\documentstyle[twoside]{article}
\pagestyle{myheadings}
\begin{document}
\markboth{\hfil OPTIMAL ORDER OF CONVERGENCE\hfil EJDE--1993/04}% 
{EJDE--1993/04\hfil C.W. Groetsch and  O. Scherzer\hfil}
\ifx\Box\undefined \newcommand{\Box}{\diamondsuit}\fi
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations}\newline
Vol. 1993(1993), No. 04, pp. 1-10.  Published October 14, 1993.\newline
ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113 
} \vspace{\bigskipamount} \\
The Optimal Order of Convergence for Stable Evaluation of Differential
Operators
\thanks{{\em 1991 Mathematics Subject Classifications:} 
Primary 47A58, Secondary 65J70.\newline\indent
{\em Key words and phrases:} Regularization, unbounded operator, 
 optimal convergence, stable.\newline\indent
\copyright 1993 Southwest Texas State 
University  and University of North Texas\newline\indent
Submitted: June 14, 1993.\newline\indent
Supported by the Austrian Fonds fur F\"orderung der
wissenschaftlichen Forschung,\newline\indent
project P7869-PHY, the Christian Doppler 
Society, Austria (O.S.), and NATO\newline\indent
project CRG-930044 (C.W.G.)
} }
\date{}
\author{C.W. Groetsch and O. Scherzer} 
\maketitle 
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\def\eps{\epsilon}
\def\Lra{\Leftrightarrow}
\def\notto{\to\!\!\!\!\!\!/}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{example}{Example}[section]

\begin{abstract}
An optimal order of convergence result, with respect to the error level in
the data, is given for a Tikhonov-like method for approximating values of an
unbounded operator. It is also shown that if the choice of parameter in the
method is made by the discrepancy principle, then the order of convergence of
the resulting method is suboptimal. Finally, a modified discrepancy principle
leading to an optimal order of convergence is developed.
\end{abstract}

\section{Introduction}

\setcounter{equation}{0}

Suppose that $L:{\cal{D}}(L) \subseteq H_1 \to H_2$ is a closed densely
defined unbounded linear operator from a Hilbert space $H_1$ into a Hilbert
space $H_2$. The problem of computing values $y=Lx$, for $x \in
{\cal{D}}(L)$, is then ill--posed in the sense that small perturbations in
$x$ may lead to data $x^\delta$ satisfying $\|x-x^\delta\| \leq \delta$, but
$x^\delta \notin {\cal{D}}(L)$, or, even if $x^\delta \in {\cal{D}}(L)$, it
may happen that $Lx^\delta \notto \;Lx$ as $\delta \to 0$, since the
operator $L$ is unbounded.

Morozov has studied a stable method for approximating the value $Lx$ when
only approximate data $x^\delta$ is available (see \cite{7} for information
on Morozov's work). This method takes as an approximation to $y=Lx$ the
vector $y_\alpha^\delta = L z_\alpha^\delta$, where $z_\alpha^\delta$
minimizes the functional

\begin{equation}
\label{eq1}
\|z-x^\delta\|^2+\alpha\|Lz\|^2\;\;\;\;\;(\alpha > 0)
\end{equation}
over ${\cal{D}}(L).$ This is equivalent to
\begin{equation}
\label{eq2}
y_\alpha^\delta = L(I + \alpha L^*L)^{-1}x^\delta.
\end{equation}
Morozov shows that if $\alpha=\alpha(\delta) \to 0$ as $\delta \to 0$, in
such a way that $\frac{\delta}{\sqrt{\alpha}} \to 0$, then $y_\alpha^\delta
\to Lx$ as $\delta \to 0$. He also develops an a posteriori method, the
{\it discrepancy principle}, for choosing the parameter $\alpha$, depending
on the data $x^\delta$, that leads to a stable convergent approximation
scheme for $Lx$.

