Earlier this year I collaborated in an extraordinary research project called The Emotional Space and The Puppet. Here is a short article about the work written by Joy Haynes, organiser of the project.
Why tell a story using puppets? For me, they provide a truly different perspective – a puppet evokes emotion yet does not have memory or  feelings of its  own. A puppet is entirely present and it is through action that an audience  empathises  with the character.
Earlier this year, I organised a five day research project in which a team of theatre professionals worked with master puppeteer and director Luca Ronga, whose background and passion is the simplicity and directness of glove puppetry. His unique approach is to create emotional and physical spaces in an audience's mind's eye through physical gesture.
In an extraordinary process, we learned not to make the glove puppet mimic human actions or to charge them with enormous emotions but rather to translate the  grand  scale of the human experience into the poetic language of the glove puppet, which is delicate and playful.
I discovered the power of starting from one clear point of reference, in which a single movement/gesture is expanded and through  its  repetition is taken away from the personal and transformed into something different and universal.  It was fascinating to see how character and situations were generated from single words, such as ' family', 'terror', restlessness', 'solitude'.
With the absence of the traditional glove puppet booth, I observed the energy and focus of the  'actor'  presence becoming the emotional counterpoint or providing a context/playboard and, most interestingly, playfully embodying landscapes and architecture.
Looking back on the research, I  realise  what a wonderful and instructive week this had been in developing my personal approach to puppet theatre and the devising process.
If you'd like to have your expectations of puppetry challenged and your theatre-making skills developed through exploration of physical space as a means to translate inner emotional landscapes, Luca Ronga will be running a masterclass entitled The 'Emotional Space' at Norwich Puppet Theatre on the 15th & 16th April. Booking on Tel: 01603 629921. Hope to see you there.


Hydrocortisone is a prescription drug. It is, however, the identical hormone to one of the naturally occurring compounds secreted by the adrenal gland cortex (outside layer) in response to stress or low blood sugar. The correct and safe use of low dose cortisol requires a doctor familiar with its administration.
Physiologic replacement doses of oral cortisol can make a dramatic difference in cases of adrenal exhaustion. Because of side-effects induced by larger doses, many doctors are reluctant to use it and many patients are thus deprived of a valuable and needed therapy. Physiologic doses of cortisol (5-25mg per day) are safe. Lab testing can confirm the diagnosis of mild adrenocortical deficiency. A doctor experienced with cortisol use should be consulted; a typical prescription is 5mg four times per day, with an emphasis on early morning use if later doses keep you up at night, or if fewer doses are taken.
Never apply a topical steroid like cortisone to rosacea unless directed to do so by your doctor for a specific reason. Cortisone treatments can worsen rosacea over the long term and make it even more resistant to treatment.


Looking For Progressive Reading Glasses Near Pekin IL?
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As we age, our eyes have greater difficulty focusing properly on various distances, a problem referred to as presbyopia. Bifocals help people who have difficulty with near- and farsightedness with lenses that have two distinct optical strengths – one for long range, one for close work.
Safeguarding your eyes while outdoors is important and is especially important if you are reading a book or newspaper in the bright sunlight. There are reading sunglasses that you can get that will protect your eyes from the glare off of pages. These are likewise available in many trendy looks so that you can enjoy the most recent novel at the beach without compromising your sense of style.
Bifocal reader sunglasses are also easily available in many designs and colors. These have the dual benefit of allowing you to read in the sunlight and enjoy the views without changing to normal sunglasses.
Magnetic readers, also known as Clics, come apart at the nose bridge and magnetically rejoin at the bridge. These have an integrated glasses keeper that enables them to drape around your neck for handiness. Clics are the solution to always having your reading glasses nearby even if you don't have a pocket to put them in.
When space is at a premium, folding readers fit the bill. These compact folding reader glasses fit in a small evening bag or pocket and are handy alternatives that generally come with a small carrying case that fits in almost just about anywhere.
Also named trifocals, progressive reading glasses have multiple strengths all in just a single pair. They have no visible lines and allow the wearer to see clearly no matter if they are doing close work, intermediate range work or need to see clearly far away.
Readers can be purchased at most any price point from $1 to hundreds of dollars for designer reading glasses. In case you are just beginning to require reading glasses, your eye specialist may well tell you to buy cheap readers until your vision deteriorates to the point where you may need customized reading glasses.
The drawbacks of cheap readers are that the frames are typically not trendy, both lenses have the identical correction and the optical center of the lenses isn't precisely customized to your eyes. If you experience headaches, nausea or notice considerable eye strain, it is recommended that you check with your eye health specialist to see if you may have to transition to a custom made prescription pair of readers.
Reading glasses specifically developed for computer monitors are referred to as computer glasses. Computer screens give off blue light which can be harmful to your eyes just as UV rays can. If you do work in front of computer screens throughout the day, or you are concerned about the consequences of blue light on your children's eyes, you should get these "blue blockers". Both prescription or non-prescription computer readers include standard anti-reflective lenses to reduce eye strain and give 100% UV protection.
The best men's reading glasses and the best women's reading glasses in Pekin, IL are the type that give you the crystal clear vision you look for without the downsides. Headaches, eye fatigue and blurriness are indicators that you don't have the best glasses to do the job right. Take your time selecting your Progressive Reading Glasses and you will be able to find the best answer to better reading vision.


Extent to which statistics are comparable between different statistical domains.
Comparability between domains refers to the comparability between different statistics which target similar characteristics in different statistical domains. Users frequently compare statistics from different domains, which are produced on the basis of different methodologies. The difference in methodologies used for the estimation of the statistics should be reported. This concerns mainly the definition of statistical characteristics, the reference period, the definition of the statistical unit and the statistical measure.
definition Extent to which statistics are comparable between different statistical domains.
Description Comparability between domains refers to the comparability between different statistics which target similar characteristics in different statistical domains. Users frequently compare statistics from different domains, which are produced on the basis of different methodologies. The difference in methodologies used for the estimation of the statistics should be reported. This concerns mainly the definition of statistical characteristics, the reference period, the definition of the statistical unit and the statistical measure.


text:
Wasim Akram spreads awareness to tackle health crisis
By Farhan Nawaz
The gruesome coronavirus pandemic has spread to all parts of the globe with Pakistan also bearing the brunt of the disease with the rapidly rising number of COVID-19 cases. The ailment has negatively impacted the world of sports as several competitions and series have been cancelled due to fears over the expansion of the virus.
Amidst these troubling times, legendary cricketer Wasim Akram has joined hands with the Pakistan Cricket Board (PCB) to spread awareness about useful tips needed to combat the impending health crisis.
In a video message uploaded by the official PCB Twitter account, Wasim can be seen seated alongside his wife Shaniera Akram as both of them relay the relevant information to curb the virus. Wasim claimed he and his wife has isolated themselves and are not making unnecessary trips outside their home to ensure their contact with infected people is as less as possible.

redpajama_set_name: RedPajamaCommonCrawl


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

In this article we propose a new {\em nonstationary Iterated Tikhonov} (NIT)
type method~\cite[Sec.~1.2]{BS87} for obtaining stable approximations of linear
ill-posed problems.
The Lagrange multiplier is chosen so as to guarantee the residual of
the next iterate to be in a \emph{range}.
Previous strategies for choosing the Lagrange multiplier in each iteration
of NIT type methods either prescribe (\emph{a priori}) a geometrical increase
of this multiplier \cite{HG98} or require (\emph{a posteriori}) the residual
at the next iterate to assume a prescribed value which depends on the current
residual.

In those NIT methods that prescribe a geometrical increase of the Lagrange
multipliers, the use of a too large geometric factor may lead to numerical
instabilities and failure of convergence, whereas the use of a too small
factor leads to a slow convergent method (see Figures~\ref{fig:compare-sIT-nIT}
and~\ref{fig:nIT-unstable}); these features are highly dependent on the problem
at hand and, in general, it is not clear how to adequately choose the geometric
factor.

In those NIT methods that require the residual at the next iterate to assume
a prescribed value, at each iteration one needs to solve a nonlinear equation
which involves the resolvent of an ill-posed operator \cite{Ha97, DoHa13}.
This is accomplished by means of iterative methods (\emph{e.g.}\ Newton)
which require, at each of their steps, the solution of a linear system
for a different operator.
Consequently, the number of iterations required for these methods
does not fully quantify their computational costs.

The main contribution in this article is the proposal of a novel
\emph{a posteriori} strategy for choosing the  Lagrange multipliers
in NIT methods.
Since  we prescribe the residual of the next iterate to be in a range,
the set of feasible Lagrange multipliers, at each iteration,
is a \emph{non-degenerate interval},  which renders feasible their economical
computation (as explained later on).
Many relevant theoretical convergence properties present at previous
\emph{a posteriori} methods~\cite{EngHanNeu96} (e.g., residual
convergence rates, stability, semi-convergence) still hold for our novel strategy.
We also explore the feature of a feasible interval for the Lagrange multiplier
to speed up a Newton-like method for computing it.
The resulting method proves, in our preliminary numerical experiments,
to be more efficient {\color{black} (with respect to computational cost)}
than the geometrical choice of the Lagrange multipliers
\cite{HG98}, typically used in implementations of NIT type methods,
for low noise levels.

The \textit{inverse problem} we are interested in consists of determining an
unknown quantity $x \in X$ from the set of data $y \in Y$, where $X$ and $Y$ are
Hilbert spaces {\color{black} with norms $\norm{\cdot}_X$ and $\norm{\cdot}_Y$ respectively}.
In practical situations, one does not know the data exactly;
instead, only approximate measured data $y^\delta \in Y$ are available with
\begin{equation} \label{eq:noisy-data}
\norm{ y^\delta - y }_Y \ \le \ \delta \, ,
\end{equation}
where $\delta > 0$ is the (known) noise level. The available data $y^\delta$
are obtained by indirect measurements of the parameter $x$, this process
being described by the ill-posed operator equation
\begin{equation} \label{eq:ip}
A \, x \ = \ y \, ,
\end{equation}
where $A: X \to Y$ is a bounded linear operator, whose inverse $A^{-1}: R(A) \to X$
either does not exist, or is not continuous.
Consequently, approximate solutions are extremely sensitive to noise in the data.

Linear ill-posed problems are commonly found in applications ranging from image
analysis to parameter identification in mathematical models.
There is a vast literature on iterative methods for the stable solution of \eqref{eq:ip}.
We refer the reader to the books
\cite{Gr84, Hof86, Bau87, Lo89, Mor93, BakKok04, Kir96, EngHanNeu96, NatWue01, KalNeuSch08}
and the references therein.
Iterated Tikhonov (IT) type methods for solving the ill-posed problem \eqref{eq:ip}
are defined by {\color{black} an} iteration formula
\begin{equation*}
  x_k^\delta \ = \
  \argmin_{x\in X} \big\{ \lambda_k {\color{black} \| A x - y^\delta \|_Y^2}
                          + {\color{black} \| x - x_{k-1}^\delta \|_X^2} \big\} \, ,
\end{equation*}
{\color{black} that} corresponds to
\begin{align} \label{eq:it}
  \begin{aligned}
  x_k^\delta
  &= x_{k-1}^{\delta} -
  (I + \lambda_k A^*A)^{-1} \, \lambda_k A^* (A x_{k-1}^\delta - y^\delta ) \, ,\\

    &=
      (\lambda_k^{-1}I + A^*A)^{-1}     \left[\lambda_k^{-1}x_{k-1}^{\delta}+
      A^* y^\delta  \right]
  \end{aligned}
\end{align}
where $A^*: Y \to X$ is the adjoint operator to $A$.
The parameter $\lambda_k > 0$ can be viewed as the Lagrange multiplier of the problem
of projecting $x^\delta_{k-1}$ onto a levelset of $\norm{Ax - y^\delta}^2$.
If the sequence $\{ \lambda_k = \lambda \}$ is constant, iteration \eqref{eq:it}
is called {\em stationary} IT (or SIT), otherwise it is denominated {\em nonstationary}
IT (or NIT).
{\color{black} To simplify the notation, from now on we will use $\norm{\cdot}$
instead of $\norm{\cdot}_X$ or $\norm{\cdot}_Y$, whenever the norm under consideration
is clearly understood.}

In the NIT methods, each $\lambda_k$ is either chosen \emph{a priori} (e.g., in geometric
progression) or it is chosen \emph{a posteriori}.
In the  \emph{a posteriori} variants, $\lambda_k$ is chosen so that the next
iterate has a prescribed residual which is either a fixed fraction of the
current residual or a fraction which depends also on the noise level
\cite[eq.\ 2.11b in Alg.\ 1]{DoHa13}.
In other words, $\lambda_k$ is chosen in~\eqref{eq:it} so that
\begin{align}
  \label{eq:it.e}
  \norm{Ax_k^\delta-y^\delta}=\Phi(\norm{Ax_{k-1}^\delta-y^\delta},\delta)
\end{align}
(see \cite{Do12} for yet another strategy). We propose $\lambda_k$ to be {\color{black} chosen} so that
\begin{align}
  \label{eq:it.e2}
  \delta\leq   \norm{Ax_k^\delta-y^\delta}\leq\Psi(\norm{Ax_{k-1}^\delta-y^\delta},\delta).
\end{align}
{\color{black}
The upper bound in \eqref{eq:it.e2} for the residual depends on the current residual
and the noise level as in \cite{DoHa13}; however we propose here a new formula
to define this upper bound (see eq.\ \eqref{eq:p-proj}).}

The SIT method for solving \eqref{eq:ip} was considered in \cite{Lo89, Gr84},
where a well developed convergence analysis can be found
{\color{black} (see also Lardy \cite{Lar75}, where the particular choice $\lambda_k = 1$ is analyzed)}.
It is worth distinguishing between the SIT method and the \emph{iterated Tikhonov
methods of order $n$} \cite{KiCh79, Eng87, Sch93a}, where the number of iterations
(namely $n$) is fixed.
In this case $\lambda_k = \lambda > 0$, $k=0,\dots,n-1$, and $\lambda$ plays the
role of the regularization parameter.

