEG-ECG1}-\eqref{EEG-ECG2} can be formulated in terms of the so-called {\em transfer matrix} ${\mathsf{A}}$, which maps discrete versions of $f$ to $(u_1,u_2,\ldots,u_m)$. Below we both define ${\mathsf{W}}$ and present our analysis in terms of Euclidean spaces. It therefore follows that our findings, in principle, are directly applicable to discrete versions of \eqref{EEG-ECG1}-\eqref{EEG-ECG2}. On the other hand, practical use would require that a number of involved technical issues are handled: one must construct suitable geometrical models, obtain EEC or ECG recordings, handle noisy data, etc.
\end{comment}

Solving inverse source problems using optimization procedures is challenging. For example, the minimizer $f_\alpha \in F_h \subset L^2(\Omega)$ of
    \begin{equation*}
        \min_{(f,u) \in F_h \times H^1(\Omega)} \left\{ \frac{1}{2}\|u - d\|_{L^2(\partial\Omega)}^2 + \frac{1}{2}\alpha\|f\|_{L^2(\Omega)}^2 \right\}
    \end{equation*}
    subject to
    \begin{equation*}
    \begin{split}
        -\Delta u + \epsilon u &= f \quad \mbox{in } \Omega, \\
        \frac{\partial u}{\partial {\mathbf{n}}} &= 0  \quad \mbox{on } \partial \Omega,
    \end{split}
    \end{equation*}
is not, in general, a good approximation of a true interior source: Even if the data $d$ is generated from a single basis function $\phi_j \in F_h$, representing an interior local source, the minimum $L^2$-norm least-squares solution $f^* = \lim_{\alpha\rightarrow0} f_{\alpha}$ will attain its maximum at the boundary $\partial \Omega$ of the domain $\Omega$, see \cite{Elv20}. However, by employing a carefully chosen "diagonal" regularization operator $W$, i.e., $$W \phi_i = w_i \phi_i, \quad i=1,2,\ldots,n,$$ with suitable weights $\{ w_i \}$, we proved in \cite{Elv20} that $W^{-1} f^*$ attains its maximum at the position of the true local source $\phi_j$. On the other hand, $W^{-1} f^*$ is very smooth and its magnitude will typically not be close to the magnitude of the true source.

The need for sparse solutions in many applications, as exemplified above, thus motivates the study of $\ell^1$-regularization for inverse source problems. We now present a brief overview of the main results of our forthcoming analysis.

Assume that the Dirichlet boundary data $d \in L^2(\partial\Omega)$ in \eqref{eq1} is generated from a basis function $\phi_j \in F_h$, i.e., $d = {K_h}} %{{\mathsf{K_h}}\phi_j$, where
\begin{equation*}
 {K_h}} %{{\mathsf{K_h}}: F_h \rightarrow L^2(\partial\Omega), \quad f \mapsto u|_{\partial\Omega},
\end{equation*}
is the forward operator associated with \eqref{eq1}-\eqref{eq2}. Provided that $W$ denotes the above-mentioned regularization operator introduced in \cite{Elv20}, we will prove that $\phi_j$ is the unique solution of
\begin{equation*}
    \min_{f \in F_h} \sum_i w_i |(f, \phi_i)_{L^2(\Omega)}| \quad \textnormal{subject to} \quad {K_h}} %{{\mathsf{K_h}} f = {K_h}} %{{\mathsf{K_h}} \phi_j.
\end{equation*}
We will further show, using a slightly modified fidelity term in \eqref{eq1}-\eqref{eq2}, that the associated minimizer of the weighted $\ell^1$-regularized problem is
\begin{equation*}
 f_\alpha = \gamma_\alpha \phi_j,
\end{equation*}
where $\gamma_\alpha = 1 - c\alpha$, and $c$ is a positive constant. That is, the correct basis function is recovered without any blurring for all values of $\alpha < \bar{\alpha}$, albeit with an error in magnitude equal to $c\alpha$. The constants $\bar{\alpha}$ and $c$ can be computed from $W$ and a projection operator.

Ideally, we would like to analyze the recovery of more general composite sources. We have not been able to do so, but we address this issue numerically in section \ref{sec:numerical_experiments}.

\section{Analysis}\label{sec:analysis}
For the sake of clarity, we will present our theoretical results in terms of Euclidean spaces, employing the standard inner product and the standard basis vectors. It should be noted that analogous results can be established for linear operators acting on finite dimensional vector spaces, provided that an orthonormal basis is employed for the domain of the operators.

\begin{comment}
\textcolor{red}{Below we will discretize the source $f \in F_h$ in \eqref{eq1}-\eqref{eq2} in terms of scaled characteristic functions $\{ \phi_i = c_i \mathcal{X}_{\Omega_i} \}$ with disjoint supports $\{ \Omega_i \}$, $f= \sum_i x_i \phi_i$. Since also our regularization operator ${\mathsf{W}}$ is "diagonal", i.e. ${\mathsf{W}} \phi_i=w_i \phi_i$ for $i=1,2,\ldots,n$, where $\{w_i \}$ are positive scalars, there is a simple connection between the weighted $L^1$- and $\ell^1$-norms:
\begin{align*}
    \|{\mathsf{W}} f\|_1 &= \int_{\Omega} |{\mathsf{W}} \sum_i x_i \phi_i| \, dx \\
    &= \sum_i \int_{\Omega_i} |w_i x_i \phi_i| \, dx \\
    &= C \sum_i |w_i x_i| \\
\end{align*}
provided that the constants $\{ c_i\}$ are chosen such that $C=\| \phi_1 \|_1 = \| \phi_2 \|_1 = \ldots \| \phi_n \|_1$. In this context, $n$ represents the number of basis functions.
}
\end{comment}

Consider the problem
\begin{equation} \label{eq:BF1}
      \min_{{\mathbf{x}}\in{\mathbb{R}^n}} \left\{ \frac{1}{2}\|{\mathsf{A}}{\mathbf{x}} - {\mathbf{b}}\|_2^2 + \alpha\|{\mathsf{W}}{\mathbf{x}}\|_1 \right\},
   \end{equation}
where ${\mathsf{A}} \in {\mathbb{R}^{m\times n}}$ has a non-trivial null space\footnote{Our analysis also holds for matrices with linearly independent columns, but this is the trivial case.} $\nullspace{{\mathsf{A}}}$. Associated with ${\mathsf{A}}$ is the orthogonal projection matrix \begin{equation}\label{eq:Pmatrix}
\mathsf{P}: {\mathbb{R}^n} \rightarrow \mathcal{N}({\mathsf{A}})^\perp.
\end{equation}
That is, $\mathsf{P} = {\mathsf{A}}^\dagger{\mathsf{A}}$, where ${\mathsf{A}}^\dagger$ denotes the Moore-Penrose inverse of ${\mathsf{A}}$.
\textcolor{black}{Appendix \ref{app:discretization} contains information about the forward matrix ${\mathsf{A}}$ associated with our model problem \eqref{eq1}-\eqref{eq2}.}

Throughout this paper, the diagonal regularization matrix ${\mathsf{W}} \in {\mathbb{R}^{n\times n}}$ is defined as
\begin{equation}\label{eq:Wmatrix}
    {\mathsf{W}}{\mathbf{e}}_i = \|\mathsf{P}{\mathbf{e}}_i\|_2 {\mathbf{e}}_i, \, i=1,2,\ldots,n,
\end{equation}
where we assume that $\mathsf{P}{\mathbf{e}}_i \neq 0, \, i=1,2,\ldots,n.$ The definition of this matrix can be motivated by the classical theory for the minimum norm least squares solution of linear systems and Tikhonov regularization, see \cite{Elv20} for further details.
Furthermore, the beneficial mathematical properties of this operator in connection with quadratic regularization are studied in \cite{Elv20,Elv20a2}. Below it will become clear that \eqref{eq:Wmatrix} also plays an important role for developing sparsity promoting regularization techniques.

It can be CPU demanding to compute $\mathsf{P}{\mathbf{e}}_i$ for large systems because $\mathsf{P}$ involves the Moore-Penrose inverse ${\mathsf{A}}^\dagger$ of ${\mathsf{A}}$. On the other hand, if the underlying problem is ill posed, such as \eqref{eq1}-\eqref{eq2}, then one would typically not use a very fine mesh for the discretization of the source, and the regularization matrix ${\mathsf{W}}$ is applicable. Below we will approximate ${\mathsf{A}}^\dagger$ using either truncated SVD (subsection \ref{subsec:altOpt}) or standard Tikhonov regularization (subsection \ref{subsec:largeCircularSource}).

The main purpose of this section is to analyze whether \textcolor{black}{the use of our weighted regularization technique enables the recovery of a standard basis vector ${\mathbf{e}}_j \in {\mathbb{R}^n}$ from the exact data ${\mathbf{b}}^\dagger = {\mathsf{A}}{\mathbf{e}}_j$.}
Our starting point is thus the following optimization problem.
\begin{itemize}
    \item \textbf{Problem 0:}
    \begin{equation}\label{eq:genmin}
      \min_{{\mathbf{x}}\in{\mathbb{R}^n}} \left\{ \frac{1}{2}\|{\mathsf{A}}{\mathbf{x}} - {\mathbf{b}}^\dagger\|_2^2 + \alpha\|{\mathsf{W}}{\mathbf{x}}\|_1 \right\}.
   \end{equation}
\end{itemize}

\subsection{Weighted basis pursuit}

Our first result concerns the solution of Problem 0 in the limit $\alpha \rightarrow 0$.
Recalling that ${\mathbf{b}}^\dagger = {\mathsf{A}}{\mathbf{e}}_j$, it is well-known that this limit problem can be formulated as
\begin{itemize}
    \item \textbf{Problem I:}
    \begin{equation*
      \min_{{\mathbf{x}}\in{\mathbb{R}^n}} \|{\mathsf{W}}{\mathbf{x}}\|_1 \quad \textnormal{subject to} \quad {\mathsf{A}}{\mathbf{x}} = {\mathsf{A}}{\mathbf{e}}_j.
   \end{equation*}
\end{itemize}
We also introduce an equivalent formulation of Problem I, which reads
 \begin{itemize}
  \item \textbf{Problem II:}
    \begin{equation*
      \min_{{\mathbf{x}}\in{\mathbb{R}^n}} \|{\mathsf{W}}{\mathbf{x}}\|_1 \quad \textnormal{subject to} \quad \mathsf{P}{\mathbf{x}} = \mathsf{P}{\mathbf{e}}_j.
  \end{equation*}

  \end{itemize}
Problems I and II are equivalent because the null spaces of ${\mathsf{A}}$ and $\mathsf{P}$ coincide, i.e., $\mathcal{N}({\mathsf{A}}) = \mathcal{N}(\mathsf{P})$. Hence, either problem can be reformulated as
  \begin{equation*}
    \min_{\mathbf{q} \in \mathcal{N}({\mathsf{A}})} \|{\mathsf{W}}({\mathbf{e}}_j + \mathbf{q})\|_1.
  \end{equation*}