As a simple concrete example of this type of approximation, consider
differentiation in $L^2({\bf R})$.  That is, the operator $L$ is defined on
$H^1({\bf R})$, the Sobolev space of functions possessing a weak
derivative in $L^2({\bf R})$, by $Lx = x^\prime$.  For a given data function
$x^\delta \in L^2({\bf R})$ satisfying $\| x-x^\delta \| \leq \delta$, the
stabilized approximate derivative (1.2) is easily seen (using Fourier
transform analysis) to be given by
\[
y^\delta_\alpha(s) = \int^\infty_{-\infty} \sigma_\alpha (s-t)x^\delta(t)
\,dt\,
\]
where the kernel $\sigma_\alpha$ is given by
\[
\sigma_\alpha(t) = -\frac{{\mbox{sign}}\,(t)}{2\alpha} \exp(-|t|/\sqrt{\alpha}).
\]

Another concrete example of this stable evaluation method is provided by the
Dirichlet to Neumann map.  Consider for simplicity the unit disk $D$ and unit
circle $\partial D$.  For a given function $g$ on $\partial D$ we denote by
$u$ the function which is harmonic in $D$ and takes boundary values $g$.  The
operator $L$ is then defined by $Lg = \frac{\partial u}{\partial n}$.  To be
more specific, $L$ is the closed operator defined on the dense subspace
\[
{\cal D}(L) = \left\{ g\in L^2(\partial D): \sum_{n\in {\bf Z}} |n|^2
|\hat{g}(n)|^2 < \infty \right\}
\]
of $L^2(\partial D)$ by
\[
(Lg)(e^{i\theta}) = \sum_{n\in {\bf Z}} |n| \hat{g}(n) \exp(in\theta)
\]
where
\[
\hat{g}(n) = \frac{1}{2\pi} \int^{2\pi}_0 g(t) e^{-int} \,dt\,.
\]
The stable approximation (1.2) for $Lx$, given approximate data $x^\delta$, is
then
\[
y^\delta_\alpha (e^{i\theta}) = \sum_{n\in {\bf Z}}
\left( \frac{n}{1+\alpha n^2} \right) \hat{x^\delta} (n) \exp (in \theta).
\]

Our aim is to provide an order of convergence result for
$\{y_\alpha^\delta\}$ and to show that this order of convergence is
essentially best possible. Our approach, which is inspired by a work of Lardy
\cite{6} on generalized inverses of unbounded operators, is based on spectral
analysis of certain {\it bounded} operators associated with $L$ (see also
\cite{5} where other consequences of this approach are investigated). We also
determine the best possible order of convergence when the discrepancy
principle is used to determine $\alpha$. This order of convergence turns out
to be suboptimal.  For results of a similar nature pertaining to Tikhonov
regularization for solving first kind equations involving bounded operators,
see \cite[Chapter 3]{4}.  Finally, we propose a modification of the
discrepancy principle for approximating values of an unbounded operator that
leads to an optimal convergence rate.

\section{Order of Convergence}

\setcounter{equation}{0}

To establish the order of convergence of (\ref{eq2}) it will be convenient
to reformulate (\ref{eq2}) as
\begin{equation}
\label{eq3}
y_\alpha^\delta = L\check{L} [ \alpha I + (1 - \alpha)\check{L}]^{-1}
x^\delta
\end{equation}
where $\check{L} = (I + L^* L)^{-1}$ and $L \check{L}$ are known to be
bounded everywhere defined linear operators and $\check{L}$ is self--adjoint
with spectrum $\sigma(\check{L}) \subseteq [0,1]$ (see, e.g.
\cite[p.307]{8}).

Because $x^\delta$ in (\ref{eq3}) is operated upon by a product of bounded
operators, we see that for fixed $\alpha > 0$, $y_\alpha^\delta$ depends
continuously on $x^\delta$, that is, the approximations $\{y_\alpha^\delta\}$
are stable. The representation (\ref{eq3}) has certain advantages in that the
dependence of $y_\alpha^\delta$ on bounded operators ($\check{L}$ and
$L\check{L}$), which are independent of $\alpha$, is explicit. To further
simplify the presentation, we introduce the functions

$$T_\alpha(t) = (\alpha + (1-\alpha)t)^{-1},\;\;\;\alpha>0,\;t \in [0,1].$$

We then have

$$y_\alpha^\delta = L\check{L} T_\alpha(\check{L}) x^\delta.$$

The approximation with no data error will be denoted by $y_\alpha$:

$$y_\alpha = L\check{L} T_\alpha(\check{L}) x.$$

\begin{theorem}
\label{th1}
{\rm
If $x \in {\cal{D}}(LL^*L)$ and $\alpha=\alpha(\delta)$ satisfies
$\frac{\alpha^3}{\delta^2} \to C > 0$ as $\delta \to 0$, then
$\|y_\alpha^\delta - Lx\| = O \bigl( \delta^{\frac{2}{3}} \bigr).$
}
\end{theorem}