The NIT method was addressed by many authors, e.g. \cite{Fak81, HG98, BS87}.
In numerical implementations of this method, the geometrical choice $\lambda_k = q^k$,
$q>1$, is a commonly adopted strategy and we shall refer to the resulting
method as gNIT method ({\em geometrical nonstationary IT method}).

The numerical performances of NIT type methods are superior to the ones
of SIT type methods
{\color{black} in terms of number of iterations and computational time
required to attain a predefined accuracy.}
This fact is illustrated by Example \ref{ex:gNIT-sIT}.

\begin{example} \label{ex:gNIT-sIT}
A linear system modeled by the Hilbert matrix $H^{25 \times 25}$ is
considered in Figure~\ref{fig:compare-sIT-nIT} (random noise of level
$\delta = 0.001\%$ is used).
{\color{black} In this benchmark problem the SIT method is tested with
$\lambda_k = 2$ (RED), while the gNIT method is tested with
$\lambda_k = 2^k$ (BLACK) and $\lambda_k = 3^k$ (BLUE).
In all these tests the same linear solver was used.
}
\end{example}

The computation $\lambda_k = q^k$ in the  gNIT is straightforward; however,
the choice of $q > 1$ is \emph{exogenous} to \eqref{eq:ip},
\eqref{eq:noisy-data} and it is not clear which are the good values for $q$.
Indeed, as shown in the next example, increasing the constant $q$ may lead
either to faster convergence or failure to converge:

\begin{example} \label{ex:gNIT-unstable}
We set $A = H^{25 \times 25}$, $X = Y = \R^{25}$ and random noisy data with
$\delta = 10^{-5}\,\%$.
{\color{black} In Figure~\ref{fig:nIT-unstable} the gNIT method is tested
with $\lambda_k = 2^k$ (BLACK), $\lambda_k = 3^k$ (BLUE) and
$\lambda_k = 4^k$ (ORANGE); in the last test there is not convergence.
}
\end{example}

The above described issues motivated  the use of \emph{a posteriori} choices
for the Lagrange multipliers, which requires the residual at the next iterate
to assume a prescribed value dependent on the current residual and
also on the noise level.
\medskip

\begin{figure}[t]
\centerline{\includegraphics[width=\textwidth]{pictures/REVfig-iT-sIT-nIT-compare2}}
\vskip-0.5cm
\caption{\small Comparison between SIT and NIT type methods:
$A = H^{25 \times 25}$ is the square Hilbert matrix; artificial random noise
($\delta = 10^{-3} \%$) is added to the data. The stopping criteria (discrepancy
principle with $\tau = 2$) is reached after 17 steps (BLACK), 11 steps (BLUE)
and 59275 steps (RED).}
\label{fig:compare-sIT-nIT}
\end{figure}

\begin{figure}[t]
\centerline{\includegraphics[width=\textwidth]{pictures/REVfig-iT-nIT-unstable2}}
\vskip-0.5cm
\caption{\small Unstable behavior of gNIT type methods:
matrix $A \in \R^{25 \times 25}$ as in Figure~\ref{fig:compare-sIT-nIT};
artificial random noise ($\delta = 10^{-5} \%$) is added to the data.
The stopping criteria (discrepancy principle with $\tau = 2$) is reached
after 27 steps (BLACK), 18 steps (BLUE). In the last implementation (ORANGE),
the gNIT method becomes unstable before the stopping criteria is reached.}
\label{fig:nIT-unstable}
\end{figure}

\noindent
Next we briefly review some relevant convergence results for IT type methods.

$\bullet$ \ In \cite{BS87} Brill and Schock proved that, in the exact data case, the
NIT method \eqref{eq:it} converges to a solution of $Ax = y$ if and only if
$\sum \lambda_k = \infty$.
Moreover, a convergence rate result was established under the additional assumption
$\sum \lambda_k^2 < \infty$.

$\bullet$ \ The assumptions needed in \cite{BS87} in order to derive convergence
rates are neither satisfied for the sIT, nor for the NIT with the geometrical choice
$\lambda_k = q^k$, $q > 1$.

$\bullet$ \ In \cite{HG98} rates of convergence are established for the stationary Lardy's
method \cite{Lar75} as well as for the NIT with geometrical choice of $\lambda_k$.
Under the source condition {\color{black} $x^\dag \in \mathop\mathrm{Range}((A^*A)^\nu)$},
where $x^\dag = A^\dag y$ is the normal solution of $Ax = y
\footnote{\color{black} I.e., $x^\dag$ is the unique vector satisfying $x^\dag \in
\mathop\mathrm{Dom}(A) \cap \mathop\mathrm{Ker}(A)^\perp$
and $A x^\dag = y$, where $\mathop\mathrm{Dom}(A)$ and $\mathop\mathrm{Ker}(A)$
stand for the domain and the kernel of $A$ respectively.},
and $\nu > 0$
\footnote{See \cite[Theorem 2.1]{HG98} for details on the positive the scalar $\nu$.}
the linear rate of convergence $\norm{x_k - x^\dag} = O(q^{k\nu})$ is proven.
\medskip

\noindent
The article is organized as follows:
In Section~\ref{sec:rrNIT} we introduce the new method (rrNIT), which is proposed and
analyzed in this manuscript.
Moreover, a detailed formulation of this method is given, and some preliminary estimates
{\color{black} for the Lagrange multipliers $\lambda_k$} are also derived.
In Section~\ref{sec:conv-anal} a convergence analysis of the rrNIT method is presented.
In Section~\ref{sec:num-implementation} we discuss the algorithmic implementation of the
rrNIT method. In particular, we address the challenging numerical issue of efficiently
computing the Lagrange multipliers $\lambda_k$.
Section~\ref{sec:num-experiments} is devoted to numerical experiments. The
{\em Image deblurring} problem and the {\em Inverse Potential
Problem} are considered in Subsections~\ref{ssec:num-id} and~\ref{ssec:num-ipp}
respectively.
Section~\ref{sec:conclusion} is dedicated to final remarks and conclusions.

\section{Range-relaxed non-stationary iterated Tikhonov method} \label{sec:rrNIT}

In this section first we propose in Subsection~\ref{sec:orrpm} a
conceptual projection method for solving~\eqref{eq:noisy-data},
\eqref{eq:ip}.
In this method, each iterate is obtained projecting the previous one
onto a level set of the residual function. The level set is
prescribed to belong to a range of level sets, instead of being a
single one.
Second, we propose in Subsection~\ref{ssec:true-alg} an implementable
version of the conceptual method where the projection is computed via
Lagrange multipliers.
Finally, we derive some basic properties of the new proposed method.

The implementable method proposed here happens to be a new NIT method
where, in each iteration, the set of feasible choices for the Lagrange
multipliers is an interval, instead of a single real
number. For this reason, we call the new method a range-relaxed Non-stationary
Iterative Tikhonov Method (rrNIT).

For the remaining of this article we suppose that the following assumptions
hold true:
\bigskip

\hfill\begin{minipage}{15cm}
{\bf (A1)} There {\color{black} exists} $x^\star \in X$ such that $A x^\star = y$, where
$y \in \mathop\mathrm{Range}(A)$ are the exact data.
\bigskip

{\bf (A2)} The operator $A: X \to Y$ is linear, bounded and ill-posed, i.e., even if the
operator $A^{-1}: R(A) \to X$ (the left inverse of $A$) exists, it is not continuous.
\end{minipage}

\subsection{A  Successive Orthogonal range-relaxed Projections Method}
\label{sec:orrpm}

We use the notation $\Omega_\mu$, for $\mu\geq 0$, to denote the $\mu$-levelset of
the residual functional $\norm{Ax-y^\delta}$, that is,
\begin{align} \label{eq:best-set}
 \Omega_\mu \ := \ \{x \in X \, : \, \norm{Ax - y^\delta}^2 \leq \mu^2\} \, .
\end{align}
The basic geometric properties of the levelsets $\Omega_\mu$, described next, are
instrumental in the forthcoming analysis.

\begin{proposition}
  \label{pr:omega-mu}
  Let $\Omega_\mu$ be as in \eqref{eq:best-set}.
  \begin{enumerate}
  \item   For each $\mu \geq 0$, the set $\Omega_\mu$  is
    closed and convex.

  \item
    If $\mu'\geq\mu>0$

    then $\Omega_\mu \subset \Omega_{\mu'}$.
  \item  If
    $\mu\geq\delta$ then $A^{-1}(y) \subset \Omega_\mu$.

  \item
    If  $\mu > \delta$
  then  $\Omega_\mu$ has non-empty interior.
  \end{enumerate}
\end{proposition}

\begin{proof}
{\color{black}
Item 1. follows from the continuity and convexity of $\norm{Ax-y^\delta}$ as a function
of $x$ together with definition \eqref{eq:best-set}.
Item 2. follows trivially from \eqref{eq:best-set}.
Item 3. follows from \eqref{eq:best-set} and assumption \eqref{eq:noisy-data}, because
$\norm{Ax^\star - y^\delta} = \norm{y - y^\delta} \leq \delta$.
The last item follows from this inequality together with definition \eqref{eq:best-set}
and the continuity of $\norm{Ax - y^\delta}$.}
\end{proof}

Notice that all available information about the solution set
$A^{-1}(y)$ is contained in \eqref{eq:noisy-data}, \eqref{eq:ip}.
Thus, in the absence of additional information, $\Omega_\delta$ is the
set of best possible approximate solutions for the inverse problem under
consideration
\footnote{I.e., given two elements in $\Omega_\delta$, it is not
  possible to distinguish which of them better approximates $x^\star$.}

Nevertheless the levelset $\Omega_\delta$ is, in general, unbounded and it is
desirable to exclude those approximate solutions with ``too large''
norms.
Moreover, very often a crude estimation $\hat x$ to the
solution of \eqref{eq:ip} is available.  In this context, it is
natural to consider the projection problem
\begin{align} \label{eq:pr-pb}
\left\{
 \begin{array}{ll}
   \min_x          & \norm{x - \hat x}^2 \\
   \mathrm{s.t.} & \norm{A x - y^\delta}^2 \, \leq \, \mu^2 ,
 \end{array} \right.
\end{align}
where $\mu \geq 0$.
Observe that if $\norm{A\hat x-y^\delta}>\mu\geq\delta$,
then the solution of this projection problem is closer to $\Omega_\delta$
than $\hat x$ and has a smaller residual than $\hat x$.