We will assume that no {\em two} columns of ${\mathsf{A}}$ are parallel in order to ensure that the solutions of our minimization problems are {\em unique}.
\begin{assumption}\label{nonpar}
 We assume that ${\mathsf{A}} \in {\mathbb{R}^{m\times n}}$ is such that ${\mathsf{A}}{\mathbf{e}}_j \neq c{\mathsf{A}}{\mathbf{e}}_i$ for all $i, j \in \{1,2,...,n\}$, $i \neq j$, and all $c \in \mathbb{R}$.
\end{assumption}
\noindent Note that this assumption implies that ${\mathbf{e}}_i \notin \mathcal{N}(A)$, \, $i \in \{1,2,...,n\}$. Furthermore, invoking the orthogonal decomposition ${\mathbf{e}}_i = \mathsf{P} {\mathbf{e}}_i + ({\mathbf{e}}_i - \mathsf{P} {\mathbf{e}}_i)$, where $({\mathbf{e}}_i - \mathsf{P} {\mathbf{e}}_i) \in \nullspace{A}$, it follows that ${\mathsf{A}} {\mathbf{e}}_i = {\mathsf{A}} \mathsf{P} {\mathbf{e}}_i$. Consequently, if ${\mathsf{A}}$ obeys Assumption \ref{nonpar},
then $\mathsf{P}$ must also satisfy
\begin{equation}
    \label{Pnonpar}
    \mathsf{P} {\mathbf{e}}_j \neq c\mathsf{P} {\mathbf{e}}_i \mbox{ for all } i, j \in \{1,2,...,n\}, \, i \neq j, \mbox{ and all } c \in \mathbb{R}.
\end{equation}

We can now formulate our first result. Let us remark that the study of sparsity promoting regularization techniques usually leads to very involved mathematical analysis. However, with the very specific choice of the weight matrix ${\mathsf{W}}$ in \eqref{eq:Wmatrix} and the data ${\mathbf{b}}^\dagger = {\mathsf{A}}{\mathbf{e}}_j$, our analysis becomes rather "transparent".
\begin{theorem}[Exact recovery of a basis vector]\label{thm:exactRecov}
    Assume that ${\mathsf{A}} \in {\mathbb{R}^{m\times n}}$ satisfies Assumption \ref{nonpar}, and let $\mathsf{P}$ and ${\mathsf{W}}$ be the matrices defined in \eqref{eq:Pmatrix} and \eqref{eq:Wmatrix}, respectively. Then ${\mathbf{e}}_j$ is the unique solution of problems I and II.
\end{theorem}
\begin{proof}
  Problem I and Problem II are equivalent. We choose to consider the latter, i.e.,
    \begin{equation*}
      \min_{{\mathbf{x}}\in{\mathbb{R}^n}} \|{\mathsf{W}}{\mathbf{x}}\|_1 \quad \textnormal{subject to} \quad \mathsf{P}{\mathbf{x}} = \mathsf{P}{\mathbf{e}}_j.
  \end{equation*}
  Define $X_j = \{{\mathbf{x}}\in{\mathbb{R}^n}: \mathsf{P}{\mathbf{x}} = \mathsf{P}{\mathbf{e}}_j\}$. Let ${\mathbf{x}} = \sum_i c_i{\mathbf{e}}_i\in X_j, {\mathbf{x}} \neq {\mathbf{e}}_j$ be arbitrary. Then, see \eqref{eq:Wmatrix},
  \begin{eqnarray*}
    \|{\mathsf{W}}{\mathbf{e}}_j\|_1 &=& \left\| \|\mathsf{P}{\mathbf{e}}_j\|_2 {\mathbf{e}}_j \right\|_1 \\
    &=& \|\mathsf{P}{\mathbf{e}}_j\|_2 \\
    &=& \left\|\mathsf{P}\left(\sum_i c_i{\mathbf{e}}_i\right)\right\|_2 \\
    &=& \left\|\sum_i c_i\mathsf{P}{\mathbf{e}}_i\right\|_2 \\
    &\leq& \sum_i |c_i| \|\mathsf{P}{\mathbf{e}}_i\|_2 \\
    &=& \left\| \sum_i c_i \|\mathsf{P}{\mathbf{e}}_i\|_2 {\mathbf{e}}_i \right\|_1\\
    &=& \|{\mathsf{W}}{\mathbf{x}}\|_1.
  \end{eqnarray*}
Using \eqref{Pnonpar}, which is a consequence of Assumption \ref{nonpar}, we get strict inequality in the third to last step, and the result follows.
\end{proof}
\noindent Theorem \ref{thm:exactRecov} states that a single basis vector ${\mathbf{e}}_j$ can be exactly recovered from the data ${\mathbf{b}}^\dagger = {\mathsf{A}}{\mathbf{e}}_j$ by solving either Problem I or Problem II. If Assumption \ref{nonpar} does not hold, ${\mathbf{e}}_j$ would still be a minimizer, but the uniqueness is not assured. (Similar statements hold for the remaining results presented in this paper.)

\subsection{Alternative optimization problems}
\label{subsec:altOpt}
Inspired by Problem II, we now suggest an alternative to \eqref{eq:BF1}.
Since $\mathsf{P} = {\mathsf{A}}^\dagger{\mathsf{A}}$, it follows that
\[
\mathsf{P} {\mathbf{e}}_j = {\mathsf{A}}^\dagger{\mathsf{A}}{\mathbf{e}}_j =  {\mathsf{A}}^\dagger {\mathbf{b}}^\dagger,
\]
where ${\mathbf{b}}^\dagger = {\mathsf{A}}{\mathbf{e}}_j$.
Consequently, for a general right-hand-side ${\mathbf{b}}$, Problem II motivates the following alternative to \eqref{eq:BF1}
\begin{equation}
    \label{eq:BF2}
    \min_{{\mathbf{x}}\in{\mathbb{R}^n}} \left\{ \frac{1}{2} \|\mathsf{P}{\mathbf{x}} - {\mathsf{A}}^\dagger {\mathbf{b}}\|_2^2 + \alpha\|{\mathsf{W}}{\mathbf{x}}\|_1 \right\}.
\end{equation}

If ${\mathsf{A}}$ has very small singular values, it may not be advisable to apply ${\mathsf{A}}^\dagger$ in practise. Therefore we want to approximate  ${\mathsf{A}}^\dagger$ with a more well-behaved matrix. This can, e.g., be accomplished by employing the truncated SVD to get an approximation $\mathsf{A}_k$ of ${\mathsf{A}}$. Here, $k \leq m$ represents the number of singular values that are unchanged in the truncation: Assuming that the singular values of ${\mathsf{A}}$ are sorted in decreasing order $\sigma_1 \geq \sigma_2 \geq \ldots \geq \sigma_m$, $\mathsf{A}_k$ will have the singular values $\sigma_1 \geq \sigma_2 \geq \ldots \geq \sigma_k \geq 0 = 0 = \ldots = 0$.

Below we need the orthogonal projection onto the orthogonal complement of the null space of $\mathsf{A}_k$,
\begin{equation}
\label{eq:defPK}
\mathsf{P}_k: \mathbb{R}^n \rightarrow \nullspace{\mathsf{A}_k}^\perp.
\end{equation}
And, analogously to \eqref{eq:Wmatrix}, we define
\begin{equation}\label{eq:WKmatrix}
    \mathsf{W}_k {\mathbf{e}}_i = \|\mathsf{P}_k {\mathbf{e}}_i\|_2 {\mathbf{e}}_i, \, i=1,2,\ldots,n.
\end{equation}

Replacing ${\mathsf{A}}^\dagger$ in \eqref{eq:BF2} with $\mathsf{A}_k^\dagger$, keeping in mind that $\mathsf{P} = {\mathsf{A}}^\dagger{\mathsf{A}}$, leads to the following alternative to \eqref{eq:BF1}
      \begin{equation} \label{eq:BF3}
        \min_{{\mathbf{x}}\in{\mathbb{R}^n}} \left\{ \frac{1}{2} \|\mathsf{A}_k^\dagger {\mathsf{A}}{\mathbf{x}} - \mathsf{A}_k^\dagger {\mathbf{b}}\|_2^2 + \alpha\|\mathsf{W}_k{\mathbf{x}}\|_1 \right\}.
      \end{equation}
Note that we have also replaced ${\mathsf{W}}$ with $\mathsf{W}_k$.

From the SVD of ${\mathsf{A}}={\mathsf{U}} {\mathsf{\Sigma}} {\mathsf{V}}^T$, we find the SVD of $\mathsf{A}_k$,
\[
\mathsf{A}_k = {\mathsf{U}} \mathsf{\Sigma}_k {\mathsf{V}}^T.
\]
Observe that
\[
\mathsf{P}_k=\mathsf{A}_k^\dagger \mathsf{A}_k = {\mathsf{V}} \mathsf{\Sigma}_k^\dagger \mathsf{\Sigma}_k {\mathsf{V}}^T,
\]
and that
\[
\mathsf{A}_k^\dagger {\mathsf{A}} = {\mathsf{V}} \mathsf{\Sigma}_k^\dagger {\mathsf{\Sigma}} {\mathsf{V}}^T.
\]
This yields, since $\mathsf{\Sigma}_k^\dagger \mathsf{\Sigma}_k = \mathsf{\Sigma}_k^\dagger {\mathsf{\Sigma}}$,
\begin{equation}
\label{eq:projectionProperty}
    \mathsf{P}_k=\mathsf{A}_k^\dagger {\mathsf{A}},
\end{equation}
and we can write \eqref{eq:BF3} in the form
\begin{equation}
\label{eq:BF4}
        \min_{{\mathbf{x}}\in{\mathbb{R}^n}} \left\{ \frac{1}{2} \|\mathsf{P}_k {\mathbf{x}} - \mathsf{A}_k^\dagger {\mathbf{b}}\|_2^2 + \alpha\|\mathsf{W}_k{\mathbf{x}}\|_1 \right\}.
\end{equation}

In the next subsections we will analyze \eqref{eq:BF2} and \eqref{eq:BF4} when ${\mathbf{b}} = {\mathsf{A}}{\mathbf{e}}_j$ and ${\mathbf{b}} = {\mathsf{A}}{\mathbf{e}}_j + \eta$, respectively, where $\eta$ represents noise.