{\bf{Proof:}} Let $w=(I+LL^*)Lx.$ Then $Lx=\hat{L}w$, where $\hat{L} =
(I+LL^*)^{-1}$. Since 
$$ L\check{L} = L(I + L^*L)^{-1} = (I + LL^*)^{-1}L=\hat{L}L $$
and $ Lx = (I+LL^*)^{-1}w = \hat{L}w$, we obtain from (\ref{eq3})
\begin{eqnarray*}
y_\alpha-Lx 
&=& 
L \left( \check{L} - [\alpha I + (1-\alpha) \check{L} ] \right)
[\alpha I + (1-\alpha) \check{L} ]^{-1} x \\
&=& 
\alpha L ( \check{L} - I )
[\alpha I + (1-\alpha) \check{L} ]^{-1} x \\
&=& 
\alpha ( \hat{L} - I )
[\alpha I + (1-\alpha) \hat{L} ]^{-1} Lx \\
&=& 
\alpha(\hat{L}-I)T_\alpha(\hat{L})\hat{L}w.
\end{eqnarray*}

Since $\|T_\alpha(\hat{L})\hat{L}\| \leq 1$, we find that

\begin{equation}
\label{eq4}
\|y_\alpha - Lx \| = O(\alpha).
\end{equation}
Also,

\begin{eqnarray}
\label{eq5}
\begin{array}{rcl}
\|y_\alpha^\delta - y_\alpha\|^2
& = &
(L^*L \check{L} T_\alpha(\check{L})(x^\delta -x),
 \check{L}T_\alpha(\check{L})(x^\delta-x)) \\
& = &
((I-\check{L})T_\alpha(\check{L})(x^\delta-x),
 \check{L}T_\alpha(\check{L})(x^\delta-x))\\
& \leq &
\|I - \check{L}\|\frac{\delta^2}{\alpha}
\end{array}
\end{eqnarray}
since $\|T_\alpha(\check{L})\| \leq \frac{1}{\alpha}.$ Therefore,

$$\|y_\alpha^\delta - y_\alpha\| =
  O \left( \frac{\delta}{\sqrt{\alpha}} \right).$$
We then have

$$\|y_\alpha^\delta - Lx\| = O(\alpha) +
  O\left( \frac{\delta}{\sqrt{\alpha}} \right) =
  O\bigl( \delta^{\frac{2}{3}} \bigr),$$
since $\frac{\alpha^3}{\delta^2} \to C > 0.$
\hfill$\Box$

This theorem shows that under the regularity condition $x \in
{\cal{D}}(LL^*L)$ on the exact data the order of convergence
$O\left(\delta^\frac{2}{3}\right)$ is attainable by the approximation
(\ref{eq3}) using approximate data with error level $\delta$. In the next
section we show that this order is best possible, except for the trivial case
when $x \in N(L)$, i.e., when $Lx=0$.

\section{Optimality}

\setcounter{equation}{0}

We begin by showing that any improvement in the order $ O\bigl(
\delta^{\frac{2}{3}} \bigr)$ entails a certain convergence rate for the
parameter $\alpha$.

\begin{lemma}
\label{le1}
{\rm
If $ x \notin N(L)$ and $\|y_\alpha^\delta-Lx\| = o \bigl(
\delta^{\frac{2}{3}} \bigr)$ for all $x^\delta$ satisfying $\|x-x^\delta\|
\leq \delta$, then $\alpha = o \bigl( \delta^\frac{2}{3} \bigr).$
}
\end{lemma}