The considerations in the preceding paragraph show that it is possible,
at least conceptually, to devise projection methods for solving our
ill-posed problem.
Let us briefly discuss the
{\color{black} conditioning of the projection problem} \eqref{eq:pr-pb}
with respect to {\color{black} the} parameter $\mu$:
\begin{enumerate}
\item[(i)] for $0 \leq \mu < \delta$ {\color{black} the projection problem}
  \eqref{eq:pr-pb} may be unfeasible, that is, it may become the
  problem of projecting $\hat x$ onto an \emph{empty set};
\item[(ii)] for $\mu = \delta$, in view of (i), problem
  \eqref{eq:pr-pb} is in general ill-posed with respect to the
  parameter $\mu$;
\item[(iii)] for $\mu > \delta$ problem
  \eqref{eq:pr-pb} is well posed, and it is natural to expect that
  the larger the $\mu$ the better conditioned it becomes.
\end{enumerate}

A compromise between reducing the residual norm $\norm{Ax-y^\delta}$ and
preventing ill-posedness of the projection problem would be to choose
\[
\hat\mu \ = \ p\norm{A\hat x-y^\delta} + (1-p)\delta \, ,
\]
where $0 < p < 1$ quantify this compromise. However,
\begin{enumerate}
\item projecting $\hat x$ onto $\Omega_{\hat\mu}$, which is a \emph{pre-defined}
level set of $\norm{Ax-y^\delta}^2$, entails an additional numerical difficulty:
the projection has to be computed by solving a linear system where the
Lagrange multiplier is implicitly defined by an \emph{algebraic equation};
\item the projection of $\hat x$ onto any levelset $\Omega_{\mu}$ with
$\delta \leq \mu \leq \hat\mu$ is as good as (or even better than) the
projection of $\hat x$ onto $\Omega_{\hat\mu}$.
\end{enumerate}
In view of these observations,  we shall generate $x_k^\delta$ from
$\hat x = x_{k-1}^\delta$ by projecting it onto any one of the range
of convex sets $(\Omega_{\mu})_{\delta\leq\mu\leq\hat\mu}$,
that is, by solving the \emph{range-relaxed} projection problem of
computing $(x,\mu)$ such that
\begin{align} \label{eq:p-proj}
\left\{
  \begin{array}{ll}
    \min_x          & \norm{x - \hat x}^2 \\
    \mathrm{s.t.} & ||Ax - y^\delta||^2 \, \leq \, \mu^2 \, , \qquad
    \delta \, \leq \, \mu \, \leq \, p\norm{A\hat x - y^\delta} + (1-p)\delta \, ,
  \end{array}
\right.
\end{align}
whenever $x_{k-1}^\delta \notin \Omega_\delta$.
Observe that this problem has multiple solutions.
The advantage of this strategy is that the set of feasible
Lagrange multipliers of the above problem is an interval with
non-empty interior, as we will discuss latter, instead of a
single point.

In what follows we use the notation ${\Proj}_\Omega$ to denote the orthogonal
projection onto $\Omega$, for $\emptyset\neq\Omega\subset X$ closed and convex.
The discussion in the previous paragraph leads us to propose the
\emph{conceptual} successive orthogonal range-relaxed projection
Method for problem \eqref{eq:noisy-data}, \eqref{eq:ip} described in
Algorithm~\ref{alg:rrNIT-concept}.

\begin{algorithm}[b]
\begin{center}
\fbox{\parbox{13.3cm}{
$[1]$ choose an initial guess \ $x_0 \in X$;
\smallskip

$[2]$ choose \ $p \in (0,1)$ \ and \ $\tau > 1$;
\smallskip

$[3]$ for \ $k \geq 1$ \ do
\smallskip

\ \ \ \ $[3.1]$ compute\, $(x_{k}^\delta,\mu_k)$,

\smallskip

\ \ \ \ \ \ \ \ \ \
$x_{k}^\delta={\Proj}_{\Omega_{\mu_k}}(x_{k-1}^\delta)$,
\ $\delta \, \leq \, \mu_k \, \leq \,
                      p\norm{Ax_{k-1}^\delta - y^\delta} + (1-p) \delta$;
\smallskip

\ \ \ \ $[3.2]$ stop to iterate at step\, $k^* \geq 1$\, s.t.\,
        $\norm{Ax_{k^*}^\delta - y^\delta} < \tau\delta$ \ for the first time.
} }
\end{center} \vskip-0.5cm
\caption{Successive orthogonal range-relaxed projection method.} \label{alg:rrNIT-concept}
\end{algorithm}

\noindent
Since $\norm{Ax_k^\delta-y^\delta}=\mu_k$, the variable $\mu_k$ is
redundant, nevertheless, its use in the conceptual Algorithm~\ref{alg:rrNIT-concept}
{\color{black} clarifies} the kind of projection problem used to compute $x_k^\delta$.

By its definition, Algorithm 1 is a method of successive orthogonal
projections onto level sets of $\norm{Ax-y^\delta}$. As the level set
used in each iteration shall be in a range, we {\color{black} call} it a successive
orthogonal \emph{range-relaxed} projection method.
Each iterate is obtained from the previous one by projecting it onto
a convex set that contains the solution set and in which the residual in
any point is strictly smaller that the residual at the previous iterate.
Therefore the errors as well as the residuals are strictly decreasing
along the iterates and the sequence of iterates is bounded.

\subsection{A Range-relaxed non-stationary iterated Tikhonov algorithm}
\label{ssec:true-alg}

In order to derive an implementable version of the conceptual method
(Algorithm~\ref{alg:rrNIT-concept}) discussed in the previous section, we need to specify
how to compute the range-relaxed projections (at Step [3]).
With this aim, recall that the canonical Lagrangian of problem \eqref{eq:pr-pb} is
\begin{align}
  \label{eq:lg}
  \mathcal{L}(x,\lambda) \ = \
  \dfrac{\lambda}{2}(\norm{Ax-y^\delta}^2-\mu^2) +
  \dfrac12\norm{x-\hat x}^2 \, .
\end{align}
For each $\lambda > 0$,  $\mathcal{L}(\cdot,\lambda) : X \to \R$
has a unique minimizer $x'$ which is also characterized as
the unique solution of $\nabla_x\mathcal{L}(x,\lambda)=0$, that is,
\begin{align*}
  x' =\hat x-\lambda(I+\lambda A^*A)^{-1}A^*(A\hat x-y^\delta).
\end{align*}
The next lemma summarizes the solution theory for the projection problem
\eqref{eq:pr-pb} by means of Lagrange multiplier~\cite[Sec.~5.7]{TR96}.
Recall that ${\Proj}_\Omega$ denotes the orthogonal projection onto $\Omega$.

\begin{lemma} \label{lem:p1}
Suppose $\norm{A\hat x-y^\delta}>\mu>\delta$. The following assertions are
equivalent
\begin{enumerate}
\item\label{it:p1-1} $x'=\Proj_{\Omega_\mu}(\hat x)$;
\item\label{it:p1-2} $x'$ is \emph{the} solution of \eqref{eq:pr-pb};
\item\label{it:p1-3a} $x'=\hat x-\lambda^*(I+\lambda^* A^*A)^{-1}A^*(A\hat x-y^\delta)$,
  $\lambda^*>0$ and
  \begin{align*}
    \norm{Ax'-y^\delta}=\mu
  \end{align*}
\end{enumerate}
\end{lemma}

\begin{proof}
  Equivalence between items \ref{it:p1-1} and \ref{it:p1-2} follows
  from the definition of orthogonal projections onto closed convex
  sets.  Equivalence between items~\ref{it:p1-2} and~\ref{it:p1-3a} is
  a classical Lagrange multipliers result (see, e.g.,
  \cite[Theorem~5.15]{TR96}).
\end{proof}

In the next lemma we address the range-relaxed projection problem~\eqref{eq:p-proj};
its proof follows from Lemma~\ref{lem:p1}.
\begin{lemma} \label{lem:p2}
Suppose $\norm{A\hat x-y^\delta}>\delta$ and $0<p<1$. The following assertions are
equivalent
\begin{enumerate}
\item\label{it:p2-1} $x'=\Proj_{\Omega_\mu}(\hat x)$ and
  $\delta\leq \mu\leq p\norm{A\hat x-y^{\delta}}+(1-p)\delta$;
\item\label{it:p2-2} $(x',\mu)\in X\times \R$ is a solution of \eqref{eq:p-proj};
\item\label{it:p2-3a} $x'=\hat x-\lambda(I+\lambda A^*A)^{-1}A^*(A\hat x-y^\delta)$,
  $\lambda>0$,
  \begin{align*}
    \delta\leq \norm{Ax'-y^\delta}
    \leq  p\norm{A\hat x - y^\delta} + (1-p)\delta,
  \end{align*}
  and $\mu=\norm{Ax' - y^\delta}$.
\end{enumerate}
\end{lemma}

It follows from Lemma~\ref{lem:p2}
that
solving the range-relaxed projection problem in \eqref{eq:p-proj} boils down to solving the inequalities
\begin{align}
  \label{eq:rg}
  \begin{aligned}

    \delta\leq\norm{Ax'-y^\delta} \leq
  p\norm{A\hat x-y^\delta}+(1-p)\delta,
  \\ &\qquad\qquad\text{where }
  x' =\hat x-\lambda(I+\lambda A^*A)^{-1}A^*(A\hat x-y^\delta)
\end{aligned}
\end{align}
and defining
$x=x'$ and $\mu= \norm{Ax'-y^\delta}$.
We use this result to propose an implementable version of
Algorithm~\ref{alg:rrNIT-concept} as follows:

\begin{algorithm}[h!]
\begin{center}
\fbox{\parbox{14.3cm}{
$[1]$ choose an initial guess \ $x_0 \in X$;
\smallskip

$[2]$ choose \ $p \in (0,1)$, \ $\tau > 1$ and set \ $k := 0$;
\smallskip

$[3]$ while \ $\big( \norm{Ax_k^\delta-y^\delta}\, > \, \tau\delta \big)$ \ do
\smallskip

\ \ \ \ $[3.1]$ $k := k + 1$;
\smallskip

\ \ \ \ $[3.2]$ compute\, $\lambda_k$ and $x^\delta_k$ such that

\smallskip

\ \ \ \ \ \ \ \ \ \ \,$x_k^\delta \ = \ x_{k-1}^\delta - \lambda_k \,
        (I + \lambda_k A^*A)^{-1} \, A^*(Ax_{k-1}^\delta - y^\delta)$,\;\;

        \smallskip

        \ \ \ \ \ \ \ \ \ \ \,$\delta\leq \norm{Ax_k^\delta-y^\delta}
        \leq p\norm{Ax_{k-1}^\delta - y^\delta} + (1-p) \delta$

} }
\end{center} \vskip-0.5cm
\caption{The rrNIT method.} \label{alg:rrNIT-true}
\end{algorithm}

\noindent
As in Algorithm~\ref{alg:rrNIT-concept}, the stopping index in the above algorithm
is defined by the discrepancy principle
\begin{equation} \label{def:k-star}
k^* \ := \ \min \{ k \geq 1 \, ; \ \norm{A x_j - y^\delta} > \tau\delta , \
                   j = 0, \dots, k-1 \ \ \ {\rm and} \ \ \
                   \norm{A x_{k} - y^\delta} \leq \tau\delta \} \, .
\end{equation}
The computational burden of the above algorithm resides in the
computation of step
[3.2], which requires the solution of a linear system whose
corresponding residual shall be in a given range.  In other words,
$\lambda_k$ shall be a solution of \eqref{eq:rg} with
$\hat x=x_{k-1}^\delta$.

