\def\papertitle{uniqueness and stability of parameter identification in elliptic boundary value problem}
\def\paperauthorA{\underline{ Benyoucef Abir}, Alem Le\"ila, Chorfi Lahc\`ene}
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\title{\papertitle}
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{LMA, Badji Mokhtar University, Annaba.}

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    \maketitle
    \begin{abstract}
This talk concerns uniqueness and stability of an inverse problem
in PDE. Our problem consists in identifying two parameters $b(x)$
and $c(x)$ in the following boundary-value problem \cite{kern}.
\begin{equation}\label{boundary pb}
\begin{cases}
Lu:=-b(x)u''(x)+c(x)u'(x)=f(x),\\
u(0)=u(1)=0,
\end{cases}
\end{equation}
from distributed observations $u_1$ (resp. $u_2$) associated to the
source $f_1$ (resp. $f_2$). For one observation the solution is
not unique but we prove, under some conditions, the uniqueness of
the solution $p=(b,c)$ in the case of two observations.
 Furthermore, we derive a H\"older type stability result.
  The algorithm of reconstruction uses the least squares method.
   Finally, we present some numerical examples in the case of exact and noisy data to illustrate our method.
\end{abstract}
\par
 \textbf{Key-words:} Inverse problem, Least squares method.
Levenberg-Marquardt algorithm. \\ \textbf{MSC 2010:} 47J06, 90C30.
\section{Setting of the problem}
\textbf{The direct problem} is to find the weak solution of the
problem \eqref{boundary pb}: given $b(x)$, $c(x)$, and $f(x)$,
find $u\in H^{1}_{0}(0,1)$, such that
\begin{equation}
\displaystyle\int_{0}^{1}[u'(x)v(x)(b'(x)+c(x))+u'(x)v'(x)b(x)]dx=\displaystyle\int_{0}^{1}f(x)v(x)dx,
\quad \forall v\in H^{1}_{0}(0,1).
\end{equation}
The direct problem is well-posed under standard conditions $b\in
C^1[0,1]$; $b(x)\geq b_0>0$, $c\in C[0,1]$; $c(x)\geq 0$ and $f
\in L^2(0,1)$.\par Our \textbf{inverse problem} is the parameter
identification. Given $(u,f)$ (for one observation) or $(u_1,f_1)$
and $(u_2,f_2)$ (for two observations), we reconstruct the pair of
coefficients $p=(b,c)$.\par
It is well-known that such problem is typically ill-posed problem i.e: the solution can be non-unique and unstable.\\ %There a
\section{Stability}
In this section, we give a condition for which the solution of the
inverse problem is unique.
 The equation $\Phi(p)=(u_1,u_2)$ has a unique solution $p=(b,c)$
 if and only if the linear system $\{Lu_1=f_1;\ Lu_2=f_2\}$ has a unique solution
  with respect to $(b,c)$.\\
\subsection{Notations}
First, we introduce some notations that will be used throughout
this paper.
\begin{itemize}
\item In order to provide an abstract formulation of the inverse
problem, we introduce the parameter space $M_{ad}= \{(b,c);\ b\in
C^1[0,1],\ b(x)\geq b_0>0,\,\ c\in C[0,1],\ c(x)\geq0\}.$ \item
The parameter is the pair $p_1=(b_1,c_1),\ p_2=(b_2,c_2)\in
M_{ad}.$ \item  Consider $f_1,f_2\in L^2(0,1).$ \item The mapping
$\Phi$ that relates the parameter to the observation is defined by
$\Phi(p_1)=(u_1,u_2),\ \Phi(p_2)=(v_1,v_2)\in Y=H^2(0,1)\times
H_0^1(0,1)$, with \\ $u_j$ (resp. $v_j$) solution of
$-b_1u_j''+c_1u_j'=f_j$ (resp. $-b_1v_j''+c_1v_j'=f_j$), $j=1,2$.
\item We set $\Delta_1(x)=f_1u'_2-f_2u'_1$ and
$\Delta_2(x)=f_1v'_2-f_2v'_1$.
\end{itemize}
\vspace{0.2cm} The determinant of the  previous system for
$p_1=(b_1,c_1)$ is $\dfrac{\Delta_1(x)}{b_1(x)}$.\\ \\
\textbf{Proposition 1:} Suppose that $\Delta(x)\equiv
f_1(x)u_2'(x)-f_2(x)u_1'(x)\neq 0$, $\forall x\in (0,1).$ Then,
the equation $\Phi(p)=(u_1,u_2)$ has a unique solution
$p=(b_1,c_1)$.\\ \\
\textbf{Proposition 2:} Assume that:  $\exists \alpha_1>0$, such
that
\begin{equation}\label{a1}
|\Delta_1(x)|\geq \alpha_1>0, \quad \forall x \in[0,1].
\end{equation}
Then we have the following estimate
\begin{equation}
\|p_1-p_2\|_{L^2}\leq C(p_1)\|\Phi(p_1)-\Phi(p_2)\|_{Y\times Y}.
\end{equation}
\\
 Proposition 2 means that the operator $\Phi: p\mapsto u=(u_1,u_2)$ from
$M_{ad}\subset L^2\times L^2$ to $H^2\times H^2$ is invertible in
a neighbourhood of $p_1$, moreover $\Phi^{-1}$ is continuous
(locally Lipschitz).