{\bf Proof:}
Let $x^\delta = x - \delta u$, where $u$ is a unit vector and let
$e_\alpha^\delta = y_\alpha^\delta - Lx$. 
Then
\begin{eqnarray*}
[\alpha I + (1-\alpha)\hat{L}] e_\alpha^\delta 
&=& 
[\alpha I + (1-\alpha)\hat{L}] 
\left( L\check{L} (\alpha I + (1-\alpha)\check{L})^{-1}x - Lx \right) - \\
& &\quad  \delta [\alpha I + (1-\alpha)\hat{L}] 
          L\check{L} (\alpha I + (1-\alpha)\check{L})^{-1}u \\
&=& 
\alpha (\hat{L} - I)Lx - \delta L\check{L} u.
\end{eqnarray*}
Since $\|e_\alpha^\delta\| = o \bigl( \delta^{\frac{2}{3}} \bigr)$, by
assumption, and since
$$ \|\delta L \check{L} u\| \leq \delta \|L \check{L}\| =
  o\bigl( \delta^{\frac{2}{3}} \bigr),$$
we find that
$$ \frac{\alpha}{\delta^\frac{2}{3}} \|(\hat{L}-I)Lx\| \to
0,\mbox{\rm{ as }} \delta \to 0.$$
But $x \notin N(L)=N((\hat{L}-I)L)$ and hence $\alpha = o \left(
\delta^\frac{2}{3} \right).$ \hfill$\Box$

We now show that for a wide class of operators the order of convergence
$O\left( \delta^\frac{2}{3} \right)$ can not be improved. We will consider
the important class of operators $L^*L$ which have a divergent sequence of
eigenvalues. Such is the case if $L$ is the derivative operator, when $-L^*L$
is the Laplacian operator, or more generally whenever $L$ is a differential
operator for which $\check{L}$ is compact.

\begin{theorem}
\label{th2}
{\rm
If $L^*L$ has eigenvalues $\mu_n \to \infty$ and $\|y_\alpha^\delta-Lx\| =
o\left( \delta^\frac{2}{3} \right)$ for all $x^\delta$ with $\|x-x^\delta\|
\leq \delta$, then $x \in N(L)$.
}
\end{theorem}

{\bf{Proof:}} If $x \notin N(L)$, then $\alpha = o \bigl(
\delta^{\frac{2}{3}} \bigr)$, by Lemma \ref{le1}.  Let $e_\alpha^\delta =
y_\alpha^\delta - Lx$, then
$$\|e_\alpha^\delta\|^2 = \|y_\alpha -
Lx\|^2+2(y_\alpha-Lx,y_\alpha^\delta-y_\alpha) +
\|y_\alpha^\delta-y_\alpha\|^2$$
and by hypothesis $\frac{\|y_\alpha-Lx\|^2}{\delta^{\frac{4}{3}}} \to 0$ as
$\delta \to 0$ (since $x^\delta=x$ satisfies $\|x-x^\delta\| \leq \delta$).
Therefore we must have
\begin{equation}
\label{eq6}
\frac{2(y_\alpha -
Lx,y_\alpha^\delta-y_\alpha)+\|y_\alpha^\delta-y_\alpha\|^2}
{\delta^\frac{4}{3}} \to 0 \mbox{ as } \delta \to 0.
\end{equation}
Suppose that $\{u_n\}$ are orthonormal eigenvectors of $L^*L$ associated with
$\{\mu_n\}$. Then $\{u_n\}$ are eigenvectors of $\check{L}$ associated with
the eigenvalues $\lambda_n=\frac{1}{1+\mu_n}$ and $\lambda_n \to 0$ as $n \to
\infty$. Now let $x^\delta=x+\delta u_n$. Then
\begin{eqnarray*}
\|y_\alpha^\delta - y_\alpha\|^2 
&=& 
\delta^2 
\left( \check{L} (\alpha I + (1-\alpha)\check{L})^{-1} u_n,
       L^*L\check{L} (\alpha I + (1-\alpha)\check{L})^{-1} u_n \right)\\
&=& 
\delta^2 \lambda_n^2 \mu_n (\alpha+(1-\alpha)\lambda_n)^{-2}\\
&=&
\delta^2 \lambda_n(1-\lambda_n)(\alpha+(1-\alpha)\lambda_n)^{-2}.
\end{eqnarray*}
Therefore, if $\delta=\delta_n=\lambda_n^{\frac{3}{2}}$, then $\delta_n \to
0$ as $n \to \infty$ and
\begin{equation}
\label{eq7}
\frac{\|y_\alpha^{\delta_n} - y_\alpha\|^2}{\delta_n^{\frac{4}{3}}} = (1 -
\lambda_n) \left( \frac{\alpha}{\delta_n^{\frac{2}{3}}} + 1 -\alpha
\right)^{-2} \to 1 \mbox{ as } n \to \infty.
\end{equation}
Finally, we have
$$\frac{|(y_\alpha - Lx,y_\alpha^{\delta_n} - y_\alpha)|}{\delta_n^\frac{4}{3}}
  \leq \frac{\|y_\alpha - Lx\|}{\delta_n^\frac{2}{3}}
\frac{\|y_\alpha^{\delta_n} - y_\alpha\|}{\delta_n^\frac{2}{3}} \to
0.$$
This, along with (\ref{eq7}), contradicts (\ref{eq6}) and hence $x \in N(L)$.
\hfill$\Box$