The next lemma provides a lower bound for  $\lambda=\lambda_k$
which will be used in the convergence analysis
of Algorithm~\ref{alg:rrNIT-true}, presented in Section~\ref{sec:conv-anal}.
This lower bound can also be used for
used as initial guess to compute $\lambda_k$.

\begin{lemma} \label{lm:lbound}
Under the assumptions of Lemma~\ref{lem:p1},
\begin{align*}
  \lambda^*\geq \dfrac{(\norm{A\hat x-y^\delta}-\mu)\norm{A\hat x-y^\delta}}
  {\norm{A^*(A\hat x-y^\delta)}^2}.
\end{align*}
\end{lemma}
\begin{proof}
  To simplify the notation, let
  \[
    z \ := \ \Proj_{\Omega_\mu} \hat x,\qquad b \ := \ A\hat x-y^\delta.
  \]

  From the assumption $\norm{A \hat x - y^\delta} > \mu > \delta$, it
  follows that $\hat x \notin \Omega_\delta$.
  Therefore $\norm{Az-y^\delta} = \mu$,
  \begin{align}\label{eq:Azx}
    \norm{A(z-\hat x)}\geq\norm{A\hat x-y^\delta}-\norm{Az-y^\delta}
    =\norm{b}-\mu,
  \end{align}
  and
  \[
    \mu^2 \ = \ \norm{A(z - \hat x) + b}^2
    \ = \ \norm{A(z - \hat x)}^2 \, + \, \norm{b}^2
    \, + \, 2 \ipl A(z - \hat x) , \, b \ipr \, .
  \]

Direct combination of the above equation with the previous inequality yields
\begin{eqnarray*}
- 2 \big\ipl A(z - \hat x) , \, b \big\ipr
&   =  & \norm{A(z - \hat x)}^2 \, + \, \norm{b}^2 - \mu^2 \\
& \geq & \big( \norm{b} - \mu \big)^2 \, + \, \norm{b}^2 - \mu^2 \\
&   =  & 2 \, \norm{b} \, \big( \norm{b} - \mu \big).
\end{eqnarray*}
Therefore, if follows from \eqref{eq:Azx}, Cauchy-Schwartz inequality,
and the definition of $\Proj_{\Omega_\mu}$ that
\begin{eqnarray*}
\big( \norm{b} - \mu \big) \, \norm{b}
& \leq & \big\ipl -(z - \hat x) , \, A^* b \big\ipr \\
&  =   & \big\ipl \lambda^* (I + \lambda^* A^*A)^{-1} A^* b, \, A^* b \big\ipr \\
& \leq & \lambda^* \, \norm{(I + \lambda^* A^*A)^{-1}} \, \norm{A^* b}^2 \\
& \leq & \lambda^* \, \, \norm{A^* b}^2 \, ,
\end{eqnarray*}
proving the lemma.
\end{proof}

\begin{corol} \label{cor:lbd-estim}
Let the sequences $(x_k^\delta)$ and $(\lambda_k)$ be defined by the rrNIT method
(Algorithm~\ref{alg:rrNIT-true}), with $\delta \geq 0$ and $y^\delta \in Y$ as in
\eqref{eq:noisy-data} and let
$\mu_k :=\norm{Ax_k^\delta-y^\delta}$. Then
\begin{equation} \label{eq:lbd-k-estim}
\lambda_k \ \geq \
\frac{\big( \norm{A x_{k-1}^\delta - y^\delta} - \mu_k \big)
            \, \norm{A x_{k-1}^\delta - y^\delta}}
     {\norm{A^* (A x_{k-1}^\delta - y^\delta)}^2} \, , \
     k = 1, \dots, k^*.
\end{equation}
In the exact data case (i.e., $\delta = 0$) the above estimate simplifies to\,
$\lambda_k \, \geq \, (1-p) \, \norm{A}^{-2}$.
\end{corol}
\begin{proof}
From \eqref{eq:rg} and the definition of $\mu_k$ it follows that
$\delta < \mu_k < \norm{A x_{k-1}^\delta - y^\delta}$. Thus, \eqref{eq:lbd-k-estim}
follows from Lemma~\ref{lm:lbound} with $\hat x = x_{k-1}^\delta$,
$\lambda^* = \lambda_k$ and $\mu = \mu_k$ (in the proof of that lemma
it holds $z = x_k^\delta$).

In the exact data case, it follows from \eqref{eq:lbd-k-estim}, together with Assumption
\textbf{(A2)}, that $\lambda_k \, \geq \, \big( \norm{A x_{k-1} - y} - \mu_k \big)
\norm{A}^{-2} \norm{A x_{k-1} - y}^{-1}$.
Moreover, since $\delta = 0$ we have $\mu_k \leq p \norm{A x_{k-1} - y}$.
Combining these two facts, the second assertion follows.
\end{proof}

\section{Convergence Analysis} \label{sec:conv-anal}

We begin this section by establishing an estimate for the decay of the residual
$\norm{A x_k^\delta - y^\delta}$.

\begin{proposition} \label{prop:decay-residual}
Let $(x_k^\delta)$ be the sequence defined by the rrNIT method (Algorithm~\ref{alg:rrNIT-true}),
with $\delta \geq 0$ and $y^\delta \in Y$ as in \eqref{eq:noisy-data}. Then
\[
\big[ \norm{A x_k^\delta - y^\delta} - \delta \big] \ \leq \  p\,
\big[ \norm{A x_{k-1}^\delta - y^\delta} - \delta \big] \ \leq \ p^k\,
\big[ \norm{A x_0 - y^\delta} - \delta \big] \, , \ k = 1, \dots, k^* \, ,
\]
where $k^* \in \N$ is defined by \eqref{def:k-star}.
\end{proposition}
\begin{proof}
It is enough to verify the first inequality.
Recall that $x_k^\delta \in \Omega_{\mu_k}$, where $\delta \leq \mu_k \leq
p \norm{A x_{k-1}^\delta - y^\delta} + (1-p)\delta$.
Consequently,
$\norm{A x_k^\delta - y^\delta} \leq p\norm{A x_{k-1}^\delta - y^\delta} + (1-p)\delta$,
and the first inequality follows.
\end{proof}

Now we are ready to prove finiteness of $k^*$ and to provide an upper
bound for it, whenever $\delta>0$.

\begin{corol} \label{cor:estim-k*}
Let $(x_k^\delta)$ be the sequence defined by the rrNIT method (Algorithm~\ref{alg:rrNIT-true}),
with $\delta > 0$ and $y^\delta \in Y$ as in \eqref{eq:noisy-data}.
Then the stopping index $k^*$ defined in \eqref{def:k-star} satisfies
\[
k^* \ \leq \ |\ln p|^{-1} \,
             \ln \left[ \frac{\norm{A x_0 - y^\delta}-\delta}{(\tau-1)\delta} \right] + 1 \, .
\]
\end{corol}
\begin{proof}
We may assume $\norm{A x_0 - y^\delta} > \tau\delta$
\footnote{Otherwise the iteration does not start, i.e.,\, $k^* = 0$.}
From \eqref{def:k-star} follows $\tau\delta < \norm{A x_{k}^\delta - y^\delta}$,
$k=0, \dots\ k^*-1$. This inequality (for $k = k^*-1$), together with Proposition
\ref{prop:decay-residual} imply that
\[
(\tau-1)\delta \   <  \ \norm{A x_{k^*-1}^\delta - y^\delta} - \delta
               \ \leq \ p^{k^*-1} \big[ \norm{A x_0 - y^\delta} - \delta \big] \, ,
\]
completing the proof (recall that $p \in (0,1)$).
\end{proof}

Monotonicity of the iteration error $\norm{x^\star - x_k^\delta}$ was already established
in Section~\ref{sec:rrNIT}. In the next proposition we estimate the ``gain''\,
$\norm{x^\star - x_{k-1}^\delta}^2 - \norm{x^\star - x_k^\delta}^2$\, in the rrNIT method.

\begin{proposition} \label{prop:gain-estim}
Let $(x_k^\delta)$ be the sequence defined by the rrNIT method (Algorithm~\ref{alg:rrNIT-true}),
with $\delta > 0$ and $y^\delta \in Y$ as in \eqref{eq:noisy-data}. Then
\begin{equation} \label{eq:gain-estim}
\norm{x^\star - x_{k-1}^\delta}^2 - \norm{x^\star - x_k^\delta}^2
\  =  \ \norm{x_k^\delta - x_{k-1}^\delta}^2
\, + \, \lambda_k \norm{A (x^\star - x_k^\delta)}^2
\, + \, \lambda_k \, \Big[ r(x_k^\delta) - r(x^\star) \Big] \, ,
\end{equation}
for\, $k = 1, \dots, k^*$, where $r(x) := \norm{A x - y^\delta}^2$.\, Consequently,
\begin{equation} \label{eq:gain-estim-le}
\norm{x^\star - x_{k-1}^\delta}^2 - \norm{x^\star - x_k^\delta}^2
\ \geq \ \lambda_k^2 \, \norm{A^* (A x_k^\delta - y^\delta)}^2 \, + \, \lambda_k
         \Big[ \norm{A (x^\star - x_k^\delta)}^2 + (\tau^2 - 1) \, \delta^2 \Big] ,
\end{equation}
for\, $k = 1, \dots, k^*-1$; moreover,
\begin{equation} \label{eq:gain-estim-last}
\norm{x^\star - x_{k^*-1}^\delta}^2 - \norm{x^\star - x_{k^*}^\delta}^2
\ \geq \ \lambda_{k^*}^2 \, \norm{A^* (A x_{k^*}^\delta - y^\delta)}^2 \, + \,
         \lambda_{k^*} \, \norm{A (x^\star - x_{k^*}^\delta)}^2 .
\end{equation}
\end{proposition}
\begin{proof}
First we derive \eqref{eq:gain-estim}. Due to the definition of
$(x_k^\delta,\,\lambda^\delta)$ in Algorithm~\ref{alg:rrNIT-true}, the
Lagrangian $\mathcal{L}$ in \eqref{eq:lg}
(with $\mu = \mu_k$ and $\hat x = x_{k-1}$) satisfies
$$
{\cal L}(x, \lambda_k) \, = \,
{\cal L}(x_k^\delta, \lambda_k) \, + \, {\cal L}_x(x_k^\delta, \lambda_k) (x - x_k^\delta)
\, + \,
\frac 1 2 \, \big\ipl (x-x_k^\delta) ,\, H(\lambda_k) (x-x_k^\delta) \big\ipr ,
$$
where $H(\lambda_k) \,=\, (I + \lambda_k A^* A)$ is the Hessian of
${\cal L}(\cdot,\lambda_k)$ at $x=x_k^\delta$.\, Since ${\cal L}_x(x_k^\delta, \lambda_k) = 0$,
we have
\begin{align*}
  {\cal L}(x, \lambda_k)
  =
   {\cal L}(x_k^\delta, \lambda_k) \, + \,
  \frac{1}{2}
  \big\ipl (x-x_k^\delta) , \, (I + \lambda_k A^* A) (x-x_k^\delta) \big\ipr,
\end{align*}
that is,
\begin{align*}
\norm{x-x_{k-1}^\delta}^2  +  \lambda_k \big[ r(x) - \mu_k^2 \big]
=
   \norm{x_k^\delta-x_{k-1}^\delta}^2 + \lambda_k \big[ r(x_k^\delta) - \mu_k^2 \big]
   + \norm{x - x_k^\delta}^2  +  \lambda_k \norm{A (x - x_k^\delta)}^2 .
\end{align*}
Now, choosing $x = x^\star$, one establishes \eqref{eq:gain-estim}.
Inequality \eqref{eq:gain-estim-le}, on the other hand, follows from
\eqref{eq:gain-estim} together with $r(x^\star) \leq \delta^2$\,
and\, $r(x_k^\delta) > \tau^2\delta^2$, for\, $k = 1, \dots, k^*-1$.
Analogously, \eqref{eq:gain-estim-last} follows from \eqref{eq:gain-estim}
together with\, $r(x_{k^*}^\delta) > \delta^2$ (see Algorithm~\ref{alg:rrNIT-true}).
\end{proof}

\begin{corol} \label{cor:gain-estim-exactdata}
In the exact data case, i.e., $\delta = 0$ and $y^\delta = y \in R(A)$,
then \eqref{eq:gain-estim} becomes
\begin{equation*}
\norm{x^\star - x_{k-1}}^2 - \norm{x^\star - x_k}^2 \ = \
\norm{x_k - x_{k-1}}^2 \, + \, 2 \lambda_k \, \norm{A x_k - y}^2 ,
\end{equation*}
from which follows\,
$\sum_{k\geq 1} \, \lambda_k \, \norm{A x_k - y}^2 \, < \, \infty$ \, and \,
$\sum_{k\geq 1} \, \norm{x_k - x_{k-1}}^2 \, < \, \infty$\,
(the last inequality means that the operator describing the rrNIT iteration
is a reasonable wanderer in the sense of \cite{BrPe67}).
\end{corol}

Next we prove strong convergence of the rrNIT method (in the exact data case)
to a solution of the inverse problem \eqref{eq:ip}. The estimate in
Lemma~\ref{lm:lbound} plays a key role in this proof.

\begin{theorem} \label{theor:converg}
Let $(x_k)$ and $(\lambda_k)$ be the sequences defined by the rrNIT method
(Algorithm~\ref{alg:rrNIT-true}), with $\delta = 0$ and $y^\delta = y \in R(A)$.
Then $(x_k)$ converges strongly to some $x^* \in X$. Moreover, $A x^* = y$.
\end{theorem}
\begin{proof}
From the second assertion in Corollary~\ref{cor:lbd-estim} it follows that\,
$\sum_{k\geq 1}\, \lambda_k \, = \, \infty$.
The proof now follows from \cite[Theor.\,1.4]{BS87}.
\end{proof}

\section{Numerical Implementation} \label{sec:num-implementation}

In this section the implementation of Algorithm~\ref{alg:rrNIT-true} is reviewed.
We discuss the implementation of step [3.2] of that algorithm by means of a Newton-like
method, and how we accelerated this computation.