\begin{comment}
If ${\mathsf{A}}$ has very small singular values, it may not be advisable to apply ${\mathsf{A}}^\dagger$ in practise. Therefore we want to approximate  ${\mathsf{A}}^\dagger$ with a more well-behaved matrix. This can, e.g., be accomplished by considering the Tikhonov regularized problem
\begin{equation*}
    \min_{{\hat{\mathbf{x}}}\in{\mathbb{R}^n}} \frac{1}{2}\|{\mathsf{A}}{\hat{\mathbf{x}}}-{\mathbf{b}}\|_2^2 + \frac{1}{2}{\beta}\|{\hat{\mathbf{x}}}\|_2^2,
\end{equation*}
which has the first-order optimality condition
\begin{equation*}
    {\mathsf{A}}^T{\mathsf{A}}{\hat{\mathbf{x}}} + {\beta}{\hat{\mathbf{x}}} = {\mathsf{A}}^T{\mathbf{b}}.
\end{equation*}
Using the singular value decomposition ${\mathsf{A}} = \mathsf{U}\Sigma\mathsf{V}^T$ of ${\mathsf{A}}$, we get the expression
\begin{equation*}
  {\hat{\mathbf{x}}}_{\beta} = \mathsf{V}(\Sigma^2 + {\beta}\mathsf{I})^{-1}\mathsf{V}^T {\mathsf{A}}^T {\mathbf{b}}
  = {\mathsf{S}_{\beta}}{\mathbf{b}},
\end{equation*}
where
\begin{equation}\label{eq:Sa}
{\mathsf{S}_{\beta}} = \mathsf{V}(\Sigma^2 + {\beta}\mathsf{I})^{-1}\mathsf{V}^T {\mathsf{A}}^T \approx {\mathsf{A}}^\dagger.
\end{equation}
Replacing ${\mathsf{A}}^\dagger$ in \eqref{eq:BF2} with ${\mathsf{S}_{\beta}}$, keeping in mind that $\mathsf{P} = {\mathsf{A}}^\dagger{\mathsf{A}}$, leads to the following alternative to \eqref{eq:BF1} and \eqref{eq:BF2}
      \begin{equation} \label{eq:BF3}
        \min_{{\mathbf{x}}\in{\mathbb{R}^n}} \|{\mathsf{S}_{\beta}}{\mathsf{A}}{\mathbf{x}} - {\mathsf{S}_{\beta}} {\mathbf{b}}\|_2^2 + \alpha\|{\mathsf{W}}{\mathbf{x}}\|_1.
      \end{equation}
In the next subsection we will analyze \eqref{eq:BF2} and \eqref{eq:BF3} when ${\mathbf{b}} = {\mathsf{A}}{\mathbf{e}}_j$.
\end{comment}

\subsection{Analysis of regularized problems}
 In this subsection we will make use of the following maximum property derived in \cite{Elv20}:
 \begin{equation*}
     j \in \argmax_{i \in \{1,2,...,n\}} [{\mathsf{W}}^{-1}\mathsf{P}{\mathbf{e}}_j]_i,
 \end{equation*}
 where $[{\mathsf{W}}^{-1}\mathsf{P}{\mathbf{e}}_j]_i$ denotes the $i$'th component of the vector ${\mathsf{W}}^{-1}\mathsf{P}{\mathbf{e}}_j$, and $\mathsf{P}$ and ${\mathsf{W}}$ are defined in \eqref{eq:Pmatrix} and \eqref{eq:Wmatrix}, respectively.
 The proof of this is short, so we include it for the sake of completeness:
 \begin{eqnarray*}
  {\mathsf{W}}^{-1}\mathsf{P}{\mathbf{e}}_j &=& {\mathsf{W}}^{-1}\sum_{i=1}^n (\mathsf{P}{\mathbf{e}}_j,{\mathbf{e}}_i){\mathbf{e}}_i \\
  &=& \sum_{i=1}^n(\mathsf{P}{\mathbf{e}}_j,{\mathbf{e}}_i)\|\mathsf{P}{\mathbf{e}}_i\|^{-1}{\mathbf{e}}_i \\
  &=& \sum_{i=1}^n(\mathsf{P}{\mathbf{e}}_j,\mathsf{P}{\mathbf{e}}_i)\|\mathsf{P}{\mathbf{e}}_i\|^{-1}{\mathbf{e}}_i \\
  &=& \|\mathsf{P}{\mathbf{e}}_j\|\sum_{i=1}^n\left(\frac{\mathsf{P}{\mathbf{e}}_j}{\|\mathsf{P}{\mathbf{e}}_j\|}, \frac{\mathsf{P}{\mathbf{e}}_i}{\|\mathsf{P}{\mathbf{e}}_i\|}\right){\mathbf{e}}_i.
 \end{eqnarray*}
Furthermore, it is shown in \cite{Elv20a2} that, whenever Assumption \ref{nonpar} holds, we have the stronger property
 \begin{equation}\label{eq:maxprop}
     j = \argmax_{i \in \{1,2,...,n\}} |[{\mathsf{W}}^{-1}\mathsf{P}{\mathbf{e}}_j]_i|.
 \end{equation}

With ${\mathbf{b}} = {\mathsf{A}} {\mathbf{e}}_j$, \eqref{eq:BF2} reads
\begin{itemize}
    \item \textbf{Problem III:}
      \begin{equation} \label{eq:PPmin}
        \min_{{\mathbf{x}}\in{\mathbb{R}^n}} \left\{ \frac{1}{2} \|\mathsf{P}{\mathbf{x}} - \mathsf{P}{\mathbf{e}}_j\|_2^2 + \alpha\|{\mathsf{W}}{\mathbf{x}}\|_1 \right\}.
      \end{equation}
\end{itemize}
\noindent We will now see that the maximum property \eqref{eq:maxprop} allows an analysis of this problem which only involves   classical convex optimization theory.

\begin{theorem}\label{thm:Palpha}
    Assume that the matrix ${\mathsf{A}} \in {\mathbb{R}^{m\times n}}$ satisfies Assumption \ref{nonpar}, and let $\mathsf{P}$ and ${\mathsf{W}}$ be the matrices defined in \eqref{eq:Pmatrix} and \eqref{eq:Wmatrix}, respectively. Then $${\mathbf{x}}^*_\alpha = {\gamma_{j,\alpha}} {\mathbf{e}}_j$$ is the unique solution of Problem III, where
    \begin{equation}\label{eq:gamma}
        {\gamma_{j,\alpha}} = 1 - \frac{\alpha}{[{\mathsf{W}}^{-1}\mathsf{P}{\mathbf{e}}_j]_j} \quad \mbox{for } 0 < \alpha < [{\mathsf{W}}^{-1}\mathsf{P}{\mathbf{e}}_j]_j.
    \end{equation}
\end{theorem}
\begin{proof} $ $ \newline
  \textit{Existence:} Let us define the cost-functional $\mathcal{J}: {\mathbb{R}^n} \rightarrow \mathbb{R}$ associated with \eqref{eq:PPmin},
  \begin{equation} \label{eq:def_J_g_h}
      \mathcal{J}({\mathbf{x}}) = \underbrace{\frac{1}{2}\|\mathsf{P}{\mathbf{x}}-\mathsf{P}{\mathbf{e}}_j\|_2^2}_{=g({\mathbf{x}})}
      + \underbrace{\alpha\|{\mathsf{W}}{\mathbf{x}}\|_1}_{=\alpha h({\mathsf{W}}{\mathbf{x}})},
  \end{equation}
  where $g(\cdot)$ and $h({\mathsf{W}}\cdot)$ represent the fidelity and regularization terms, respectively.
  According to standard convex optimization theory, ${\mathbf{x}}$ is a minimizer of $\mathcal{J}$ if and only if
  \begin{eqnarray*}
      \mathbf{0} &\in& \partial\mathcal{J}({\mathbf{x}}) \\
      &=& \nabla g({\mathbf{x}}) + \alpha{\mathsf{W}}^T\partial h({\mathsf{W}}{\mathbf{x}}),

  \end{eqnarray*}
  where "$\partial$" denotes the subgradient.
  Since ${\mathsf{W}}^T = {\mathsf{W}}$, we can multiply with ${\mathsf{W}}^{-1}$ to obtain
  \begin{equation*}

     -{\mathsf{W}}^{-1}\nabla g({\mathbf{x}}) \in \alpha\partial h({\mathsf{W}}{\mathbf{x}}),
  \end{equation*}
  and from the expression for $g$ we find, keeping in mind that $\mathsf{P}^T \mathsf{P} = \mathsf{P} \mathsf{P} = \mathsf{P}$,
  \begin{equation}\label{eq:gradG}
     {\mathsf{W}}^{-1}\mathsf{P}({\mathbf{e}}_j - {\mathbf{x}}) \in \alpha \partial h({\mathsf{W}}{\mathbf{x}}).
  \end{equation}
  We also observe, using the fact that $h({\mathbf{y}})=\| {\mathbf{y}} \|_1$ and that ${\mathsf{W}}$ is a diagonal matrix with positive entries at its diagonal,
  \begin{equation*}
     [\partial h({\mathsf{W}}{\mathbf{x}})]_i =  [\partial h({\mathsf{W}} [x_1 \, x_2 \, \ldots \, x_n]^T)]_i =
        \begin{cases}
            \{1\}, & x_i > 0, \\
            \{-1\}, & x_i < 0, \\
            [-1,1], & x_i = 0.
        \end{cases}
  \end{equation*}

  We will now investigate whether there exists a scalar $\gamma$ such that ${\mathbf{x}} = \gamma{\mathbf{e}}_j$ satisfies the optimality criterion  \eqref{eq:gradG}. Note that, for $\gamma >0$,
  \begin{equation} \label{eq:BFslow2}
  [\partial h({\mathsf{W}}\gamma {\mathbf{e}}_j)]_i =
        \begin{cases}
            \{1\}, & i = j, \\
            [-1,1], & i \neq j,
        \end{cases}
  \end{equation}

  and the condition \eqref{eq:gradG}, with ${\mathbf{x}} = \gamma{\mathbf{e}}_j$, becomes
  \begin{equation} \label{eq:BFneedsThisToUnderstand}
      (1-\gamma)[{\mathsf{W}}^{-1}\mathsf{P}{\mathbf{e}}_j]_i \in \alpha
        \begin{cases}
            \{1\}, & i = j, \\
            [-1,1], & i \neq j.
        \end{cases}
  \end{equation}
  Setting
  \begin{equation}
  \label{eq:gammaDef}
      \gamma = {\gamma_{j,\alpha}} = 1 - \frac{\alpha}{[{\mathsf{W}}^{-1}\mathsf{P}{\mathbf{e}}_j]_j},
  \end{equation}
  we observe from \eqref{eq:maxprop} that
    \begin{equation}\label{eq:uniquecond}
      (1-{\gamma_{j,\alpha}})[{\mathsf{W}}^{-1}\mathsf{P}{\mathbf{e}}_j]_i = \alpha \, \frac{[{\mathsf{W}}^{-1}\mathsf{P}{\mathbf{e}}_j]_i}{[{\mathsf{W}}^{-1}\mathsf{P}{\mathbf{e}}_j]_j}  \in \alpha
        \begin{cases}
            \{1\}, & i = j, \\
            (-1,1), & i \neq j,
        \end{cases}
  \end{equation}
  and we conclude that \eqref{eq:BFneedsThisToUnderstand} holds for the particular choice \eqref{eq:gammaDef} of $\gamma$.