\section{The Discrepancy Principle}

\setcounter{equation}{0}

We may write the approximation $y_\alpha^\delta$ to $Lx$ as
\begin{equation}
\label{eq8}
y_\alpha^\delta = Lz_\alpha^\delta \; \; \mbox{where} \; \; z_\alpha^\delta =
\check{L} T_\alpha(\check{L})x^\delta.
\end{equation}
Morozov \cite[p.125]{7} has shown that if $\|x^\delta\| > \delta$ (i.e., the
signal--to--noise ratio is greater than one), then there is a unique
$\alpha=\alpha(\delta)>0$ such that
\begin{equation}
\label{eq9}
\|z_{\alpha(\delta)}^\delta - x^\delta\| = \delta.
\end{equation}
Moreover, he showed that $y_{\alpha(\delta)}^\delta \to Lx$ as $\delta \to
0$.  We now provide an order of convergence for this method and show that, in
general, it can not be improved.

\begin{theorem}
\label{th3}
{\rm
If $x \in {\cal{D}}(L^*L)$ and $x \notin N(L)$, then
$\|y_{\alpha(\delta)}^\delta - Lx\| = O(\sqrt{\delta})$.
}
\end{theorem}

{\bf{Proof:}} First note that
$$[\alpha I + (1 - \alpha) \check{L}] (z_\alpha^\delta - x^\delta) =
   \alpha (\check{L} - I)x^\delta.$$
Moreover, note that (cf. (\ref{eq3}))
$ \| \alpha I + (1-\alpha) \check{L} \| \leq 1.$

Therefore, if $\alpha$ is chosen by (\ref{eq9}), then
$$\alpha\|(\check{L} - I)x^\delta\| \leq \|z_\alpha^\delta -
x^\delta\|=\delta$$
and hence
$$\|(\check{L} - I) x^\delta\| \leq \frac{\delta}{\alpha(\delta)}.$$
Since $x \notin N(L)$, we have $x \notin N(\check{L}-I)$ and hence
$$0 < \|(\check{L}-I)x\| \leq \liminf_{\delta \to 0}
\frac{\delta}{\alpha(\delta)}.$$
We therefore have
\begin{equation}
\label{eq10}
\alpha=O(\delta).
\end{equation}
Since $z_\alpha^\delta$ minimizes (\ref{eq1}) over ${\cal{D}}(L)$ and
$\alpha(\delta)$ satisfies (\ref{eq9}), it follows that
\begin{eqnarray*}
\delta^2 + \alpha(\delta) \|L z_{\alpha(\delta)}^\delta\|^2 &=&
\|z_{\alpha(\delta)}^\delta - x^\delta\|^2 +
\alpha(\delta)\|Lz_{\alpha(\delta)}^\delta\|^2 \\
&\leq& \|x-x^\delta\|^2+\alpha(\delta)\|Lx\|^2\\
&\leq& \delta^2 + \alpha(\delta)\|Lx\|^2
\end{eqnarray*}
and hence $\|Lz_{\alpha(\delta)}^\delta\| \leq \|Lx\|$. We then have
\begin{eqnarray*}
\|y_{\alpha(\delta)}^\delta - Lx\|^2 &=& \|y_{\alpha(\delta)}^\delta\|^2 -
2(y_{\alpha(\delta)}^\delta,Lx) + \|Lx\|^2\\
&\leq& 2(Lx - y_{\alpha(\delta)}^\delta,Lx) = 2
(x-z_{\alpha(\delta)}^\delta,L^*Lx)\\
&\leq& 4\delta \|L^*Lx\|
\end{eqnarray*}
and hence $\|y_{\alpha(\delta)}^\delta - Lx\| = O(\sqrt{\delta}).$
\hfill$\Box$