As discussed in Section~\ref{ssec:true-alg}, at step [3] of Algorithm~\ref{alg:rrNIT-true},
$\lambda_k \geq 0$ is to be obtained as a solution of the scalar rational inequalities
(or a inclusion)
\eqref{eq:rg} with $\hat x=x_{k-1}^\delta$, that is,
\begin{align}
  \label{eq:rg-k}
  \begin{aligned}
    &
    \delta\leq\norm{A_kx_k^\delta-y^\delta} \leq
    p\norm{Ax_{k-1}^\delta-y^\delta}+(1-p)\delta,
  \\ &\qquad\qquad\text{where }
  x_k^\delta =x_{k-1}^\delta-\lambda(I+\lambda A^*A)^{-1}A^*(Ax_k^\delta-y^\delta)
  \text{ with }\lambda>0.
\end{aligned}
\end{align}
Define, at iteration $k$, $\pi_k(\lambda)$ as the candidate for
$x_k^\delta$ obtained from $x_{k-1}^\delta$ with the Lagrange
multiplier $\lambda$ and let $G_k(\lambda)$ be the square residual at
that point, that is,
\begin{subequations}
\begin{align}
\label{eq:G-pi1}
  \pi_k(\lambda) & :=
                   x_{k-1}^\delta-
                   \lambda(I+\lambda A^*A)^{-1}A^*(A x_{k-1}^\delta-y^\delta),
  \\
   & :=
     (\lambda^{-1}I+ A^*A)^{-1}(\lambda^{-1} x_{k-1}+A^*y^\delta),
  \\
  \label{eq:G-pi2}
 G_k(\lambda) & := \norm{A\pi_k(\lambda)-y^\delta}^2 .
\end{align}
\end{subequations}
With this notation, \eqref{eq:rg-k} writes
\begin{align}
  \label{eq:rat-ineq}
  \delta^2\leq  G_k(\lambda)\leq (p\norm{Ax_{k-1}^\delta-y^{\delta}}+(1-p)\delta)^2
  \quad\text{ with }\lambda>0.
\end{align}

As earlier mentioned, the set of feasible Lagrange multipliers is a
non-degenerate interval in each iteration of the rrNIT method.

\begin{proposition}
  \label{pr:ndi}
  Suppose Algorithm~\ref{alg:rrNIT-true} reaches iteration $k$ and
  $\norm{Ax_k^\delta-y^\delta} > \tau\delta$. Then, for $\lambda \geq 0$
  \begin{align*
    \dfrac{d}{d\,\lambda}G_k(\lambda)
    &= -2 \left\langle A^*( A\pi_k(\lambda)-y^\delta) ,
      (I+\lambda A^*A)^{-1}A^*(A\pi_k(\lambda)-y^\delta) \right\rangle\\
    &= -2\lambda^{-3} \left\langle x_{k-1}-\pi_k(\lambda) ,
      (\lambda^{-1}I+A^*A)^{-1}(x_{k-1}-\pi_k(\lambda)) \right\rangle.
  \end{align*}
  Moreover, $G_k$ is strictly decreasing in $(0,\infty)$,

   the solution set of~\eqref{eq:rat-ineq} is
  $[\lambda_{\min},\lambda_{\max}]\cap\R$ where
  \begin{align*}
    \lambda_{\min}
    &:=\min\set{\lambda>0\,:\,G_k\leq
      (p\norm{Ax_{k-1}^\delta-y^{\delta}}+(1-p)\delta)^2}\\
    \lambda_{\max}&:=\sup\set{\lambda>0\,:\,\delta^2\leq G_k(\lambda)},
  \end{align*}
  and $0<\lambda_{\min}<\lambda_{\max}\leq\infty$.
\end{proposition}

We will solve \eqref{eq:rat-ineq} by means of a Newton-type method.
Newton's method for solving \eqref{eq:rat-ineq},
would be to take some $\lambda_{k,0}>$ and to iterate
\begin{align*}
  \lambda_{k,j+1} \ = \ \lambda_{k,j}-\dfrac{G_k(\lambda_{k,j})-\delta^2}
                        {G'_k(\lambda_{k,j})}
\end{align*}
as long as \eqref{eq:rat-ineq} is not satisfied; when \eqref{eq:rat-ineq}
is satisfied, the last $\lambda_{k,j}$ is used as $\lambda_k$.
We will introduce a number of modifications in this iteration to accelerate it:
\begin{enumerate}
\item[(M1)] A ``greedy'' version of Newton's method will be used, aiming at
  $G_k(\lambda)=0$, that is, the numerator on the above fraction
  will be $G_k(\lambda_{k,j})$.
\item[(M2)] Newton's step will be dynamically over-relaxed by a factor $w_j$,
  as described below.
\item[(M3)] We choose $\lambda_{k,0}$ using Lemma~\ref{lm:lbound} for $k=0$ and
  information gathered at previous iterations of $k\geq 2$, ad
  described below.
\end{enumerate}

\noindent  Regarding modifications (M1) and (M2), while $\lambda=\lambda_{k,j}$ does not
satisfy~\eqref{eq:rat-ineq}, we use the iteration
\begin{equation} \label{eq:newton-step-or}
\lambda_{k,j+1} \ := \ \lambda_{k,j}-\omega_j\dfrac{G_k(\lambda_{k,j})}
                       {G'_k(\lambda_{k,j})} \, ,
\end{equation}
where  the over-relaxation factor $\omega_j$ is chosen as follows: \\
--- for $j=0$, $\omega_0 = 1$; \\
--- for $j \geq 1$, after computing $\lambda_{k,j}$,  $\pi_k(\lambda_{k,j})$,
and
    $G_k(\lambda_{k,j})$,\\
\mbox{\ \ \ \ \ \ \ \ }
{\bf if} \ \,$G_k(\lambda_{k,j-1})>2 \big( p\norm{A x_{k-1}^\delta - y^\delta} + (1-p)\delta \big)^2$ \\
\mbox{\ \ \ \ \ \ \ \ \ \ \ \ }
     {\bf then} \  $\omega_{j} \, = \, 2\, \omega_{j-1}$ \\
     \mbox{\ \ \ \ \ \ \ \ \ \ \ \ } {\bf else} \ \
     $\omega_{j} \, = \, 1$

\noindent  Regarding modification (M3):
     \\
     --- for $k=1$, $\lambda_{1,0}$ is the lower bound provided by Lemma~\ref{lm:lbound}; \\
     --- for $k =2$, $\lambda_{2,0}=\lambda_1$,
     \\
     --- for $k \geq 3$,
we use a linear extrapolation on $\log$ $\lambda$ from the two previous iterates
as starting point, that is,  $\lambda_{k,0} = {\lambda_{k-1}^2} / {\lambda_{k-2}}$.

The acceleration effect caused by modifications (M1), (M2) and (M3)
is illustrated in Example~\ref{ex:acceleration} below.

Over-relaxation is a well established technique for accelerating iterative
methods for solving linear and non-linear equations, the SOR method being
a classical example.
This {\color{black} fact} motivated the introduction of over-relaxation as in (M2).

In our numerical experiments, we observed that the sequence $\lambda_k$
increases exponentially. This fact motivated the use of (liner) extrapolation
(in the $\log$) for its initial value from iteration 3 on (modification (M3)).

It is worth noticing that, in step 3, each ``inner iteration'' \eqref{eq:newton-step-or}
requires the solution of either two linear systems per failed inner iteration or one
linear system at the successful last inner iteration.
On the other hand, in the gNIT method the computation of $\lambda_k = q^k$
(for some {\em a priori} chosen $q > 1$) is straightforward. Consequently,
one needs to solve one linear system (modeled by $(I + q^k A^*A)$) in each
step of the gNIT method.
This facts motivated us to use the accumulated number of linear system to measure
the performance of the different NIT method by plotting the residual and the error
as a function of this quantity (see Figure~\ref{fig:rrNIT-acceleration1}).

\begin{example} \label{ex:acceleration}
The benchmark problem presented in Example~\ref{ex:gNIT-sIT} is revisited and
solved by the rrNIT method using the Newton-method \eqref{eq:newton-step-or}
with combinations of modifications (M1)-(M3):
In Figure~\ref{fig:rrNIT-acceleration1} this inverse problem is solved using:
\\ (BLUE)
modification (M1);
\\ (PINK)
modifications (M1), (M2);
\\ (RED)
modifications (M1), (M2), (M3);
\\
Notice that, in Figure~\ref{fig:rrNIT-acceleration1}, the x-axis denotes the
accumulated number of linear systems. This choice allows an better comparison
of the efficiency of the different rrNIT implementations.
\end{example}

\begin{figure}[t]
\centerline{\includegraphics[width=\textwidth]{pictures/REVfig-enIT-acceleration2}}
\vskip-0.5cm
\caption{\small Implementation of rrNIT method: acceleration caused
by modifications (M1), (M2), (M3) to iteration \eqref{eq:newton-step-or}.}
\label{fig:rrNIT-acceleration1}
\end{figure}

Figure~\ref{fig:rrNIT-acceleration1} illustrates that the \emph{cumulative}
effect of modifications (M1), (M2), and (M3) is to accelerate the computation
of $\lambda_k$ as required in step [3] of Algorithm~\ref{alg:rrNIT-true}.

A pseudo-code version of our implementation of Algorithm~\ref{alg:rrNIT-true}
with the strategies above discussed is presented in Appendix~A for
the sake of completeness.

\section{Numerical experiments} \label{sec:num-experiments}

In this section, Algorithm~\ref{alg:rrNIT-numer} (see Appendix~A)
{\color{black} is implemented} for solving two well known linear ill-posed problems.
In Section~\ref{ssec:num-id} the {\em Image Deblurring Problem}
\cite{BeBo98, Ber09} is considered, while in Section~\ref{ssec:num-ipp}
we address the {\em Inverse Potential Problem} \cite{HeRu96, Isa06} in 2D,
which is an elliptic parameter identification problem.

In both Sections~\ref{ssec:num-id} and~\ref{ssec:num-ipp}, the performance
of Algorithm~\ref{alg:rrNIT-numer} is compared to the performance of two well
established methods: The gNIT method with $\lambda_k = 2^k$; the NIT method
proposed in \cite{DoHa13}.

\subsection{Image deblurring problem} \label{ssec:num-id}

Image deblurring problems \cite{BeBo98}
are finite dimensional problems modeled, in general, by high dimensional linear
systems of the form \eqref{eq:ip}.
In this setting,
$x \in X = \R^n$ represents the pixel values of an unknown true image, while
$y \in Y = X$ contains the pixel values of the observed (blurred) image.
In practice, only noisy blurred data $y^\delta \in Y$ satisfying \eqref{eq:noisy-data}
is available.

The matrix $A$ describes the blurring phenomenon \cite{Ber09, BeBo98}. We consider the simple
situation where the blur of the image is modeled by a space invariant point spread function
(PSF). In the continuous model, the blurring process is represented by an integral
operator of convolution type and \eqref{eq:ip} corresponds to an integral equation of
the first kind \cite{EngHanNeu96}.
In our discrete setting, after incorporating appropriate boundary conditions into the model,
the discrete convolution is evaluated by means of the FFT algorithm.
We added to the exact data (the convoluted image) a normally distributed noise with zero mean
and suitable variance for achieving a prescribed relative noise level .