  This argument shows that ${\mathbf{x}}_\alpha^* = {\gamma_{j,\alpha}}{\mathbf{e}}_j$ is a minimizer of $\mathcal{J}$. The next step is to use the property that $(1-{\gamma_{j,\alpha}})[{\mathsf{W}}^{-1}\mathsf{P}{\mathbf{e}}_j]_i$ is contained in the \underline{open} interval $(-\alpha,\alpha)$, $i \neq j$, to prove the uniqueness.
  \newline \newline \noindent
  \textit{Uniqueness:}
  We have determined a minimizer ${\mathbf{x}}_\alpha^* = {\gamma_{j,\alpha}}{\mathbf{e}}_j$ of $\mathcal{J}$. Let $\mathbf{y} \in {\mathbb{R}^n}, \mathbf{y} \neq {\mathbf{x}}_\alpha^*$, be arbitrary. We will show that
  \begin{equation*}
      \mathcal{J}({\mathbf{y}}) > \mathcal{J}({\mathbf{x}}_\alpha^*).
  \end{equation*}
  If $\mathbf{y} = c{\mathbf{x}}_\alpha^*$, $c \neq 1$, it follows from the analysis presented above that this is not a minimizer of $\mathcal{J}$. Consequently, for the remaining part of the proof we assume that $\mathbf{y} \neq c{\mathbf{x}}_\alpha^*$. In particular, this implies that at least one of the components, say $y_k$, $k \neq j$, of $\mathbf{y}$ is such that $y_k \neq 0$.

 Recall the definition \eqref{eq:def_J_g_h} of $\mathcal{J}$, $g$ and $h$ and that, using the definition of the subgradient,
 \[
 h({\mathsf{W}} {\mathbf{y}}) - h({\mathsf{W}} {\mathbf{x}}_\alpha^*) \geq \mathbf{z}^T({\mathsf{W}}{\mathbf{y}} - {\mathsf{W}}{\mathbf{x}}_\alpha^*) \quad  \mbox{for all }  \mathbf{z} \in \partial h({\mathsf{W}}{\mathbf{x}}_\alpha^*).
 \]
 Therefore
 \begin{align}
 \nonumber
    \mathcal{J}(\mathbf{y}) - \mathcal{J}({\mathbf{x}}_\alpha^*)
    &= g(\mathbf{y}) + \alpha h({\mathsf{W}}\mathbf{y}) - g({\mathbf{x}}_\alpha^*) - \alpha h({\mathsf{W}}{\mathbf{x}}_\alpha^*) \\
    \nonumber
    &\geq\frac{1}{2}\|\mathsf{P}\mathbf{y}-\mathsf{P}{\mathbf{e}}_j\|_2^2 - \frac{1}{2}\|\mathsf{P}{\mathbf{x}}_\alpha^*-\mathsf{P}{\mathbf{e}}_j\|_2^2 \\
    \label{eq:BFslow1}
    &\quad +\alpha \mathbf{z}^T {\mathsf{W}} ({\mathbf{y}} - {\mathbf{x}}_\alpha^*) \quad  \mbox{for all }  \mathbf{z} \in \partial h({\mathsf{W}}{\mathbf{x}}_\alpha^*).
  \end{align}

 Since ${\mathbf{x}}_\alpha^* = {\gamma_{j,\alpha}}{\mathbf{e}}_j$, we can write \eqref{eq:uniquecond} as
    \begin{align}
    \label{eq:BFslow3}
      [{\mathsf{W}}^{-1}\mathsf{P}({\mathbf{e}}_j-{\mathbf{x}}_\alpha^*)]_i &\in \alpha
        \begin{cases}
            \{1\}, & i = j \\
            (-1,1), & i \neq j
        \end{cases} \\
        \nonumber
        &\subset \alpha
        \begin{cases}
            \{1\}, & i = j \\
            [-1,1], & i \neq j
        \end{cases} \\
        \nonumber
        &= \alpha [\partial h({\mathsf{W}}{\mathbf{x}}_\alpha^*)]_i,
  \end{align}
  see \eqref{eq:BFslow2}, i.e.,
  \begin{equation}\label{eq:insubdiff}
      \frac{1}{\alpha}{\mathsf{W}}^{-1}\mathsf{P}({\mathbf{e}}_j - {\mathbf{x}}_\alpha^*) \in \partial h({\mathsf{W}}{\mathbf{x}}_\alpha^*).
  \end{equation}

  Hence, we could choose $\mathbf{z} = \frac{1}{\alpha}{\mathsf{W}}^{-1}\mathsf{P}({\mathbf{e}}_j - {\mathbf{x}}_\alpha^*)$ in \eqref{eq:BFslow1}, but then we do not (directly) get a strict inequality.
  Recall that $\mathbf{y}$ has a component $y_k \neq 0$, where $k \neq j$. Without loss of generality, we may assume that $[{\mathsf{W}}\mathbf{y}-{\mathsf{W}}{\mathbf{x}}_\alpha^*]_k > 0$.

  Define $\mathbf{\tilde{z}}=[\tilde{z}_1 \, \tilde{z}_2 \, \ldots \tilde{z}_n]^T$ as follows\footnote{If $[{\mathsf{W}}\mathbf{y}-{\mathsf{W}}{\mathbf{x}}_\alpha^*]_k < 0$, define $\tilde{z}_k = -1$, etc.}
  \begin{equation*}
      \tilde{z}_i =
      \begin{cases}
            1, & i = k, \\
            \frac{1}{\alpha}[{\mathsf{W}}^{-1}\mathsf{P}({\mathbf{e}}_j-{\mathbf{x}}_\alpha^*)]_i, & i \neq k.
        \end{cases}
  \end{equation*}
Since \eqref{eq:BFslow3} implies that $\left|\frac{1}{\alpha}[{\mathsf{W}}^{-1}\mathsf{P}({\mathbf{e}}_j-{\mathbf{x}}_\alpha^*)]_k\right| < 1$, we find that
    \begin{equation*}
        \mathbf{\tilde{z}}^T [{\mathsf{W}}\mathbf{y}-{\mathsf{W}}{\mathbf{x}}_\alpha^*] > \frac{1}{\alpha}[{\mathsf{W}}^{-1}\mathsf{P}({\mathbf{e}}_j-{\mathbf{x}}_\alpha^*)]^T [{\mathsf{W}}\mathbf{y}-{\mathsf{W}}{\mathbf{x}}_\alpha^*].
    \end{equation*}
Due to \eqref{eq:BFslow3} and \eqref{eq:BFslow2}, $\mathbf{\tilde{z}} \in \partial h({\mathsf{W}}{\mathbf{x}}_\alpha^*)$, and therefore \eqref{eq:BFslow1} yields
 \begin{align*}

    \mathcal{J}(\mathbf{y}) - \mathcal{J}({\mathbf{x}}_\alpha^*)
    &\geq\frac{1}{2}\|\mathsf{P}\mathbf{y}-\mathsf{P}{\mathbf{e}}_j\|_2^2 - \frac{1}{2}\|\mathsf{P}{\mathbf{x}}_\alpha^*-\mathsf{P}{\mathbf{e}}_j\|_2^2 \\

    &\quad +\alpha \mathbf{\tilde{z}}^T {\mathsf{W}} ({\mathbf{y}} - {\mathbf{x}}_\alpha^*) \\
    &> \frac{1}{2}\|\mathsf{P}\mathbf{y}-\mathsf{P}{\mathbf{e}}_j\|_2^2 - \frac{1}{2}\|\mathsf{P}{\mathbf{x}}_\alpha^*-\mathsf{P}{\mathbf{e}}_j\|_2^2 \\
    &\quad + \alpha \frac{1}{\alpha}[{\mathsf{W}}^{-1}\mathsf{P}({\mathbf{e}}_j-{\mathbf{x}}_\alpha^*)]^T [{\mathsf{W}}\mathbf{y}-{\mathsf{W}}{\mathbf{x}}_\alpha^*] \\
    &= \frac{1}{2}\|\mathsf{P}\mathbf{y}-\mathsf{P}{\mathbf{e}}_j\|_2^2 - \frac{1}{2}\|\mathsf{P}{\mathbf{x}}_\alpha^*-\mathsf{P}{\mathbf{e}}_j\|_2^2 \\
    &\quad + [\mathsf{P}({\mathbf{e}}_j-{\mathbf{x}}_\alpha^*)]^T [\mathbf{y}-{\mathbf{x}}_\alpha^*].
  \end{align*}

The gradient of $g$, see \eqref{eq:def_J_g_h}, is $\nabla g(x)=\mathsf{P} ({\mathbf{x}}-{\mathbf{e}}_j)$. Consequently, the convexity of $g$ implies that
\begin{align*}

    \mathcal{J}(\mathbf{y}) - \mathcal{J}({\mathbf{x}}_\alpha^*)
    &> \frac{1}{2}\|\mathsf{P}\mathbf{y}-\mathsf{P}{\mathbf{e}}_j\|_2^2 - \frac{1}{2}\|\mathsf{P}{\mathbf{x}}_\alpha^*-\mathsf{P}{\mathbf{e}}_j\|_2^2 \\
    &\quad - \nabla g({\mathbf{x}}_\alpha^*)^T [\mathbf{y}-{\mathbf{x}}_\alpha^*] \\
    &\geq \frac{1}{2}\|\mathsf{P}\mathbf{y}-\mathsf{P}{\mathbf{e}}_j\|_2^2 - \frac{1}{2}\|\mathsf{P}{\mathbf{x}}_\alpha^*-\mathsf{P}{\mathbf{e}}_j\|_2^2 \\
    &\quad - [g(\mathbf{y})-g({\mathbf{x}}_\alpha^*)] \\
    &=\frac{1}{2}\|\mathsf{P}\mathbf{y}-\mathsf{P}{\mathbf{e}}_j\|_2^2 - \frac{1}{2}\|\mathsf{P}{\mathbf{x}}_\alpha^*-\mathsf{P}{\mathbf{e}}_j\|_2^2 \\
    &\quad - \left[ \frac{1}{2}\|\mathsf{P}\mathbf{y}-\mathsf{P}{\mathbf{e}}_j\|_2^2 - \frac{1}{2}\|\mathsf{P}{\mathbf{x}}_\alpha^*-\mathsf{P}{\mathbf{e}}_j\|_2^2 \right] \\
    &=0,
  \end{align*}
  which finishes the proof.
\end{proof}

This theorem shows that ${\mathbf{x}}_\alpha^* = {\gamma_{j,\alpha}}{\mathbf{e}}_j$ is the unique minimizer of \eqref{eq:PPmin}.
The solution of \eqref{eq:PPmin} is thus obtained by only changing the magnitude of the true source ${\mathbf{e}}_j$, where the scaling factor ${\gamma_{j,\alpha}} \rightarrow 1$ as $\alpha \rightarrow 0$.