It turns out that if the parameter is chosen by the discrepancy method
(\ref{eq9}), then the order of convergence derived in Theorem \ref{th3} can
not be improved in general. To see this, suppose that $\check{L}$ has a
sequence of eigenvalues $\lambda_n \to 0$ and that$\{u_n\}$ is a
corresponding sequence of orthonormal eigenvectors. Furthermore, let
$\lambda_n=\frac{1}{1+\mu_n}$,$x=u_1$, and \\
$x^{\delta_n}=u_1+\delta_n u_n.$ An easy calculation then gives
\begin{equation}
\label{eq11}
\|y_{\alpha}^{\delta_n} - Lx\|^2 \geq
\frac{\lambda_n^2}{(\alpha+(1-\alpha)\lambda_n)^2}\delta_n^2 \mu_n.
\end{equation}
Now set $\delta_n = \frac{\mu_n}{(1+\mu_n)^2}$, then $\delta_n \to 0$ as $n
\to \infty$. We will show that if $\alpha$ satisfies (\ref{eq10}), then
$\|y_\alpha^{\delta_n} - Lx\| = o(\sqrt{\delta_n})$ is not possible.  Indeed,
if this were the case, then by (\ref{eq11}) we have
$$ \left( \frac{\alpha}{\delta_n} +(1-\alpha)\frac{\lambda_n}{\delta_n}
\right)^{-2} = \mu_n \lambda_n^2 \delta_n (\alpha + (1-\alpha)\lambda_n)^{-2}
\to 0$$
and hence $\frac{\alpha}{\delta_n} + (1-\alpha)\frac{\lambda_n}{\delta_n} \to
\infty$. But if $\alpha$ is chosen by (\ref{eq9}), then by (\ref{eq10}),
$\frac{\alpha}{\delta_n}$ is bounded and hence $\frac{\lambda_n}{\delta_n}
\to\infty$. But $\frac{\lambda_n}{\delta_n} = \frac{1}{\mu_n} +1 \to 1,$ a
contradiction.

In the next section we show how the discrepancy principle can be modified to
recover the optimal order of convergence.

\section{Optimal Discrepancy Methods}

\setcounter{equation}{0}

Engl and Gfrerer \cite{1},\cite{2},\cite{3} have developed discrepancy
principles of optimal order for approximating solutions of bounded linear
operator equations of the first kind by Tikhonov regularization. In this
section we investigate similar principles for approximating values of
unbounded linear operators.

We begin by considering the function
$$ \rho(\alpha)=\alpha^2\|z_\alpha^\delta - x^\delta\|.$$
By using a spectral representation of the operator
$\check{L}T_\alpha(\check{L})$ which defines $z_\alpha^\delta$ via
(\ref{eq8}), it is easy to see that the function $\alpha \to \rho(\alpha)$ is
continuous, strictly increasing and satisfies
$$\lim_{\alpha \to 0+} \rho(\alpha) = 0 \mbox{ and }
  \lim_{\alpha \to \infty} \rho(\alpha) = \infty$$
(we assume that $x^{\delta} \not\in N(L)$, for otherwise the approximations
are trivial).  Therefore, there is a unique $\alpha=\alpha(\delta)>0$
satisfying
\begin{equation}
\label{eq12}
\|z_\alpha^\delta - x^\delta\|= \frac{\delta^2}{\alpha^2}.
\end{equation}
We will show that the modified discrepancy principle (\ref{eq12}) leads,
under suitable conditions, to an optimal order of convergence for the
approximations $y_\alpha^\delta$ to $Lx$.

\begin{theorem}
\label{th4}
{\rm
Suppose $x \in {\cal{D}}(L^*L)$ and $x \notin N(L)$.  If
$\alpha=\alpha(\delta)$ is chosen by condition (\ref{eq12}), then
$\frac{\delta^2}{\alpha^3(\delta)} \to \|L^*Lx\|>0$ as $\delta \to 0$.
}
\end{theorem}