The computation of our deblurring experiment was conducted using MATLAB~2012a.
The corresponding setup is shown in Figure~\ref{fig:ID-setup}:
(a) True image $x \in \R^n$, $n = 256^2$ (Cameraman $256\times256$);
(b) PSF is the rotationally symmetric Gaussian low-pass filter of size [257 257] and
standard deviation $\sigma = 4$ (command {\tt fspecial('gaussian',\,[257 257],\,4.0)});
(c) Exact data $y = Ax \in \R^n$ (blurred image).
The noise was generated used the {\tt randn} routine while the FFT was computed
using the {\tt fft2} routine.

\begin{figure}[t]
\centerline{\includegraphics[width=0.35\textwidth]{pictures/fig-ID-exact-solution}
            \includegraphics[width=0.37\textwidth]{pictures/fig-ID-psf}
            \includegraphics[width=0.35\textwidth]{pictures/fig-ID-exact-data} } \vskip0.3cm
\vskip-0.5cm \centerline{\hfill (a) \hspace{5cm} (b) \hspace{5cm} (c) \hfill}
\caption{\small Image deblurring problem: setup of the inverse problem. \
(a) Original image $x$; \ (b) Point spread function; \ (c) Blurred image $y$.}
\label{fig:ID-setup}
\end{figure}

Three distinct scenarios are considered, where the relative noise level
$\norm{y-y^\delta} / \norm{y}$ corresponds to $10^{-1} \%$, $10^{-3} \%$ and
$10^{-6}\%$ respectively (in the third scenario, the choice of the noise level
is motivated by MATLAB's double-precision accuracy).

In Figure~\ref{fig:ID-numerics} the following methods are compared for the third
scenario:
(BLACK) gNIT with $\lambda_k = 2^k$;
(RED) rrNIT method in Algorithm~\ref{alg:rrNIT-numer} (with $p = 0.2$);
(BLUE) Hanke-Donatelli NIT method in \cite{DoHa13}.
The pictures in Figure~\ref{fig:ID-numerics} show:
(TOP) relative error $\norm{x^\star - x_k^\delta} / \norm{x^\star}$;
(BOTTOM) residual $\norm{A x_k^\delta - y^\delta}$.
The x-axis in the these pictures is scaled by the accumulated number of linear systems solved.
This choice allows an easier comparison between the efficiency of the different methods.

All methods are stopped according to the discrepancy principle with $\tau = 3$.
As initial guess we choose $x_0 = y^\delta$ (the noisy blurred image).

The numerical results concerning all three scenarios are summarized in Table~\ref{tab:id}.
In this table we show, for each scenario, the total number of linear systems solved, as
well as the number of iterations needed to reach the stop criteria.

\begin{table}[h]
\begin{center}
\begin{tabular}{c  c c c}
\hline
     {$\delta$}    &    gNIT   & NIT in \cite{DoHa13} &   rrNIT   \\
\hline
$10^{-1} \%$ \ \ \ & \ 6 (\ 6) &        15 (\ 5)      & \ 7 (\ 4) \\
$10^{-3} \%$ \ \ \ &  17 (17)  &        23 (\ 7)      &  11 (\ 7) \\
$10^{-6} \%$ \ \ \ &  36 (36)  &        43 (11)       &  16 (11)  \\
\end{tabular}
\end{center}
\caption{Image deblurring problem: total number of linear system solves
for different noise levels with the number of iterations in parentheses.}
\label{tab:id}
\end{table}

\begin{figure}[t]
\centerline{\includegraphics[width=\textwidth]{pictures/REVfig-ID-delta1e-6} }
\vskip-0.5cm
\caption{\small Image deblurring problem: {\color{black} third scenario $\delta = 10^{-6} \%$}.
(TOP) Relative iteration error; (BOTTOM) Residual.}
\label{fig:ID-numerics}
\end{figure}

The restored images for the third scenario ($\delta = 10^{-6} \%$) are presented
in Figure~\ref{fig:ID-restored}. From left to right: gNIT, NIT in \cite{DoHa13},
and rrNIT.
\begin{figure}[t]
\centerline{\includegraphics[width=0.35\textwidth]{pictures/REVfig-ID-restored-a}
            \includegraphics[width=0.35\textwidth]{pictures/REVfig-ID-restored-b}
            \includegraphics[width=0.35\textwidth]{pictures/REVfig-ID-restored-c} } \vskip0.3cm
\vskip-0.5cm \centerline{\hfill (a) \hspace{5cm} (b) \hspace{5cm} (c) \hfill}
\caption{\small Image deblurring problem: restored images for the third scenario
($\delta = 10^{-6} \%$). \ (a) gNIT, \ (b) NIT in \cite{DoHa13}, \ (c) rrNIT.}
\label{fig:ID-restored}
\end{figure}

\subsection{Inverse potential problem} \label{ssec:num-ipp}

In what follows we address the {\em inverse potential problem}
\cite{FSL05, CLT09, HeRu96, DAL09}.
Generalizations of this inverse problem appear in many relevant applications
including Inverse Gravimetry \cite{Isa06, DAL09}, EEG \cite{EF06}, and EMG \cite{DAP08}.

The forward problem considered here consists in solving on a Lipschitz domain
$\Omega \subset \R^d$, for a given source function $x \in L_2(\Omega)$, the
boundary value problem
\begin{equation} \label{eq:ipp}
-\Delta u \ = \ x \, ,\ {\rm in} \ \Omega \, , \quad
u \ = \ 0 \, \ {\rm on} \ \partial\Omega\, .
\end{equation}
The corresponding inverse problem is the so called {\em inverse potential problem}
(IPP), which consists of recovering an $L_2$--function $x$, from measurements of
the Dirichlet data of its corresponding potential on the boundary of $\Omega$, i.e.,
$y := u_{\nu} |_{\partial\Omega} \in L_2(\partial\Omega)$.
This problem is modeled by the linear operator
$A: L_2(\Omega) \to L_2(\partial\Omega)$ defined by
$A x := u_{\nu} |_{\partial\Omega}$, where $u \in H_0^1(\Omega)$ is the
unique solution of \eqref{eq:ipp} \cite{HeRu96}.  Using this notation,
the IPP can be written in the abbreviated form \eqref{eq:ip}, where the
available noisy data $y^\delta \in L_2(\partial\Omega)$ satisfies
\eqref{eq:noisy-data}.

In our experiments we follow \cite{CLT09} in the experimental setup, selecting
$\Omega = (0,1) \times (0,1)$ and assuming that the unknown parameter $x^\star$
is an $H^1$-function with sharp gradients shown in Figure~\ref{fig:IPP-setup}
(a).
After solving Problem~\eqref{eq:ipp} for such $x= x^\star$, we added to the exact
Dirichlet data a normally distributed noise with zero mean and suitable variance
for achieving a prescribed relative noise level.
In our numerical implementations we set $p = 0.1$, $\tau = 3$ (discrepancy
principle constant) and the initial guess $x_0 \equiv 1.5$ (constant function in $\Omega$).

As in Section~\ref{ssec:num-id}, three distinct scenarios are considered, where
the relative noise level $\norm{y-y^\delta} / \norm{y}$ corresponds to
$10^{-1} \%$, $10^{-3} \%$ and $10^{-6}\%$ respectively.

In Figure~\ref{fig:IPP-numerics} the following methods are compared for the second
scenario:
(BLACK) gNIT with $\lambda_k = 2^k$;
(RED) rrNIT method in Algorithm~\ref{alg:rrNIT-numer} (with $p = 0.1$);
(BLUE) NIT method proposed in \cite{DoHa13}
\footnote{As before, this NIT method was implemented with $q = 0.6$ and $\rho = 10^{-4}$.
For the computation of the Lagrange multipliers, a scalar equation was solved using an
over-relaxed Newton method and a precision of $5 \%$.}
What concerns Algorithm~\ref{alg:rrNIT-numer}, the corresponding iterate $x_5^\delta$ and
iteration error $|x^\star - x_5^\delta|$ are shown in Figure~\ref{fig:IPP-setup}~(b)
and~(c) respectively.

The pictures in Figure~\ref{fig:IPP-numerics} show:
(TOP) relative error $\norm{x^\star - x_k^\delta} / \norm{x^\star}$;
(BOTTOM) residual $\norm{A x_k^\delta - y^\delta}$.
The x-axis in the these pictures is scaled by the accumulated number of linear systems solved.

The numerical results concerning all three scenarios are summarized in Table~\ref{tab:ipp}.
In this table we show, for each scenario, the total number of linear systems solved, as
well as the number of iterations needed to reach the stop criteria.

\begin{table}[h]
\begin{center}
\begin{tabular}{r  c c c}
\hline
          {}       &    gNIT   & NIT in \cite{DoHa13} &   rrNIT   \\
\hline
$10^{-1} \%$ \ \ \ & \ 6 (\ 6) &        11 (\ 3)      &   6 (\ 3) \\
$10^{-3} \%$ \ \ \ &  10 (10)  &        34 (\ 5)      &  10 (\ 5) \\
$10^{-6} \%$ \ \ \ &  13 (13)  &        86 (\ 7)      &  12 (\ 6)  \\
\end{tabular}
\end{center}
\caption{Inverse potential problem: total number of linear systems for
different noise levels with the number of iterations in parentheses.}
\label{tab:ipp}
\end{table}

\subsection{Remarks} \label{ssec:num-remarks}

In the two above discussed inverse problems, for all scenarios, both {\em a posteriori}
NIT type methods (rrNIT in Algorithm~\ref{alg:rrNIT-numer} and NIT in \cite{DoHa13})
require similar number of steps to reach discrepancy.
However, the total numerical effort of rrNIT is much smaller than the one of
NIT in \cite{DoHa13}, and is comparable to the total numerical effort of the
gNIT method (see Tables\ref{tab:id} and~\ref{tab:ipp}).

Specially in the third scenario (small noise level), the NIT method \cite{DoHa13}
needs several Newton steps to compute the Lagrange multipliers in the final iterations.
Algorithm~\ref{alg:rrNIT-numer}, on the other hand, needs only {\color{black} a few}
Newton steps to compute each one of the Lagrange multipliers solving \eqref{eq:rat-ineq}.
\smallskip

Notice the exponential decay of the residual in the rrNIT method (Figures~\ref{fig:ID-numerics}
and~\ref{fig:IPP-numerics}), which is in accordance to Proposition~\ref{prop:decay-residual}.
We also observed exponential growth of the corresponding Lagrange multipliers.
\smallskip

The NIT method proposed in \cite{DoHa13} was implemented with $q = 0.6$ and
$\rho = 10^{-4}$ as described in \cite[Sec.5]{DoHa13}. For the computation of
the Lagrange multipliers, a scalar equation was solved using an over-relaxed Newton
method with a precision of $1\%$ (Section~\ref{ssec:num-id}) and $5 \%$
(Section~\ref{ssec:num-ipp}).

\begin{figure}[t]
\centerline{\includegraphics[width=1.25\textwidth]{pictures/REVfig-IPP-setup} }
\vskip-0.4cm \centerline{\hfil (a) \hspace{5.0cm} (b) \hspace{5.0cm} (c) \hfil}
\vskip-0.2cm
\caption{\small Inverse potential problem: second scenario $\delta = 10^{-3} \%$.
\ (a) Exact solution $x^\star$;
\ (b) Approximate solution $x_5^\delta$ (rrNIT Algorithm~\ref{alg:rrNIT-numer});
\ (c) Iteration error $|x^\star - x_5^\delta|$ {\color{black} (absolute error pixel-wise)}.}
\label{fig:IPP-setup}
\end{figure}

\begin{figure}[ht]
\centerline{\includegraphics[width=\textwidth]{pictures/REVfig-IPP-delta1e-3} }
\vskip-0.5cm
\caption{\small Inverse potential problem: second scenario $\delta = 10^{-3} \%$.}
\label{fig:IPP-numerics}
\end{figure}

\section{Conclusions} \label{sec:conclusion}

We investigate NIT type methods for computing stable approximate
solutions to ill-posed linear operator equations.  The main
contributions of this article is a novel strategy for choosing a
sequence of Lagrange multipliers for the NIT iteration, allowing each
of this multipliers to belong to a non-degenerate interval.
We also
derived an efficient numerical algorithm based on this strategy
for computing the required Lagrange multipliers.

We prove monotonicity of the proposed rrNIT method, and exponential
decay of the residual $\norm{A x_k^\delta - y^\delta}^2$.
Moreover, we provide estimates to the``gain''\,
$\norm{x^\star - x_{k-1}^\delta}^2 - \norm{x^\star - x_k^\delta}^2$,
and to the Lagrange multipliers $\lambda_k$. A convergence proof
in the case of exact data is provided.

An algorithmic implementation of the rrNIT method is proposed
(Algorithm~\ref{alg:rrNIT-numer}, where the computation of Lagrange
multipliers are accomplished using an over relaxed Newton-like method,
with appropriate choice of the initial guess.
The resulting rrNIT method is competitive with gNIT and also with other
commonly used \emph{a posteriori} method ; not only from the point of
view of the total number of iterations, but also from the point of
view of the overall numerical effort required.