\subsubsection{Noisy observation data}
As mentioned in subsection \ref{subsec:altOpt}, it may not be advisable to apply the pseudo-inverse ${\mathsf{A}}^\dagger$ in practical computations. We therefore now want to study \eqref{eq:BF3} in more detail. Setting ${\mathbf{b}}={\mathsf{A}} {\mathbf{e}}_j + \eta$ in \eqref{eq:BF4}, where $\eta \in \mathbb{R}^m$ represents noise, leads to
\[
\min_{{\mathbf{x}}\in{\mathbb{R}^n}} \left\{ \frac{1}{2} \|\mathsf{P}_k {\mathbf{x}} - (\mathsf{A}_k^\dagger {\mathsf{A}}{\mathbf{e}}_j + \mathsf{A}_k^\dagger \eta)\|_2^2 + \alpha\|\mathsf{W}_k {\mathbf{x}}\|_1 \right\}.
\]
Since $\mathsf{A}_k^\dagger {\mathsf{A}} = \mathsf{P}_k$, see \eqref{eq:projectionProperty}, this problem can also can be written in the form
\begin{itemize}
    \item \textbf{Problem IV:}
      \begin{equation} \label{eq:PPminnoise}
        \min_{{\mathbf{x}}\in{\mathbb{R}^n}} \left\{ \frac{1}{2} \|\mathsf{P}_k {\mathbf{x}} - (\mathsf{P}_k {\mathbf{e}}_j + \mathsf{A}_k^\dagger \eta )\|_2^2 + \alpha\|\mathsf{W}_k {\mathbf{x}}\|_1 \right\}.
      \end{equation}
\end{itemize}

\begin{theorem}\label{thm:Pnoise}
    Assume that $\mathsf{A}_k \in {\mathbb{R}^{m\times n}}$ satisfies Assumption \ref{nonpar}, and let $\mathsf{P}_k$ and $\mathsf{W}_k$ be the matrices defined in \eqref{eq:defPK} and \eqref{eq:WKmatrix}, respectively. Then $${\mathbf{x}}^*_{\alpha,\eta} = {\gamma_{j,\alpha,\eta}} {\mathbf{e}}_j$$ is the unique solution of Problem IV, where
    \begin{equation}\label{eq:gammanoise}
        {\gamma_{j,\alpha,\eta}} = 1 - \frac{\alpha + [\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_j}{[\mathsf{W}_k^{-1}\mathsf{P}_k{\mathbf{e}}_j]_j}.
    \end{equation}
    In order for this to hold, $\alpha$ must obey
  \begin{equation}
  \label{ineq:bounds}
      \max_{i \neq j} \frac{1 + |\tau_{ij}|}{1-|\tau_{ij}|} \max_i \left|[\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_i\right| < \alpha < [\mathsf{W}_k^{-1}\mathsf{P}_k{\mathbf{e}}_j]_j - [\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_j,
  \end{equation}
  where $$\tau_{ij} = \frac{[\mathsf{W}_k^{-1}\mathsf{P}_k{\mathbf{e}}_j]_i}{[\mathsf{W}_k^{-1}\mathsf{P}_k{\mathbf{e}}_j]_j} \in \left(-1,1\right), \quad i\neq j.$$
\end{theorem}
\begin{proof}
  Following the same reasoning as in the proof of Theorem \ref{thm:Palpha}, we derive that $x_\alpha^* = \gamma{\mathbf{e}}_j$ is the unique minimizer of  \eqref{eq:PPminnoise} if
  \begin{equation}\label{eq:condnoise}
        (1-\gamma)[\mathsf{W}_k^{-1}\mathsf{P}_k{\mathbf{e}}_j]_i - [\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_i \in \alpha
        \begin{cases}
            \{1\}, & i = j \\
            (-1,1), & i \neq j,
        \end{cases}
  \end{equation}
  where we have used the fact that $\mathsf{P}_k^T=\mathsf{P}_k=\mathsf{A}_k^\dagger \mathsf{A}_k$, and consequently
  \[
  \mathsf{P}_k^T \mathsf{A}_k^\dagger = \mathsf{A}_k^\dagger \mathsf{A}_k \mathsf{A}_k^\dagger = \mathsf{A}_k^\dagger.
  \]

  The criterion \eqref{eq:condnoise} holds for $i = j$ if
  \begin{equation*}
      \gamma = {\gamma_{j,\alpha,\eta}} = 1 - \frac{\alpha + [\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_j}{[\mathsf{W}_k^{-1}\mathsf{P}_k{\mathbf{e}}_j]_j} > 0.
  \end{equation*}
  Consequently, $\alpha$ must satisfy the upper bound
  \begin{equation*}
      \alpha < [\mathsf{W}_k^{-1}\mathsf{P}_k{\mathbf{e}}_j]_j - [\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_j.
  \end{equation*}

  Furthermore, setting $\gamma = {\gamma_{j,\alpha,\eta}}$ in \eqref{eq:condnoise}, the condition \eqref{eq:condnoise} for $i \neq j$ reads
  \begin{equation}
  \label{eq:BFextraStep}
      \left(\alpha + [\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_j\right)\frac{[\mathsf{W}_k^{-1}\mathsf{P}_k{\mathbf{e}}_j]_i}{[\mathsf{W}_k^{-1}\mathsf{P}_k{\mathbf{e}}_j]_j}
      \in \left(-\alpha + [\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_i, \, \alpha + [\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_i\right).
  \end{equation}
  Recall that $\tau_{ij} := \frac{[\mathsf{W}_k^{-1}\mathsf{P}_k{\mathbf{e}}_j]_i}{[\mathsf{W}_k^{-1}\mathsf{P}_k{\mathbf{e}}_j]_j} \in (-1, \,1)$ for $i \neq j$, see\footnote{The property \eqref{eq:maxprop} holds for any matrix ${\mathsf{A}}$ satisfying Assumption \ref{nonpar}, where $\mathsf{P}$ and ${\mathsf{W}}$ are defined in \eqref{eq:Pmatrix} and \eqref{eq:Wmatrix}, respectively. Consequently, \eqref{eq:maxprop} also holds for $\mathsf{A}_k$, replacing ${\mathsf{W}}$ with $\mathsf{W}_k$ and $\mathsf{P}$ with $\mathsf{P}_k$, provided that $\mathsf{A}_k$ also satisfies Assumption \ref{nonpar}.} \eqref{eq:maxprop}, and \eqref{eq:BFextraStep} can thus be expressed as the two inequalities
  \begin{eqnarray}
        \left(1 + \tau_{ij}\right)\alpha &> [\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_i - \tau_{ij}[\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_j, \label{eq:cond1} \\
        \left(1-\tau_{ij}\right)\alpha &> \tau_{ij}[\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_j - [\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_i. \label{eq:cond1b}
  \end{eqnarray}
 It turns out that both of these inequalities are satisfied, for $i \neq j$, if
   \begin{equation}
      \max_{i \neq j} \frac{1 + |\tau_{ij}|}{1-|\tau_{ij}|} \max_i \left|[\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_i\right| < \alpha. \label{eq:cond1_satisfied}
  \end{equation}

 Let us end the proof by verifying that \eqref{eq:cond1} holds if \eqref{eq:cond1_satisfied} is satisfied. Since $(1+\tau_{ij}) > 0$, the requirement \eqref{eq:cond1} can be written in the form
 \begin{eqnarray*}
  \alpha > \frac{[\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_i - \tau_{ij}[\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_j}{1+\tau_{ij}}.
 \end{eqnarray*}
 We derive the following inequalities, considering the case $i \neq j$,
 \begin{eqnarray*}
   \frac{[\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_i - \tau_{ij}[\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_j}{1+\tau_{ij}}
   &\leq&   \frac{\left|[\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_i - \tau_{ij}[\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_j\right|}{\left|1+\tau_{ij}\right|} \\
   &\leq& \frac{\left|[\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_i\right| + |\tau_{ij}|\left|[\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_j\right|}{1-\left|\tau_{ij}\right|} \\
   &\leq&       \max_{i \neq j} \frac{1 + |\tau_{ij}|}{1-|\tau_{ij}|} \max_i \left|[\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_i\right|,
 \end{eqnarray*}
 and we conclude that: If $\alpha$ satisfies \eqref{eq:cond1_satisfied}, then  \eqref{eq:cond1} holds. Similarly, one verifies that \eqref{eq:cond1_satisfied} implies  \eqref{eq:cond1b}.  This finishes the proof.
\end{proof}

Roughly, the left inequality in \eqref{ineq:bounds} ensures that the error amplification caused by the inverse solution procedure does not become too dominate, and the right inequality prevents the regularization term from becoming too "strong" and thereby yielding a poor recovery of the source.

\begin{remark}[\bf{Several sources}]
 Let us mention that the methods introduced in this paper can not, in general, guarantee the recovery of multiple sources. To show this, assume that the exact data is ${\mathbf{b}}^\dagger = {\mathsf{A}}{\mathbf{e}}_m + {\mathsf{A}}{\mathbf{e}}_n$ and that there exist a constant $c$ and an index $j$ such that ${\mathsf{A}}{\mathbf{e}}_m + {\mathsf{A}}{\mathbf{e}}_n = c{\mathsf{A}}{\mathbf{e}}_j$.

 Recall that $\mathsf{P} = {\mathsf{A}}^\dagger{\mathsf{A}}$. Hence, multiplying ${\mathsf{A}}{\mathbf{e}}_m + {\mathsf{A}}{\mathbf{e}}_n = c{\mathsf{A}}{\mathbf{e}}_j$ with ${\mathsf{A}}^\dagger$ yields
 \begin{equation*}
     \mathsf{P}{\mathbf{e}}_m + \mathsf{P}{\mathbf{e}}_n = c\mathsf{P}{\mathbf{e}}_j.
 \end{equation*}
 The weighted basis pursuit problem, cf. Problem II, then reads
  \begin{equation}
  \label{problemSS}
     \min_{{\mathbf{x}}\in{\mathbb{R}^n}} \|{\mathsf{W}}{\mathbf{x}}\|_1 \quad \textnormal{subject to} \quad \mathsf{P}{\mathbf{x}} = \mathsf{P}{\mathbf{e}}_m + \mathsf{P}{\mathbf{e}}_n.
 \end{equation}
 Following the proof of Theorem \ref{thm:exactRecov}, we get, see \eqref{eq:Wmatrix},
 \begin{eqnarray*}
  \|c{\mathsf{W}}{\mathbf{e}}_j\|_1 &=& |c|\|\mathsf{P}{\mathbf{e}}_j\|_2 \\
  &=& |c| \left\|\frac{\mathsf{P}{\mathbf{e}}_m+\mathsf{P}{\mathbf{e}}_n}{c}\right\|_2 \\
  &=& \|\mathsf{P}{\mathbf{e}}_m + \mathsf{P}{\mathbf{e}}_n\|_2 \\
  &<& \|\mathsf{P}{\mathbf{e}}_m\|_2 + \|\mathsf{P}{\mathbf{e}}_n\|_2 \\
  &=& \|{\mathsf{W}}({\mathbf{e}}_m + {\mathbf{e}}_n)\|_1,
 \end{eqnarray*}
 where the strict inequality is a consequence of Assumption \ref{nonpar}, see also \eqref{Pnonpar}.
 This argument shows that the two true sources ${\mathbf{e}}_m$ and ${\mathbf{e}}_n$ will not be recovered by solving \eqref{problemSS}. See Figure \ref{fig:1d} for an illustration.