{\bf{Proof:}} To simplify notation we set $\alpha=\alpha(\delta)$ in the
proof.  First we show that $\alpha \to 0$ as $\delta \to 0$. Since
$$[\alpha I + (1 - \alpha)\check{L}](z_\alpha^\delta-x^\delta) =
\alpha(\check{L} - I)x^\delta \mbox{ and } \|\check{L}\| \leq 1,$$
we have
\begin{equation}
\label{eq13}
\alpha \|(\check{L} - I)x^\delta\| \leq \|z_\alpha^\delta - x^\delta\| =
\frac{\delta^2}{\alpha^2}.
\end{equation}
Also, $(\check{L}-I)x^\delta \to (\check{L}-I)x \neq 0$ as $\delta \to 0$
since $\check{L}x=x$ implies $L^*Lx=0$, i.e., $x \in N(L)$.\\
Therefore, from (\ref{eq13}), we find that $\alpha \to 0$ as $\delta \to 0$.

Next we show that $\frac{\delta}{\alpha} \to 0$ as $\delta \to 0$. In fact,
$$\|z_\alpha-z_\alpha^\delta\| = \|
\check{L}T_\alpha(\check{L})(x-x^\delta)\| \leq \delta $$
and
$$x-z_\alpha = x - \check{L}T_\alpha(\check{L}) x \to 0 \; \; \mbox{as} \; \;
\alpha \to 0$$
since $N(\check{L})=\{0\}$ and $tT_\alpha(t) \to 1$ as $\alpha \to 0$ for
each $t \neq 0$. Therefore $\|x-z_\alpha^\delta\| \to 0$ as $\delta \to 0$.
We then have
$$ \frac{\delta^2}{\alpha^2} = \|z_\alpha^\delta - x^\delta\| \leq
   \|z_\alpha^\delta - x\| + \delta \to 0  \mbox{ as } \delta \to 0$$
and hence $\frac{\delta}{\alpha} \to 0$ as $\delta \to 0.$\\
We can now show that $L^*Lz_\alpha^\delta \to L^*Lx$ as $\delta \to 0$.\\
Indeed,
\begin{equation}
\label{eq14}
L^*Lz_\alpha - L^*Lx = (\check{L}T_\alpha(\check{L}) - I)L^*Lx \to 0 \; \;
\mbox{as} \; \; \delta \to 0
\end{equation}
and
$$ L^*L(z_\alpha^\delta - z_\alpha) =
   (I - \check{L})T_\alpha(\check{L})(x^\delta - x),$$
therefore,
\begin{equation}
\label{eq15}
\|L^*L(z_\alpha^\delta-z_\alpha)\| \leq
\|(I-\check{L})\|\frac{\delta}{\alpha}.
\end{equation}
Since $\|T_\alpha(\check{L})\| \leq \frac{1}{\alpha}$. But, since
$\frac{\delta}{\alpha} \to 0,$ we find from (\ref{eq14}) and (\ref{eq15})
that
$$L^*Lz_\alpha^\delta \to L^*Lx \mbox{ as } \delta \to 0.$$
Finally, we have
$$x^\delta - z_\alpha^\delta = \alpha L^*Lz_\alpha^\delta$$
and hence, by (\ref{eq12})
$$\frac{\delta^2}{\alpha^3} = \|L^*L z_\alpha^\delta \| \to \|L^*L x\|
\mbox{ as } \delta \to 0.$$
\hfill$\Box$

From Theorem \ref{th1} and \ref{th4} we immediately obtain

\begin{corollary}
\label{cor1}
{\rm If $x \in {\cal{D}}(LL^*L)$, $x \notin N(L)$ and
$\alpha=\alpha(\delta)$
is chosen by (\ref{eq12}), then $\|y_{\alpha(\delta)}^\delta - Lx\| =
O\bigl( \delta^{\frac{2}{3}} \bigr).$
}
\end{corollary}

The Corollary requires the ``smoothness'' condition $x \in {\cal{D}}(LL^*L)$
in order to guarantee the optimal convergence rate, but it is possible to
obtain a ``quasi--optimal'' rate without any additional smoothness
assumptions on the data $x$. It follows from the proof of Theorem \ref{th1}
(specifically, from (\ref{eq5})), that
\begin{equation}
\label{eq16}
\frac{1}{2}\| y_\alpha^\delta - Lx \|^2 \leq
           \| y_\alpha - Lx \|^2 + C\frac{\delta^2}{\alpha}.
\end{equation}
Let $m(x,\delta)$ be the infimum, over $\alpha > 0$, of the right hand side
of (\ref{eq16}). It is possible, following ideas of Engl and Gfrerer
\cite{2}, to choose a parameter $\alpha = \alpha(\delta)$ such that
$\|y_{\alpha(\delta)}^\delta - Lx \|^2$ has the same order as $m(x,\delta)$
which we call the quasi-optimal rate. In fact, minimizing the right hand side
of (\ref{eq16}) leads to a condition of the form
$$f(\alpha,x):=\left( [\alpha(I-\check{L})T_\alpha(\check{L})]^3x,x \right) =
C\delta^2.$$