Our algorithm is tested for two well known applications with three noise
levels: the inverse potential problem, and the image deblurring
problem. The results obtained validate the efficiency of our method.

\section*{Acknowledgments}

The work of R.B. is supported by the research council of  the Alpen-Adria-Universit\"at
Klagenfurt (AAU) and by the Karl Popper Kolleg ``Modeling-Simulation-Optimization''
funded by the AAU and by the Carinthian Economic Promotion Fund (KWF).
A.L. acknowledges support from the research agencies CAPES, CNPq
(grant 311087/2017-5), and from the AvH Foundation.
The work of B.F.S. was partially supported by CNPq (grants 474996/2013-1,
302962/2011-5) and FAPERJ (grant E-26/102.940/2011).

We thank the anonymous referees for the constructive criticism and corrections
which improved the original version of this work.

redpajama_set_name: RedPajamaArXiv


text:
California Governor Steps Up for LGBT Youth
"Late last night, California Gov. Jerry Brown signed into law Senate Bill 1172, a bill to protect young people from dangerous and disproved efforts designed to change their sexual orientation or gender expression that have proven to inflict great emotional harm, which we believe to be child abuse."
From: TransEpiscopal
To: Susan Russell
"In honor and memory of Louise Brooks and with deepest gratitude for the support and encouragement offered by both of you in and through Out of the Box."
Just completed a three-day "strategic planning meeting" with a great cloud of Integrity witnesses. (more on that later.) Deeply grateful for hard work of claiming legacy of past while imagining new frontiers in the future. That and moved and grateful to TransEpiscopal for the gift of this icon as an outward and visible sign of the ongoing transfiguration at work in the Episcopal Church through the work and witness of those committed to proclaiming the Good News of God's inclusive love available to absolutely everybody!
An Evening w/Gene Robinson at All Saints Church!
Friday, October 5 | 7:30 p.m.
In his new book "God Believes in Love: Straight Talk on Gay Marriage," Bishop Gene Robinson makes the case for same-sex marriage through the lens of faith and morality — helping readers to better understand why full marriage equality is vitally important — and why it is a violation of civil rights to deny access to marriage because of sexual orientation.
In his visit to Live Talks Los Angeles, he'll address important, frequently-asked questions like: "What's wrong with civil unions?" and "Why should straight people care about gay marriage?" … in a conversation moderated by actor, activist and All Saints Church member Bradley Whitford.
Plan to come and spend an evening with one of our generation's most prophetic voices for inclusion and welcome Bishop Robinson back to All Saints Church. Details here.
Space Shuttle over Pasadena
[photo credit: Jeff from CPK]
I missed the shuttle flyover here in Pasadena which happened while I was in New York ... but here's a great shot Jeff @CPK got ... which he had displayed in the restaurant this afternoon when I was picking up my lunch and I asked him to send me a copy and he did.
Cool, eh?
What Mitt Romney & Mark Lawrence have in common:
You know how Mitt Romney has that secret plan for America he can't tell us any details yet but we should trust him? Turns out Mark Lawrence has one of those for the Diocese of South Carolina, too!
We announced last month on August 20th that the Standing Committee and I were in agreement on a course of action regarding the future of the Diocese of South Carolina and the challenges many of us face because of decisions by the recent General Convention of the Episcopal Church.
However, for many reasons it was then and is now, imprudent to reveal that course of action. Things are progressing—we have not stopped or dropped the ball.
Please know that I understand the level of anxiety and concern of many in the diocese. Nevertheless I must ask you all for your continued patience and prayers as we seek to deal wisely and carefully with a fluid situation that requires great discernment and sensitivity on a regular basis. I will communicate to you the details at the very earliest moment such a communication is prudent.
Faithfully yours in Christ,
--(The Rt. Rev.) Mark J. Lawrence, Bishop of South Carolina
h/t Titusonenine
In NYC ...
Earth to Mitt Romney:
Away from the news all day I'm sitting here in near disbelief at the callous disregard of and utter contempt for 47% of the American people demonstrated by the words of presidential candidate Mitt Romney.
Ask my Iraq War veteran Obama supporting son if he's a victim. Ask my working overtime in the auto industry son if he's a victim. As for me, the only "victim" in this sad scenario is the truth that John 8:32 tells us will set us free ... Where do I order my "We Are The 47%" t-shirt, coffee mug and bumper sticker?
Homecoming 2012 ... What WILL you do with your one wild, precious life?
Today was my eleventh Homecoming Sunday at All Saints Church -- I celebrated my 10th anniversary on August 1st -- and as always it was a grand conglomeration of festive Eucharistic celebration, food, fellowship and faith in action. Here are a few random snapshots:
And here's the rector's sermon -- arguably a conversation between Mary Oliver and Mark 8:27-38. The quote that's getting the most "re-tweets" over on twitter is: "Religion is for people trying to stay out of hell. Spirituality is for people who've been through hell."
Finally, in answer to all the "how are you doing?" questions all I can really say is "I'm doing." One thing at a time -- an inch at a time. As I told friends at dinner last night, the best analogy I can come up with is I feel like I'm waiting for the novocaine to wear off: everything is a little numb; everything is a little slow; everything is a little weird. And one minute I'm kind of OK and then next minute I'm totally not. And that's just going to be how it is until it isn't that way anymore.
I am more grateful than I can find words for all the love, support and care received -- and even more grateful for all the tributes and affirmation of Louise's life, work and witness. Of course I thought about that when Ed preached about our "one wild, precious life" ... and thought that nobody summed it up better than Michael Hopkins did in the words he spoke at her memorial service on September 8 ... words he shared on his blog which included:
Louise's great gift was enabling people who have no voice to have one. She took flesh and enabled it to have a word, and in doing so, justice was made. You might notice that is backward from "the Word was made flesh," but that is how it works. The Word was made flesh so that flesh might speak the word, or, to use another important concept from John's Gospel, The Truth became flesh so that those in the flesh might speak the truth. And when that happens we see glory and justice is made.
Happy Homecoming!
The Gospel According to Hillary Clinton
Let me be abundantly clear: There is absolutely no excuse, rationale or alibi that justifies violence against other members of the human family -- and faith based violence is categorically antithetical to the core values of ALL world religions.
That said, the events of this week in Libya, Egypt and across the Middle East call us to look at the reality of our interconnectedness as a global community -- and consider how and where we can be agents of peace, justice and compassion: the truly "traditional values" of all three Abrahamic faiths. Here's how journalist Joseph Hanania explains the challenge in a story filed yesterday with Aslan Media:
In what businessmen might call cross-promotion, Muslim fundamentalists spread the word about this Christian fundamentalist video. And Hollywood, with its multi-million dollar promotional budgets, could not have been more successful. As word spread, the video scored over one million hits. The results were just as predictable as shouting "fire!" in a crowded theater.
Amid the smoke, Muslim fundamentalists launched an apparently planned military attack. Four American diplomats were murdered, including our popular ambassador, J. Christopher Stevens, who had helped overthrow the Libyan dictator. American fundamentalists lit the match and tossed that match into the pile. Libyan (and other Arab) fundamentalists fanned the flame as cover for their planned 9/11 attack. The two groups had worked hand in hand to create this result.
So what can we do? Are we powerless against these forces of fundamentalism that infect all our faith traditions? Are we doomed to stand by helplessly as flames are fanned and fingers are pointed while God's dream of a human family as beloved community is unrealized at best and undermined at worst?
Not according to the U.S. Secretary of State -- who offered nothing less than The Gospel According to Hillary in her speech marking the end of Ramadan on September 13:
"Refraining from violence is not a sign of weakness in one's faith; it is absolutely the opposite, a sign that one's faith is unshakable. We can pledge that whenever one person speaks out in ignorance and bigotry, ten voices will answer. They will answer resoundingly against the offense and the insult; answering ignorance with enlightenment; answering hatred with understanding; answering darkness with light."
"In times like these, it can be easy to despair that some differences are irreconcilable, some mountains too steep to climb; we will therefore never reach the level of understanding and peacefulness that we seek, and which I believe the great religions of the world call us to pursue. But that's not what I believe, and I don't think it's what you believe. Part of what makes our country so special is we keep trying. We keep working. We keep investing in our future."
This is not apologizing for America's values -- this is preaching them. This is the kind of leadership we need to debunk what Ed Bacon calls "the myth of the separate self" and instead build up authentic communities of shared values of justice, peace and compassion. This IS the change we can not only believe in but live into. And it is ours to choose.
So let us "choose this day" to proclaim the Gospel According to Hillary: to answer ignorance with enlightenment; to answer hatred with understanding and to answer darkness with light.
In today's mail: City of Los Angeles Tribute to Louise Emerson Brooks
Posted by SUSAN RUSSELL at 10:21 PM 1 comment:
From Our Archives: Interfaith Peace Service
In 2007 over 600 Muslims, Christians and Jews gathered together to pray for peace and to celebrate the common heritage of their shared Abrahamic faiths. This video is offered today in the ongoing hope that the prayers for peace will overcome the calls for violence and we will one day become the human family we were created to be.
Whoever you are and wherever you find yourself ...
... there is a place for you here!
This video was just "launched" for use in our newcomer/welcome ministry at All Saints Church in Pasadena -- and I thought it WAY too fabulous to hide under a bushel basket. (See also: Matthew 5:16)
And ... just in case you're in the neighborhood ... next Sunday (September 16th) is Homecoming Sunday ... Join Us!
Reflecting on 9/11: Strength for the Journey and Hope for the Future
Ten years ago today I celebrated for the first time at the All Saints Church altar. The occasion was the first anniversary of what has come to be known as simply "9/11" -- and this is the reflection I wrote following that service.
The candles massed in front of the altar burn in tribute to the names being read from the lectern – Naomi Leah Solomon, Daniel W. Song, Michael C. Soresse, Fabian Soto – as other names scroll above the altar projected on a video screen – John Bentley Works, William Wren, Sandra Wright, Myrna Yashkulka.
The church is silent save for the reading of the names and the careful footsteps of those who come forward to light a candle -- the gentle thud of a kneeler lowered for prayer --the quiet rustle of pages turned in a prayer book.
"American Airline Flight 11"– Anna Allison, David Lawrence Angell, Lynn Edwards Angell, Seima Aoyamma. The names began at 5:46 – the west coast moment when the first plane struck – and will continue through the morning until we gather for Eucharist at noon. The table is already set. The red frontal – blood of martyrs – covers the altar. The chalice is vested, the missal marked. The credence table is ready, too: flagons of wine, silver chalices and ciborium lined up – ready to hold the holy food and drink of new and unending life we will share here at All Saints Church.
"All Saints." Charles' deep voice breaks the silence as he begins reading the next segment of the list of names: "World Trade Center, continued" – Paul Riza, John Frank Rizzo, Stephen Luis Roch, Leo Roberts. I remember the ancient words of comfort from the prophet Isaiah, "I have called you by name and you are mine." As Charles tolls the names of the dead that assurance echoes again and again in my head. These names I do not know – some I cannot even pronounce – each and every one known to God. Beloved of God.
"United Airlines Flight 93": Christine Adams, Lorraine Berg, Todd Beamer, Alan Beaven. Gone from our sight yet gathered into God's embrace -- seated at the heavenly banquet we can but glimpse through the sacrament we are preparing to share -- the offering of praise and thanksgiving we will present at this altar.
I look again at the ciborium massed on the credence table – the candles flickering in the polished silver – the light of lives lost reflected in the vessels holding the bread of life. It staggers the mind to consider what they represent – the magnitude of the collective loss of love, joy, hope and possibilities taken on that day a year ago with such sudden unexpectedness.
Takashi Ogawa. Albert Ogletree. Gerald Michael Olcott. The pain of death and loss mingles mysteriously in the promise of life and hope. Body and Blood. Bread and Wine. Strength for the journey and hope for the future. Hope for a world where differences enrich rather than divide. Hope for the end of wars waged in the name of the God who created us not to destroy but to love each other.
Dipti Patel. James Matthew Patrick. Sharon Christina Millan Paz. "Whoever you are and wherever you find yourself on your journey of faith there is a place for you here." Thanks be to God. Alleluia. Amen.
How can I thank you? Let me count the ways ...
While every day is arguably the first day of the rest our lives, some days are more "loaded" with significance in that regard than others... and so I want to start out this particular first day of the rest of my life with this expression of gratitude.