\section{Algorithm}
One of the most commonly used approaches for solving the inverse
problem is by setting it as a least squares problem
\cite{chavent}. The solution $p=(b,c)$ realize the minimum of the
functional
\begin{equation}
J(p)=\dfrac{1}{2}\big[\|\Phi_1(p)-d_1\|^{2}+\|\Phi_2(p)-d_2\|^{2}\big],
\quad \text{for}\,\ p\in M_{ad},
\end{equation}
where  $\Phi_j(p)=u_j(p)$ is the operator solution  and
$(d_1,d_2)\in L^2\times L^2$ is the measured data.  To solve the
least squares problem, we apply the \textbf{Levenberg-Marquardt}
method \cite{more} which
consists in iterating the procedure:  \\
$p_0$: initial approximation, $p_{n+1}=p_n+h_n$, where $h_n$ is
the solution of the linearized equation
\begin{equation}
\Phi'^{*}(p_n)\Phi'(p_n)h_n+\alpha_{n}h_{n}=\Phi'^{*}(p_n)(d-\Phi(p_n)),
\end{equation}
where $\Phi^\prime(p)$ is the Fréchet-derivative of $\Phi$  \cite{neubauer} given by the theorem:\\
\textbf{Theorem:} $\Phi$ is Fréchet-differentiable. The partial
derivatives are given by:
\begin{equation}
\begin{cases}
\dfrac{\partial \Phi}{\partial b}(p;h)=\big(A(p)\big)^{-1}(hu''),\\ \\
\dfrac{\partial \Phi}{\partial c}(p;k)=-\big(A(p)\big)^{-1}(ku').
\end{cases}
\end{equation}
Where $A(p)$ is the differential operator defined by:
$A(p):\mathcal{D}(A)\longrightarrow H=L^2(0,1)$
\begin{equation}
\begin{cases}
\mathcal D(A)=H^2(0,1)\cap H^{1}_{0}(0,1),\\
A(p)\varphi=L\varphi, \quad L\varphi=-b(x)\varphi''+c(x)\varphi'.
\end{cases}
\end{equation}
\section{Numerical examples}
As an example, we choose the coefficients $b(x)=1+0.5\sin(\pi x)$ and $c(x)=1+x-x^2$.\\
\textbf{Example 1:} $f_1(x)=\cos(\pi x)$ and $f_2(x)=x-x^2$. The
solution $u_1$ and $u_2$ are computed with Finite Element
Method.\\ \\
The following figure shows the variation of $\Delta(x)$
\begin{figure}[H]
\begin{center}
\includegraphics[height=3cm]{Figs/delta1en.png}
%\caption{Variation of $\Delta(x)$}
\end{center}
\end{figure}
The following figures show reconstructions without noise
\begin{figure}[H]
\begin{center}
\includegraphics[width=4cm,height=4cm]{FIGs/ben.png}
\includegraphics[width=4cm,height=4cm]{FIGs/cen.png}
%\caption{ Reconstruction without noise, $p_0=(1,1)$, $n=80$ and
%$itermax=10$.}
\end{center}
\end{figure}
The following figures show reconstructions with noise-level $\delta=10^{-5}$
\begin{figure}[H]
\begin{center}
\includegraphics[width=4cm,height=4cm]{FIGs/bben.png}
\includegraphics[width=4cm,height=4cm]{FIGs/cben.png}
%\caption{ Reconstruction with noise-level $\delta=10^{-5}$,
%$p_0=(1,1)$, $n=80$ and $itermax=10$.}
\end{center}
\end{figure}
{\bf Example 2:} $f_1(x)=\left\{\begin{array}{ccc} x &\text{if}&
x\leq 0.5\\ 0.5 &\text{if}& x>0.5\end{array}\right.\quad ,\quad
f_2(x)=x-x^2.$\\    
The following figure shows the variation of $\Delta(x)$
\begin{figure}[H]
\begin{center}
\includegraphics[height=4cm]{Figs/delta2en.png} 
%\caption{Variation of $\Delta(x)$}
\end{center}
\end{figure}
The following figures show reconstructions without noise
\begin{figure}[H]
\begin{center}
\includegraphics[width=4cm,height=4cm]{FIGs/ex2ben.png}
\includegraphics[width=4cm,height=4cm]{FIGs/ex2cen.png}
%\caption{ Reconstruction without noise, $p_0=(1,1)$, $n=80$ and
%$itermax=10$.}
 \end{center}
\end{figure}

\subsection{Commentaries}
\begin{itemize}
\item In example 1, $\Delta_1$ does not change the sign
($\Delta_1(x)\geq 0.02$); which confirm the numerical stability.
\item In example 2, $\Delta_1$ changes the sign, it vanishes at
$x=0.5$. For the parameter $c(x)$, the algorithm converges to
another solution (lack of uniqueness). However the parameter $b$
is relatively stable.
\end{itemize}
\begin{thebibliography}{99}
\bibitem{chavent} 
G. Chavent, Nonlinear Least Squares for Inverse problems. Series
Scientific Computation, Springer, 2009.
\bibitem{neubauer}
H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse
Problems, Kluwer Academic Publishers, 1996.
\bibitem{kern}
M. Kern, Probl\`emes inverses, École superieure d’ingénieurs
Léonard de Vinci, (2002- 2003).
\bibitem{more}
J.J. Mor\'{e}, The Levenberg-Marquardt algorithm: implementation
and theory, in: G.A. Watson (Ed.), Lecture Notes in Mathematics
630: Numerical Analysis, Springer-Verlag, Berlin, 1978, pp.
105–116.
\end{thebibliography}
\end{document}