 On the other hand, if there do not exist an index $j$ and a constant $c$ such that ${\mathsf{A}}{\mathbf{e}}_m+{\mathsf{A}}{\mathbf{e}}_n = c{\mathsf{A}}{\mathbf{e}}_j$, it is an open problem whether the weighted basis pursuit formulation can recover a 2-sparse vector. Or, more generally, can we recover a  $s$-sparse solution if its image under ${\mathsf{A}}$ is not equal to the image under ${\mathsf{A}}$ of any $s'$-sparse solution with $s' < s$?
\end{remark}
\begin{figure}[h]
    \centering
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/counter_ex/true_source.png}
        \caption{True source.}

    \end{subfigure}
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/counter_ex/inverse_source.png}
        \caption{Inverse solution.}

    \end{subfigure}\par
    \caption{Panel (b) shows the solution $f^*$ of \eqref{eq1}-\eqref{eq2} when $\Omega = (0,1), \epsilon = 1$ and $\alpha = 0.001$. The observation data $d$ was in this case generated by the two true local sources depicted in Panel (a).}
    \label{fig:1d}
\end{figure}

\begin{comment}
\begin{remark}
\textcolor{red}{Fjerne denne kommentaren?}
Note that if the data is generated by a scaling of a basis vector ${\mathbf{e}}_j$, i.e. ${\mathbf{b}}^\dagger = {\mathsf{A}}(c{\mathbf{e}}_j)$, then all the results above will hold using $\hat{{\mathsf{A}}} = c{\mathsf{A}}$.
\end{remark}

\section{Counterexample for two sources}
In Section \ref{sec:analysis}, we proved that exact recovery of a source consisting of a single basis vector is always possible provided Assumption \ref{nonpar} holds. We will now illustrate that for multiple sources, there is no general guarantee for exact recovery within this framework.

Let us start by considering a uniform grid on the interval $(-1,1)$ using $2N - 1$ cells. Furthermore, assume $\phi_i$ is the characteristic function of cell $i$, where the cells are numbered from left to right. Define $F_h = \textnormal{Span}(\{\phi_i\}_{i=1}^{2N-1})$.
Now, let us consider the problem
\begin{equation*}
    \min_{f \in F_h, u \in H^1((-1,1))} \frac{1}{2} [u^2(1) + u^2(-1)] + \int_{-1}^1 {\mathsf{W}} f,
\end{equation*}
subject to
\begin{eqnarray*}
    - u'' + u &=& f, \quad \textnormal{in} \quad  (-1,1), \nonumber \\
    u'(-1) = u'(1) &=& 0.
\end{eqnarray*}
Due to symmetry, the solutions of
\begin{equation*}
 -u_1 '' + u_1 = \phi_n \quad \textnormal{and} \quad -u_2'' + u_2 = \phi_{2N-n},
\end{equation*}
both with homogeneous Neumann boundary conditions, are such that $u_2(t) = u_1(-t)$. Consequently, if we define $\hat{u} := u_1 + u_2$, we obtain
\begin{equation*}
    \hat{u}(1) := \hat{u}(-1).
\end{equation*}
For a proper choice of c, the solution $\tilde{u}$ of
\begin{equation*}
 -u'' + u = c\phi_N
\end{equation*}
with homogeneous Neumann boundary conditions, will also be such that $\hat{u}(1) = \tilde{u}(1) = \hat{u}(-1) = \tilde{u}(-1)$. Hence, it is impossible to separate the two sources $\phi_n$ and $\phi_{2N-n}$ from the single source $\phi_N$ given Dirichlet measurements on the boundary. From the analysis in Section \ref{sec:analysis}, we can conclude that it is the single source $\phi_N$ that would be recovered in the inverse formulation. A numerical visualization of the inverse problem above is shown in Figure \ref{fig:1d}.
\end{comment}

\section{Numerical experiments}
\label{sec:numerical_experiments}
We committed the so-called "inverse crime" in Example 1 below: The same grid was used to both generate the boundary observation data $d$ in \eqref{eq1} and for solving the inverse problem.
In all the other experiments, the data $d$ was generated using a finer grid for the state $u$ than was used in the inverse computations. More specifically, $h_\textnormal{forward} = 0.5h_\textnormal{inverse}$, and we performed experiments on the unit square with $129\times129$ and $65\times65$ nodes for the forward and inverse computations of the state $u$, respectively. The source $f$ was discretized in terms of a $16\times16$ mesh in all the simulations presented in examples 1-2 and 4. In Example 3, however, the true source was discretized using a finer $129\times129$ grid. Note that we employed the basis functions
\begin{equation*}
{\mathbf{\phi}}_i = \frac{1}{\| \mathcal{X}_{\Omega_i}  \|_{L^2(\Omega)}} \mathcal{X}_{\Omega_i},
\quad i=1,2, \ldots, n,
\end{equation*}
for $F_h$, where $\Omega_1, \, \Omega_2, \ldots, \Omega_n$ are uniformly sized disjoint grid cells and $\mathcal{X}_{\Omega_i}$ denotes the characteristic function of $\Omega_i$.

We employed the FEniCS software to generate the matrices involved in our experiments. Thereafter, the matrices were exported to MATLAB, where the optimization problems were solved with the split-Bregman algorithm \cite{goldstein09}. \textcolor{black}{Some details about the forward/transfer matrix ${\mathsf{A}}$ is presented in appendix \ref{app:discretization}}. We do not present a detailed description of the well-known mappings between the finite element spaces arising from the discretization of \eqref{eq1}-\eqref{eq2} to the Euclidean spaces used in \eqref{eq:BF1}, \eqref{eq:BF2} and \eqref{eq:BF4}. Note, however, that ${\mathbf{e}}_j \in {\mathbb{R}^n}$ is associated with the FE basis function $\phi_j \in F_h \subset L^2(\Omega)$.

In all the simulations $\epsilon = 1$, see \eqref{eq2}, and no noise was added to the data $d$, except in Example 2.

Figure \ref{fig:weights} contains visualizations of the entries of the weight matrix $\mathsf{W}_k$ defined in \eqref{eq:WKmatrix}. More specifically, each panel shows a plot of $\|\mathsf{P}_k{\mathbf{e}}_i\|, \, i=1,2,\ldots,n$. We observe that the weights are largest for indexes associated with basis functions positioned close to the boundary of the domain $\Omega$.

\begin{figure}[h]
    \centering
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/weights/weightsK7.png}
        \caption{$k = 7$.}

    \end{subfigure}
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/weights/weightsK70.png}
        \caption{$k = 70$.}

    \end{subfigure}\par
    \caption{Visualizations of the matrix $\mathsf{W}_k$. Panels (a) and (b) show plots of $\|\mathsf{P}_k{\mathbf{e}}_i\|, \, i=1,2,\ldots,n$, see \eqref{eq:WKmatrix}, for two different choices of the truncation parameter $k$.}
    \label{fig:weights}
\end{figure}

\subsection{Example 1: Exact recovery of a single source}
Figures \ref{fig:ex1_1} and \ref{fig:ex1_2} show numerical solutions of \eqref{eq:BF2}. In these problems we recover a single source and ${\mathbf{b}} = {\mathsf{A}} {\mathbf{e}}_j$. The theory developed for Problem III is therefore applicable, see Theorem \ref{thm:Palpha}.
Figure \ref{fig:ex1_comp}  contains a comparison of the size of ${\gamma_{j,\alpha}}$, cf. \eqref{eq:gamma}, and the maximum value, $\max_i [{\mathbf{x}}_\alpha^*]_i$, of the solution ${\mathbf{x}}_\alpha^*$ of Problem III.
 We observe that the outcome of these experiments is as one could have anticipated from Theorem \ref{thm:Palpha}.

\begin{figure}[h]
    \centering
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex1/true_interior.png}
        \caption{True source.}

    \end{subfigure}
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex1/inverse_interior.png}
        \caption{Inverse solution.}

    \end{subfigure}\par
    \caption{Comparison of a true \textit{interior} source and the inverse solution computed by solving \eqref{eq:BF2}, Example 1. The size of the regularization parameter was $\alpha = 10^{-4}$.}
    \label{fig:ex1_1}
\end{figure}

\begin{figure}[h]
    \centering
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex1/true_boundary.png}
        \caption{True source.}

    \end{subfigure}
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex1/inverse_boundary.png}
        \caption{Inverse solution.}

    \end{subfigure}\par
    \caption{Comparison of a true source located at the \textit{boundary} and the inverse solution computed by solving \eqref{eq:BF2}, Example 1. The size of the regularization parameter was $\alpha = 10^{-3}.$}
    \label{fig:ex1_2}
\end{figure}

\begin{figure}[h]
    \centering
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex1/gamma_interior.png}
        \caption{Interior source.}

    \end{subfigure}
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex1/gamma_boundary.png}
        \caption{Source at the boundary.}

    \end{subfigure}\par
    \caption{Example 1. The red curve shows the size of ${\gamma_{j,\alpha}}$, see \eqref{eq:gamma}, as a function of the regularization parameter $\alpha$, and the asterisks represent $\max_i [{\mathbf{x}}_\alpha^*]_i$, where ${\mathbf{x}}_\alpha^*$ is the solution of \eqref{eq:BF2}.}
    \label{fig:ex1_comp}
\end{figure}

\subsection{Example 2: Noise}
If the observation data contains noise, it is natural to solve \eqref{eq:BF4}. Throughout this example, $k = 7$ in the truncated SVD employed to obtain the approximation $\mathsf{A}_k$ of ${\mathsf{A}}$, see subsection \ref{subsec:altOpt}. The experiment was executed as follows:

\begin{enumerate}
    \item Generate the data
       \begin{equation*}
           {\mathbf{b}} = {\mathbf{b}}^\dagger + \eta = {\mathsf{A}}{\mathbf{e}}_j + \eta, \quad \eta = \delta\rho,
       \end{equation*}
       where $\delta$ is a scalar and $\rho$ is a vector containing normally distributed numbers with zero mean and standard deviation equal to $1$. See \cite[Example 6]{Elv20} for a thorough discussion of the noise.
    \item
        Compute ${\mathbf{x}}_k^* = \mathsf{A}_k^\dagger{\mathbf{b}} = \mathsf{A}_k^\dagger({\mathsf{A}}{\mathbf{e}}_j + \eta)$, see \eqref{eq:BF4}.
    \item
        Set
          \begin{equation*}
              \bar{\alpha} = \max_{i \neq j} \frac{1 + |\tau_{ij}|}{1-|\tau_{ij}|} \max_i \left|[\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_i\right|,
          \end{equation*}
        cf. Theorem \ref{thm:Pnoise}.
    \item Compute, see \eqref{eq:BF4},
        \begin{equation*}
            {\mathbf{x}}_\alpha^* = \argmin_{{\mathbf{x}}\in{\mathbb{R}^n}} \left\{ \frac{1}{2}\|\mathsf{P}_k{\mathbf{x}} - {\mathbf{x}}_k^*\|_2^2 + \alpha\|\mathsf{W}_k{\mathbf{x}}\|_1 \right\}
        \end{equation*}
    for both $\alpha = 0.3\bar{\alpha}$ and $\alpha = 3\bar{\alpha}$.
\end{enumerate}
This "setup" is such that the theory, presented in Theorem \ref{thm:Pnoise}, for Problem IV is applicable.