And if there was world-enough-and-time (being the well brought up young lady my mother hoped I would be) this would not be a blog but individual handwritten (on Crane's informal stationery) notes to absolutely everybody who surrounded Louise with prayers during her illness and surgery and blessed us in our grief at her passing.
So for her family -- for myself, for Louise's brother Fred and her sister-friend Julena, let me offer our profound thanks and gratitude ...
To all who prayed without ceasing ... in the blogosphere and at the bedside. We felt your love and support and I know Louise did, too.
To the phenomenal doctors, nurses and health care teams at the Norris Cancer Center and at Keck/USC Hospital. who did absolutely everything they could until there was nothing left to do -- and then with grace and abundant sensitivity enabled us to send her off with peace and dignity.
To the entire All Saints Church community for gifts too many to number -- prayers, support, space, grace and love. For mobilizing on what was arguably one of the busiest weeks of the launching-program year to make Saturday's celebration of her life so exactly what she wanted it to be that we shattered our own oft-repeated mantra of "progress not perfection." It was so perfect I don't believe even Louise would have had any "post-production notes" to go over ... other than to say thank you to everybody.
To those who dropped everything and came to be with us -- especially Michael and John Clinton, who are family in ways that transcend DNA; colleagues Lis Jacobs, Caroline Hall, Sharon Groves, Neil Thomas, Denise Eger, Joel Kushner, Ellie Steinman; Roland Stringfellow, Mel White & Gary Nixon; Jay Johnson; Lynn Jay; Mark Kowalewski & Dan Ade and so many others; Deacon Vicki Gray for blessing us with her ministry; Mary B and April; Kay Sylvester for her presence and prayer beads and for all the diocesan colleagues who reached out and showed up; my brother Bill who schlepped from Santa Ynez; Carissa and Pam for their incarnational and pastoral "being there" for us; Christina for her ministry of presence; Bear and Susan for sharing so much of the journey of this last 10 months; Susan for her ministry of healing and friendship; Rose for her faithful friendship to Louise and Julena over so many years and for remembering we all needed to eat something; Edna and Stasia for tending our confused Luna and Juno during the vortex and taking that anxiety away;
To those who made the service exactly what she wanted it to be ... the All Saints staff -- driven by Melissa and Zelda as the Liturgy and Pastoral Care "leads; Gary who did the amazingly lovely flowers; the reflectors who helped tell the story of Louise in the context of the liturgy: Julena, Jacqui and Michael; the readers who made the words of Steven Charleston and Ralph Waldo Emerson "come alive" -- Jim and Christina and Alma for being such a powerful prayer-warrior; Candace who sent the Charleston piece that was the perfect "first reading;" the musicians who helped us "pray twice" -- James, Liz and our dear friend Dr. Bobby Rodriquez; vergers, ushers, acolytes and communion ministers and Randy, the best thurifer ever; Ed for the heartfelt and "on message" homily and +Gene for being there in both words and spirit; +Mary for the power of her presence and prayers -- in the service on Saturday and at the bedside the Sunday before; all who stepped up and told stories and shared what they loved and appreciated about Louise during the reception; Chris for organizing every last detail of the reception and all who served, schlepped, and re-scheduled in order to make it happen and to Becki for knowing when we needed what we didn't even know to ask for.
To all those members of Louise's extended family who brought their love and support into the circle of light and love that surrounded those of us in her "immediate family" in the stress and grief; cousins Fred and Chris who made the trek from Colorado and Wisconsin and their families who held down the fort at home; Rose and family who have been "familia" to Louise for so many years; long time friends Arlana and Andrea and Tim who was with us in spirit from Seoul and Colleen and Kelli who were with us in fact. And a particular shout out to Leon and Jackson who she loved so very much and who carried Julena and me through the last few weeks on the wings of their love and support.
To all who wrote tributes to Louise's work and witness providing such rich tapestry of a life well-lived and cut too short. Cameron Partridge; Caroline Hall; Elizabeth Kaeton; California Faith for Equality; Episcopal Cafe; and many others.
To all who contributed in her memory to Integrity for the "Out of the Box" project or to Husky Camp for the rescue of at-risk huskies and malamutes.
Finally, to President Barack Obama for the Affordable Care Act that made Louise's battle with kidney cancer in 2000 a part of her medical history that informed her treatment in 2012 -- NOT a pre-existing condition that prevented her treatment in 2012. Because of the ACA, Louise not only had options for her fight with cancer -- she had the health insurance she needed to exercise those options.
And so today as we -- those who loved her most and still can't believe she's really gone -- spend this first day of the rest of our lives figuring out what to do next, please, please, PLEASE ... help honor Louise's life by committing to continue to make health care available for all as we live out Ralph Waldo Emerson's credo: "Give all to love; obey thy heart."
Louise Emerson Brooks: This is to have succeeded
THE GOSPEL ACCORDING TO RALPH WALDO EMERSON
To laugh often and much; To win the respect of intelligent people and the affection of children; To earn the appreciation of honest critics and endure the betrayal of false friends; To appreciate beauty, to find the best in others; To leave the world a bit better, whether by a healthy child, a garden patch, or a redeemed social condition; To know even one life has breathed easier because you have lived. This is to have succeeded. --Ralph Waldo Emerson.
A descendent of the author, Louise Emerson Brooks months ago selected this piece of his to be read at her memorial. It was part of the liturgy celebrating her life at All Saints Church -- and I wanted to share it here as I am overwhelmed with both sadness and gratitude. Sunday I will be back at church for a morning which will include the launch of Ed Bacon's new book "8 Habits of Love" -- a handbook for living a spiritual life from the house of love rather than the house of fear ... and a recipe for the kind of success Ralph Waldo Emerson wrote about and Louise Emerson Brooks lived. Finally, here's a link to the slide show of slide show that was part of the celebration of her life: photos represent the tip of the iceberg of a life well lived and cut too short.
"Give all to love; obey thy heart."
Surrounded by love and light, Louise Emerson Brooks died on September 2 from complications following cancer surgery. Born in upstate New York to Carson and Patricia Brooks, Louise was brought up in Richmond, Virginia with brothers Judson, Carson and Frederick. Raised in the Episcopal Church, her childhood rector was John Shelby Spong. Like her mother and grandmother before her, Louise attended the Rogers Hall School for Girls in Lowell, Massachusetts.
She fondly recalled summer visits to Los Angeles with her Uncle Hal – actor Hall Smith who played Otis Campbell on The Andy Griffith Show – and was "bit by the show business bug" at an early age. After earning her degree in theater arts from the University of Richmond, Louise moved to Southern California to pursue an eclectic media career in radio, television and filmmaking.
She landed in Santa Monica where she met her best-friend-for-life Julena Lind -- and they began compiling an impressive resume of shared road trips, board games, holidays and major life events. In 1980 she broke ground as one of the first women producers for ABC's Wide World of Sports at the Lake Placid Olympics – and whether it was producing a radio talk show or daytime dating show, writing a screenplay or consulting on an international game show, Louise brought her gifts and skills as a consummate media professional.
Her credits included "The Love Connection" and "Leeza" – for which she received an Emmy nomination. A member of the Screen Writer's Guild she also taught screenwriting and earned a law degree from Loyola Law School of Los Angeles. One of her great career delights was combining her experience as a producer with her love of travel in her work for Warner Brothers as an international format production consultant – work that took her from Paris to Mumbai to Kuala Lumpur to Santorini.
Louise found her way back to the church of her childhood, becoming an active member at All Saints Episcopal Church in Beverly Hills where she was a founding member of the congregation's gay and lesbian ministry – and also served as a lay Eucharistic minister. It was in the context of that work that she met Susan. After knowing each other for a number of years as friends and diocesan colleagues they began dating in early 2003 – and they celebrated with great joy the blessing of their relationship … surrounded by friends and family … on February 18, 2006 at All Saints Church in Pasadena.
In recent years, Louise applied her vocational gifts to the work of media training and consulting for justice concerns, serving as the communication director for California Faith for Equality during the Prop 8 campaign and working with the Human Rights Campaign nationally. In the Episcopal Church she was a valiant leader in the movement for full inclusion for LGBT people and her legacy includes the Via Media teaching videos and the award winning "Voices of Witness" documentary series. She was extremely proud of the work she did when the IRS came after All Saints Church in 2005 and she counted as one of her greatest lifetime achievements equipping Ed Bacon to stay on message.
Her most recent project – "Voices of Witness: Out of the Box" -- was released in June and was influential in securing legislation to include gender identity and expression in the non-discrimination canons of the Episcopal Church. It will continue to change hearts and minds as it moves into wider distribution with study guides in both English and Spanish and a Spanish translation is in the works.
Louise's passions included politics, travel and football; spending time with her beloved Luna, Juno and Baby; beating Julena at Scrabble, the Name Game or basically any game; and pointing out to Susan that the other freeway lane was moving faster. She was a loyal friend, loving sister and partner and faithful truth teller. She loved good wine, action movies, and using her gifts to make a difference.
Louise is survived by her partner of nine years Susan Russell and stepsons Jim and Brian; her brother Fred Brooks, his wife Joy and son Clayton; her decades-long sister-friend Julena Lind, her husband Leon and son Jackson; along with scores of colleagues and friends.
In our grief at her loss, we take comfort in the words of Bishop Steven Charleston, which will be part of the celebration of her life at All Saints Church in Pasadena on Saturday, September 8 at 11:00 a.m.
I rise to sing the joy of simply being human. I do not deny the pain I have known, the losses and the grief, but I will not let bitterness be all that I taste of life. There are too many sunsets that leave me standing still before their beauty, too many babies wheeled by in strollers that make me smile, too much love in the eyes I see from faces wrinkled by time but timeless in their wisdom. All of creation calls me to celebrate. All of life invites me to dance. Therefore I choose to carry my pain with head held high and walk the blessing way until the last lights fade.
Posted by SUSAN RUSSELL at 7:29 AM 23 comments:
Louise Emerson Brooks: The work goes on ...
As Louise was undergoing chemotherapy in early 2012, she completed her last major project, a video on transgender people of faith entitled "Voices of Witness: Out of the Box."
In her final weeks, Louise had a strong desire to extend the video's message to Spanish-speaking people of faith. She began to fundraise in order to create an edition that had Spanish subtitles, accompanied by a Spanish-language study guide. We invite you to help us take her vision forward and extend Louise's vision and ministry to a wider audience.
On the Integrity website, you can donate to Integrity USA and designate the donation to "Brooks video production fund." This fund will support the production and distribution of OOTB in Spanish and English versions, along with their study guides. Thank you for your action in support of Louise's continuing vision and ministry.
In Celebration of the Life of Louise Emerson Brooks
There are no words to express the depths of our gratitude for your words of support and love for us and in tribute to Louise's extraordinary life. We will celebrate her life with a service at All Saints Church in Pasadena on Saturday, September 8th at 11am with a reception to follow.
Please come prepared to grieve her loss, celebrate her life -- and to tell stories about the part of Louise Emerson Brooks that touched your life. That is quite literally what she said she wanted -- as the producer in her was very clear that she wanted a "production credit" for the liturgical celebration of her life. For more details call All Saints at 626.796.1172
For those who have asked about memorial gifts, two of the great passions of her life were the work of LGBT inclusion and the work of animal rescue. Contributions to Integrity will help support the distribution of her last documentary "Voices of Witness: Out of the Box" and contributions to Husky Camp will support the ongoing Siberian Husky Rescue that brought us our beloved Luna and Juno.
Integrity USA in memory of Louise Brooks
http://integrityusa.org/
838 East High St. #291 Lexington, KY 40502
Husky Camp -- Siberian Husky Rescue
http://www.HuskyCamp.com/
SHRRCA 303 Cameo Dr., Danville, CA 94526
Louise Emerson Brooks: 1946-2012
After a valiant fight and surrounded by love and light my Louise left us on Sunday, September 2nd shortly after 3:30pm. We are devastated and so very grateful for the tremendous care she received, the abundant prayers that surrounded her and for the peaceful and painless end of a long, courageous struggle.
Give rest, O Christ, to your servant with your saints: where sorrow and pain are no more; neither sighing but life everlasting.
Today's Bible Verse for the Campaign Trail
Luke 16:10 "Whoever can be trusted with very little can also be trusted with much, and whoever is dishonest with very little will also be dishonest with much."
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