If we denote the spectral resolution of the identity generated by the operator
$\check{L}$ by $\{ E_\lambda : \lambda \in [0,1]\}$, then
\[
f(\alpha,z) = \int^1_0 \left[ \frac{\alpha(1-\lambda)}{\alpha(1-\lambda) +
\lambda} \right]^3 \,d\|E_\lambda z\|^2\,.
\]
>From this it follows that for any $z \not\in N(L)$, $f(\cdot,z)$ is a
monotonically increasing continuous function satisfying
\[
\lim_{\alpha\rightarrow 0} f(\alpha,z) = 0 \; \; \mbox{and} \; \;
\lim_{\alpha\rightarrow\infty} f(\alpha,z) = \|Pz\|^2,
\]
where $P$ is the orthogonal projector of $H_1$ onto $N(L)^\perp$.  Therefore,
for any $\delta > 0$ and $x^\delta \not\in N(L)$ and any positive constant
$\gamma$ which is dominated by the signal-to-noise ratio of data $x^\delta$,
that is, satisfying
\[
0 < \gamma < \|Px^\delta\|/\delta,
\]
there is a unique choice of the parameter $\alpha = \alpha(\delta)$ satisfying
\[
f(\alpha(\delta),x^\delta) = (\gamma\delta)^2.
\]
It can be shown, but we will not provide the details, that, this a posteriori
choice of the parameter always leads to the quasi-optimal rate
$\|y_\alpha^\delta - Lx\|^2=O(m(x,\delta))$, without any additional smoothness
assumptions on the data $x$.

\begin{thebibliography}{99}

\bibitem{1} H.W.~Engl,~~Discrepancy principles for Tikhonov regularization of
ill-posed problems leading to optimal convergence rates,  J. Optimiz. Theory
Appl. 49(1987), 209--215.  MR 88b:49045.

\bibitem{2} H.W.~Engl and H.~Gfrerer,~~A posteriori parameter choice for
general regularization methods for solving linear ill--posed problems, Appl.
Numer. Math. 4(1988), 395--417.  MR 89i:65060.

\bibitem{3} H.~Gfrerer,~~An a posteriori parameter choice for ordinary and
iterated Tikhonov regularization of ill--posed problems leading to optimal
convergence rates, Math. Comp. 49(1987), 507--522.  MR 88k:65049.

\bibitem{4} C.W.~Groetsch,~~The Theory of Tikhonov Regularization for
Fredholm Equations of the First Kind, Pitman, London, 1984.

\bibitem{5} C.W.~Groetsch,~~Spectral methods for linear inverse problems with
unbounded operators, J. Approx. Th. 70(1992), 16-28.

\bibitem{6} L.J.~Lardy,~~A series representation of the generalized inverse
of a closed linear operator,Att. Accad. Naz. Lincei Cl. Sci. Mat. Natur., Ser
VIII, 58(1975), 152--157.  MR 48\# 13540.

\bibitem{7} V.A.~Morozov,~~Methods for Solving Incorrectly Posed Problems,
Springer--Verlag, New York, 1984.

\bibitem{8} F.~Riesz and B.~Sz--Nagy,~~Functional Analysis, Ungar, New York,
1955.

\end{thebibliography}

\vspace{0.1in}
\noindent
\begin{tabular}{ll}
Department of Mathematical Sciences \qquad & Institut f\"ur Mathematik \\
University of Cincinnati & Universit\"at Linz\\
Cincinnati, OH 45221-0025 & A--4040 Linz\\
USA & Austria \\
E-mail: groetsch\@ucbeh.san.uc.edu & scherzer\@indmath.uni-linz.ac.at
\end{tabular}

\end{document}