Note that the problem is regularized with both standard truncated SVD (i.e., the choice of the truncation parameter $k$), and $\ell^1$-regularization (i.e., the choice of $\alpha$). How to optimally choose these parameters in relation to each other is a complicated matter and left for future research.

\begin{figure}[H]
    \centering
    \includegraphics[width=0.45\linewidth]{images/ex2/5pct/true.png}
    \caption{Example 2, true source.}\label{fig:ex2true}
\end{figure}

Figure \ref{fig:ex2a} compares over-regularized ($\alpha = 3\bar{\alpha}$) and under-regularized ($\alpha = 0.3\bar{\alpha}$) solutions of \eqref{eq:BF4} with observation data containing 5\% noise. Similar comparisons are presented in figures \ref{fig:ex2b} and \ref{fig:ex2c} for 10\% and 15\% noise, respectively. The true source is displayed in Figure \ref{fig:ex2true}.

When $\alpha = 3\bar{\alpha}$, the plots of ${\mathbf{x}}_\alpha^*$, displayed in panels \ref{fig:ex2a}a), \ref{fig:ex2b}a) and \ref{fig:ex2c}a), show that the true source is successfully recovered in all three cases, albeit with an underestimated magnitude.

We can, however, as a post-processing step, improve the magnitude of the solutions using \eqref{eq:gammanoise}: Assuming $[\mathsf{W}_k^{-1}\mathsf{A}_k^\dagger\eta]_j << [{\mathsf{W}}^{-1}\mathsf{P}_k{\mathbf{e}}_j]_j$, we compute
\begin{equation*}
    {\mathbf{x}}_{\alpha, \textnormal{SCALED}}^* = \frac{{\mathbf{x}}_\alpha^*}{1-\frac{\alpha}{[\mathsf{W}_k^{-1}\mathsf{P}_k{\mathbf{e}}_j]_j}},
\end{equation*}
where the index $j$ in the denominator is known since $\argmax_i {\mathbf{x}}_\alpha^* = j$, see Theorem \ref{thm:Pnoise}. By re-scaling the solution displayed in panel (a) of figures \ref{fig:ex2a}-\ref{fig:ex2c}, the magnitude of the rescaled source becomes 0.88, 1.0 and 0.93, respectively. This post-processing step can only be mathematically justified if the assumptions needed in Theorem \ref{thm:Pnoise} hold, but it can "always" be applied in practice. (We have not explored its success when the assumptions in Theorem \ref{thm:Pnoise} are violated.)

When the problem is under-regularized, i.e., $\alpha = 0.3\bar{\alpha}$, the inverse solutions are still good visual approximations of the true source, see figures \ref{fig:ex2a}b), \ref{fig:ex2b}b) and \ref{fig:ex2c}b). However, the inverse solutions also contain small contributions from other basis vectors than ${\mathbf{e}}_j$. Both the magnitude and the number of incorrect active basis vectors appear to increase as the noise level increases.

\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex2/5pct/inv_3.png}
        \caption{Inverse solution, $\alpha = 3\bar{\alpha}$.}

    \end{subfigure}
        \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex2/5pct/inv_03.png}
        \caption{Inverse solution, $\alpha = 0.3\bar{\alpha}$.}

    \end{subfigure}\par
    \caption{Example 2, 5\% noise, $\bar{\alpha} = 0.0031$.}
    \label{fig:ex2a}
\end{figure}

\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex2/10pct/inv_3.png}
        \caption{Inverse solution, $\alpha = 3\bar{\alpha}$.}

    \end{subfigure}
        \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex2/10pct/inv_03.png}
        \caption{Inverse solution, $\alpha = 0.3\bar{\alpha}$.}

    \end{subfigure}\par
    \caption{Example 2, 10\% noise, $\bar{\alpha} = 0.0067$.}
    \label{fig:ex2b}
\end{figure}

\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex2/15pct/inv_3.png}
        \caption{Inverse solution, $\alpha = 3\bar{\alpha}$.}

    \end{subfigure}
        \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex2/15pct/inv_03.png}
        \caption{Inverse solution, $\alpha = 0.3\bar{\alpha}$.}

    \end{subfigure}\par
    \caption{Example 2, 15\% noise, $\bar{\alpha} = 0.0122$.}
    \label{fig:ex2c}
\end{figure}

In most applications, $\bar{\alpha}$ is not available because it requires full knowledge of the noise $\eta$. We therefore performed a numerical study under the common assumption that only (an estimate of) $\|\eta\|$ is known. This allows the use of Morozov's discrepancy principle for choosing the truncation parameter $k$. Figure \ref{fig:ex2many} compares solutions of \eqref{eq:BF4} for different choices of $k$ and $\alpha$, where the choice $k = 5$ is obtained by applying Morozov's discrepancy principle with the threshold $1.05\|\eta\|$. The values $k = 3$ and $k = 15$ are chosen simply to compare the choice $k = 5$ with a smaller and larger truncation parameter.

In this particular example, we observe that using a relatively strong regularization in the truncated SVD step, i.e., choosing $k = 3$, gives good reconstruction of the source for all the tested values of $\alpha$. If the regularization by truncated SVD is reduced, i.e., when $k$ increases, it appears that $\alpha$ must be chosen more carefully to obtain a good reconstruction.

\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.32\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex2_many/k3a100.png}
        \caption{$k=3$, $\alpha = 10^{-2}$.}

    \end{subfigure}
    \begin{subfigure}[b]{0.32\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex2_many/k5a100.png}
        \caption{$k=5$, $\alpha = 10^{-2}$.}

    \end{subfigure}
    \begin{subfigure}[b]{0.32\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex2_many/k15a100.png}
        \caption{$k=15$, $\alpha = 10^{-2}$.}

    \end{subfigure}\par
    \begin{subfigure}[b]{0.32\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex2_many/k3a1000.png}
        \caption{$k=3$, $\alpha = 10^{-3}$.}

    \end{subfigure}
    \begin{subfigure}[b]{0.32\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex2_many/k5a1000.png}
        \caption{$k=5$, $\alpha = 10^{-3}$.}

    \end{subfigure}
    \begin{subfigure}[b]{0.32\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex2_many/k15a1000.png}
        \caption{$k=15$, $\alpha = 10^{-3}$.}

    \end{subfigure}\par
    \begin{subfigure}[b]{0.32\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex2_many/k3a10000.png}
        \caption{$k=3$, $\alpha = 10^{-4}$.}

    \end{subfigure}
    \begin{subfigure}[b]{0.32\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex2_many/k15a10000.png}
        \caption{$k=5$, $\alpha = 10^{-4}$.}

    \end{subfigure}
    \begin{subfigure}[b]{0.32\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex2_many/k15a10000.png}
        \caption{$k=15$, $\alpha = 10^{-4}$.}

    \end{subfigure}\par
    \caption{Example 2, 10\% noise. Comparison of inverse solutions computed with different choices of the regularization parameters $k$ and $\alpha$. The choice $k=5$ was the outcome of using Morozov's discrepancy principle with the threshold $1.05\|\eta\|$. Figure \ref{fig:ex2true} shows the true source.}
    \label{fig:ex2many}
\end{figure}

\subsection{Example 3: Large circular source}
\label{subsec:largeCircularSource}
So far we have considered examples covered by our analysis. We will now depart from this and explore more involved cases: The true source depicted in Figure \ref{fig:ex3true}(a) does not belong to the finite element space associated with the coarse mesh used to represent the source in the inverse computations.

We first note that Figure \ref{fig:ex3true}(b) shows that classical Tikhonov regularization
\begin{equation}\label{eq3tikh}
\min_{{\mathbf{x}}\in{\mathbb{R}^n}} \left\{ \frac{1}{2}\|{\mathsf{A}}{\mathbf{x}} - {\mathbf{b}}\|_2^2 + \zeta \|{\mathbf{x}}\|_2^2 \right\},
\end{equation}
with $\zeta=10^{-4}$, fails to yield an adequate solution to this problem: Compare panels (a) and (b) of Figure \ref{fig:ex3true}. The mathematical explanation for this is presented in \cite{Elv20}.

Next, we observe in Figure \ref{fig:ex3}, left column, that the sparsity structure of the inverse solution computed by solving \eqref{eq:BF4} deteriorates as the $\ell^1$-regularization parameter $\alpha$ tends to zero. However, for all three values of $\alpha$, the position of the source is quite well recovered. In these simulations, the truncated SVD employed to obtain the approximation ${\mathsf{A}}_k^\dagger$ of the pseudo inverse ${\mathsf{A}}^\dagger$ in \eqref{eq:BF4}, was obtained by choosing the truncation parameter $k=5$.

As an alternative to the truncated SVD, we also used standard Tikhonov regularization to obtain an approximation of ${\mathsf{A}}^\dagger$, see subsection \ref{subsec:altOpt}. More precisely, employing the singular value decomposition ${\mathsf{A}} = \mathsf{U}\Sigma\mathsf{V}^T$ of ${\mathsf{A}}$, the solution of the minimization problem
\begin{equation*}
    \min_{{\hat{\mathbf{x}}}\in{\mathbb{R}^n}} \left\{ \frac{1}{2}\|{\mathsf{A}}{\hat{\mathbf{x}}}-{\mathbf{b}}\|_2^2 + \frac{1}{2}{\beta}\|{\hat{\mathbf{x}}}\|_2^2 \right\},
\end{equation*}
can be expressed as
\begin{equation*}
  {\hat{\mathbf{x}}}_{\beta} = \mathsf{V}(\Sigma^2 + {\beta}\mathsf{I})^{-1}\mathsf{V}^T {\mathsf{A}}^T {\mathbf{b}}
  = {\mathsf{S}_{\beta}}{\mathbf{b}},
\end{equation*}
where
\begin{equation}\label{eq:Sa}
{\mathsf{S}_{\beta}} = \mathsf{V}(\Sigma^2 + {\beta}\mathsf{I})^{-1}\mathsf{V}^T {\mathsf{A}}^T \approx {\mathsf{A}}^\dagger.
\end{equation}
Replacing ${\mathsf{A}}^\dagger$ in \eqref{eq:BF2} with ${\mathsf{S}_{\beta}}$, keeping in mind that $\mathsf{P} = {\mathsf{A}}^\dagger{\mathsf{A}}$, leads to the following alternative to \eqref{eq:BF4}
      \begin{equation} \label{eq:OLE5}
        \min_{{\mathbf{x}}\in{\mathbb{R}^n}} \left\{ \frac{1}{2}\|{\mathsf{S}_{\beta}}{\mathsf{A}}{\mathbf{x}} - {\mathsf{S}_{\beta}} {\mathbf{b}}\|_2^2 + \alpha\|{\mathsf{W}}{\mathbf{x}}\|_1 \right\}.
      \end{equation}
\begin{comment}
\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.5\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex2/5pct/inv_3_scaled.png}
        \caption{Re-scaling of the solution depicted in Figure \ref{fig:ex2a}a).}

    \end{subfigure}\par
    \begin{subfigure}[b]{0.5\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex2/10pct/inv_3_scaled.png}
        \caption{Re-scaling of the solution depicted in Figure \ref{fig:ex2b}a).}

    \end{subfigure}\par
    \begin{subfigure}[b]{0.5\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex2/15pct/inv_3_scaled.png}
        \caption{Re-scaling of the solution depicted in Figure \ref{fig:ex2c}a).}

    \end{subfigure}
    \caption{Example 2, re-scaled versions ${\mathbf{x}}_{\alpha,\textnormal{SCALED}}^*$ of the inverse solutions ${\mathbf{x}}_\alpha^*$ for $\alpha = 3\bar{\alpha}$.}
    \label{fig:ex2Scaled}
\end{figure}
\end{comment}

\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex5/true_sourceR.png}
        \caption{True source.}

    \end{subfigure}
        \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex3/tikh.png}
        \caption{Solution of \eqref{eq3tikh} with $\zeta = 10^{-4}$.}

    \end{subfigure}\par
    \caption{Example 3. Panel (a) shows the true source, and panel (b) displays the numerical solution of \eqref{eq1}-\eqref{eq2} using Tikhonov regularization $\zeta \| f \|_{L^2(\Omega)}^2$ instead of the weighted $\ell^1$-regularization $\alpha \sum_i w_i |(f, \phi_i)_{L^2(\Omega)}|$.}
    \label{fig:ex3true}
\end{figure}

The right column in Figure \ref{fig:ex3} contains the results obtained by solving \eqref{eq:OLE5}.
We observe, particularly for the two largest values of the $\ell^1$-regularization parameter $\alpha$, that truncated SVD and Tikhonov regularization yield visually rather similar results.

\subsection{Example 4: Multiple sources}
Our last examples concern several sources. To solve the problems, we did the following:
\begin{enumerate}
    \item For $N$ true sources, we computed
       \begin{equation*}
           {\mathbf{b}}^\dagger = {\mathsf{A}}\left(\sum_{i=1}^N {\mathbf{e}}_{q_i}\right).
       \end{equation*}
    \item Thereafter, we solved the problem
        \begin{equation*}
            \min_{{\mathbf{x}}\in{\mathbb{R}^n}} \left\{ \frac{1}{2}\|{\mathsf{S}_{\beta}}{\mathsf{A}}{\mathbf{x}} - {\mathsf{S}_{\beta}}{\mathbf{b}}^\dagger\|_2^2 + \alpha\|{\mathsf{W}}{\mathbf{x}}\|_1 \right\},
        \end{equation*}
        where ${\mathsf{S}_{\beta}}$ is defined in \eqref{eq:Sa}.
\end{enumerate}

Figure \ref{fig:ex4} contains results obtained with 2, 4 and 8 true sources. Panel b) shows that the two sources are nearly perfectly recovered. For the case with 4 sources, the inverse solution recovers three of the sources almost perfectly, while the fourth is 'split' into two adjacent sources with less magnitude, cf. Panel d). Panel f) shows the inverse solution for the case with 8 true sources, where we observe that the three sources located at the boundary are recovered very well (note the color bar), whereas the interior sources are merged somewhat into two clusters in the inverse solution.

\section{Conclusions}
If the exact data is generated from a single basis vector ${\mathbf{e}}_j$, our weighted $\ell^1$-regularization technique is able to exactly recover the true solution. When noise is present, we have obtained estimates for the size of the regularization parameter $\alpha$ which yield an inverse solution in the form ${\gamma_{j,\alpha,\eta}}{\mathbf{e}}_j$, where ${\gamma_{j,\alpha,\eta}}$ is a positive scalar. Numerical experiments suggest that our method also can, in many cases, identify several local sources, but we do not have a thorough mathematical understanding of this. We only know with certainty that it is possible to construct scenarios for which our scheme will fail to recover two sources.

Concerning the practical use of our weighted $\ell^1$-regularization method, it seems reasonable to expect that the method can recover two or three well-separated sources. Nevertheless, the definition of "well-separated" is problem dependent since it must depend on the smoothing properties of the involved forward operator and the geometry of the solution domain $\Omega$. This means that one should, for each concrete application, perform a simulation study to explore which source patterns that can be identified by the weighted $\ell^1$-regularization method.

We defined our regularization operator and presented our analysis in terms of Euclidean spaces. Consequently, the methodology can be applied whenever a discrete version of a source identification task can be formulated in terms of a transfer matrix with a significant null space. For example, the use is not restricted to PDE-constrained optimization problems with elliptic state equations, but can also be applied when the state equation is parabolic or hyperbolic.

This study was motivated by inverse problems arising in connection with EEG and ECG recordings. In principle, our scheme can be applied to these problems, but a number of challenging engineering issues must be handled: one must construct suitable geometrical models (using, e.g., MR images), obtain EEC or ECG recordings, handle noisy data, etc. We intend to explore the EEG and ECG applications in forthcoming investigations.

\textcolor{black}{In this paper we search for a source in a finite dimensional space. From a pure mathematical perspective, an interesting problem would be to develop an analogous theory using an infinite dimensional source space. The authors believe that this {\em might} be possible following the approach presented in \cite{casas2012approximation}.}

\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex5/inverse_sourceAlpha001k.png}
        \caption{Truncated SVD. $\alpha = 10^{-2}$.}

    \end{subfigure}
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex5/inverse_sourceAlpha001.png}
        \caption{Tikhonov. $\alpha = 10^{-2}$.}

    \end{subfigure}\par
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex5/inverse_sourceAlpha00001k.png}
        \caption{Truncated SVD. $\alpha = 10^{-4}$.}

    \end{subfigure}
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex5/inverse_sourceAlpha00001.png}
        \caption{Tikhonov. $\alpha = 10^{-4}$.}

    \end{subfigure}\par
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex5/inverse_sourceAlpha0000001k.png}
        \caption{Truncated SVD. $\alpha = 10^{-6}$.}

    \end{subfigure}
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex5/inverse_sourceAlpha0000001.png}
        \caption{Tikhonov. $\alpha = 10^{-6}$.}

    \end{subfigure}\par
    \caption{Example 3, employing weighted $\ell^1$-regularization $\alpha \| {\mathsf{W}} {\mathbf{x}} \|_1$. The inverse solutions were computed by solving \eqref{eq:BF4} (approximating ${\mathsf{A}}^\dagger$ with truncated SVD, $k = 5$) and \eqref{eq:OLE5} (approximating ${\mathsf{A}}^\dagger$ with Tikhonov regularization, $\beta = 10^{-6}$). The true source is displayed in Figure \ref{fig:ex3true}(a).}
    \label{fig:ex3}
\end{figure}

\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex3/true_twosource.png}
        \caption{True sources.}

    \end{subfigure}
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex3/inverse_twosource.png}
        \caption{Inverse solution.}

    \end{subfigure}\par
        \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex3/true_foursource.png}
        \caption{True sources.}

    \end{subfigure}
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex3/inverse_foursource.png}
        \caption{Inverse solution.}

    \end{subfigure}\par
        \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex3/true_eightsource.png}
        \caption{True sources.}

    \end{subfigure}
    \begin{subfigure}[b]{0.45\linewidth}
        \centering
        \includegraphics[width=\linewidth]{images/ex3/inverse_eightsource.png}
        \caption{Inverse solution.}

    \end{subfigure}\par
    \caption{Example 4, recovering several local sources. Here, $\alpha = 0.01$ and ${\beta} = 10^{-6}$ in all the three experiments.}
    \label{fig:ex4}
\end{figure}

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Another camera has found it's way into my bag. After much debate I decided to sell my Canon 5D and grab a Canon 1D Mark III. I have been very happy with my 5D but after I got a 5D Mark II in March of 2009 the 5D took a back seat and has mainly been a back up camera. I got the chance to grab a used 1D Mark III, which had both AF fixes done and was indistinguishable from a new camera. For some time I had been eying out the release of Canon's 1D Mark IV but disappointingly it comes in at $5000.00 US and as a 1.3x crop camera. The 1D III currently goes for about half of this and although the 1D III also is a 1.3x crop and doesn't shoot video it still gives shooters many of the benefits of a 1D series camera.
I think the 1D III will work better for me as a compliment to my 5D II. Better than the 5D due to the higher fps. and AF tracking. I have now had some time to give it a test and have shot volleyball, hockey, theatre, portraits in my studio and also an engagement sessions.
If you have tried a 1 series camera before you will know that these beasts just feel different. They are built to withstand some abuse and long term use in a variety of climates. We are talking about better weather sealing, which may come in handy for me while shooting ski races. With that said I can't say that I have ever had trouble with any of my Canon bodies in the past. I do live in a dry climate and even when it snows you can just brush the snow off the camera. The 1D III however does feel very solid and is a pleasure to use. It is very responsive from the get go and shooting 10 frames per second inevitably leads to more keepers while shooting sports or wild life.
My findings on the auto focus tracking have not disappointed, nor have they exceeded my expectations. Much have been written about the AF issues of the 1D III. Even I have expressed my concerns prior to making a jump for one. The first thing you discover is the fact that the camera has quite a few custom settings and so far I find that it really pays off to adjust and experiment with these until you are comfortable and find a usable setting for what you are shooting. The ability to pick different and better individual focusing points is a real asset and I find myself doing this more and more as the outer points are more reliable than on the 5D cameras. Shooting hockey and volleyball I got more keepers than when I shoot with the 5D cameras.
[As for image quality even I had concerns shooting with a 10.1 mega pixel camera versus the 5D II with 21.1. The trick is not to compare the files but find how the two cameras can compliment each other. For cropping you cannot beat the 5D II but if you can fill the frame the AF tracking and frames per second makes up for it on the 1D III and you should end up with shots you otherwise wouldn't have been able to capture. I have noticed, and this did not come as a surprise to me, that the portraits taken in my studio has a wonderful pop and pleasant skin tones using the 1D III. For landscapes or wide angle shots in general there is little doubt that the the 5D II creates nicer images. Again it comes down to using these two camera where best applied.
(Update, July 12th, 2010) I have now used the 1D III extensively for cross country skiing, soccer, weddings, grads and wild life. I have updated the photos above and mainly posted sports photos. I do first and foremost think of the 1D III as a sports camera and paired with my 300 2.8L IS it is a stellar performer and I really enjoy the look of the files. AF is spot on and using the various focusing points increases the usability. For weddings and grads I use it when autofocus becomes crucial and it does not disappoint. For wild life I only use it for the most demanding tracking. For birds for instance I have had just as much luck if not more luck using my 5D II. This in in large due to the fact that I have more pixels on the 5D II and this works wonders when you need to do heavy cropping. For wildlife I have also had access to a 7D a couple of times. The 7D gives you even better reach and more PPB (pixels per bird).


Roger Joslin arrived in Bentonville, Arkansas, in 2006 as a newly ordained deacon to become the vicar of All Saints Episcopal Church, a mission of the Episcopal Diocese of Arkansas. All Saints has grown to be the fifth-largest congregation in the Diocese of Arkansas, with an average Sunday attendance of over 230, and is well on its way to parish status. All Saints' location in Bentonville, the home of Walmart, adds another layer to the story. Joslin makes a powerful case for the role of a progressive Episcopal church in a conservative culture and for the appeal of preaching and practicing a gospel of inclusion that reaches out to groups or individuals who have been marginalized by churches.
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