\documentclass[twoside]{article} \usepackage{amssymb} % used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil An Inverse Problem in a Parabolic Equation\hfil}% {\hfil Zhilin Li \& Kewang Zheng\hfil} \begin{document} \setcounter{page}{203} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Differential Equations and Computational Simulations III}\newline J. Graef, R. Shivaji, B. Soni, \& J. Zhu (Editors)\newline Electronic Journal of Differential Equations, Conference~01, 1997, pp. 193--199. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp 147.26.103.110 or 129.120.3.113 (login: ftp)} \vspace{\bigskipamount} \\ An Inverse Problem in a Parabolic Equation \thanks{ {\em 1991 Mathematics Subject Classifications:} 35K05, 65R30. \hfil\break\indent {\em Key words and phrases:} Coefficients identification, ill-posedness, regularization. \hfil\break\indent \copyright 1998 Southwest Texas State University and University of North Texas. \hfil\break\indent Published November 12, 1998. \hfil\break\indent Supported by NSF grant DMS-96-26703, ORAU Junior Faculty Enhancement Award, \hfil\break\indent and North Carolina State Univ. FR\&PD Fund. } } \date{} \author{ Zhilin Li \& Kewang Zheng} \maketitle \begin{abstract} In this paper, an inverse problem in a parabolic equation is studied. An unknown function in the equation is related to two integral equations in terms of heat kernel. One of the integral equations is well-posed while another is ill-posed. A regularization approach for constructing an approximate solution to the ill-posed integral equation is proposed. Theoretical analysis and numerical experiment are provided to support the method. \end{abstract} \newtheorem{theorem}{Theorem} \section{Introduction} Consider a parabolic equation of the form: \begin{eqnarray} \label{eq1} u_t & =& u_{xx} + a(t) u, \quad 0 0, \quad 0 < t < T, \label{eq4} \end{eqnarray} where $f(t)$ and $g(t)$ are assumed to be known and strictly increasing functions. We want to find the unknown function $u(x,t)$ and a function $a(t)$ to satisfy the equations above. So this is an inverse problem. Since the publication of the AMS monograph \cite{mclaughlin} in 1984, hundreds of research papers on inverse problems have been published. For the problem studied here, we refer the readers to the references in \cite{aless,cannon,cannon1,cannon2,kohn,kohn1,tikhonov}. Assumed that (\ref{eq1})-(\ref{eq4}) has a solution $u(x,t)$, then it can be shown that \begin{eqnarray} \label{eq5} u(x,t) &=& 2 \int_{0}^{t} K(x,t-\tau) \, g(\tau)\, e^{\theta(t)-\theta(\tau)} d \tau, \\ [5pt] \mbox{where} \quad K(x,t) &=& \frac{ e^{-x^2/4t} }{\sqrt{4 \pi t}}, \\ [5pt] \mbox{and} \quad \theta(t) &=& \int_{0}^{t} a(\tau) d \tau. \label{eq6} \end{eqnarray} Let $x$ approach to zero in (\ref{eq5}). Using the boundary condition (\ref{eq3}), we have an integral equation for $y(t)$: \begin{eqnarray} \label{eq7} \int_{0}^{t} \frac{ g(\tau)}{\sqrt{t - \tau} } \, y(\tau) \, d \tau= \sqrt{\pi} f(t) y(t), \end{eqnarray} from which we can obtain $\theta(t)$ as follows, \begin{eqnarray} \label{eq7b} y(t) = e^{-\theta(t)}, \quad y(0)=1. \end{eqnarray} Ideally we can solve the Volterra's equation of the second type (\ref{eq7}) to get $y(t)$, and then $\theta(t)$ from (\ref{eq7b}). Once we have $\theta(t)$, the inverse problem can be solved from the second integral equation (\ref{eq6}) to get $a(t)$. With given data $f(t)$ and $g(t)$ in $C[0,T]$, the integral equation (\ref{eq7}) for $y(t)$ is well-posed, so is $\theta(t)$ from (\ref{eq7b}). However, the other integral equation (\ref{eq6}) for $a(t)$ in the space $C[0,T]$ is ill-posed because $a(t)$ does not depend on the data $\theta(t)$ continuously. \section{Regularization approach} From now on we shall focus our attention to the integral equation (\ref{eq6}). The idea of regularization method for ill-posed problem can be found in \cite{tikhonov}. Let \begin{eqnarray*} A[a]=\int_0^{t} a(\tau) \,d\tau = \theta(t), \end{eqnarray*} where \begin{eqnarray} \label{eq8} \theta(t) = -\ln y(t). \end{eqnarray} We start with the so-called smoothing functional \begin{eqnarray} \label{eq9} M^\alpha[a,\theta] =|| A[a] -\theta||_{L^2 [0,T]}^2 + \alpha \, ||a||_{W_2^1[0,T]}^2. \end{eqnarray} The solution of the minimization problem of the functional above will serve as an an approximate solution to the ill-posed integral equation (\ref{eq6}) with appropriate choice of $\alpha$. We have the following theorem. \begin{theorem} For every element $\theta(t)$ in $L^2 [0,T]$ and every parameter $\alpha>0$, there exists a unique element $a_{\alpha}(t)\in C[0,T]$ for which the functional (\ref{eq9}) attains its greatest lower bound: \begin{eqnarray*} M^\alpha[a_{\alpha},\theta] = \inf_{a\in C[0,T]} M^\alpha[a,\theta]. \end{eqnarray*} \end{theorem} Proof: Take the first variation of the functional (\ref{eq9}) and set it to zero to obtain the Euler integro-differential equation \begin{eqnarray} \label{eq10} \alpha \left( a'' -a \right) = \int_\tau^{T}\,dt \int_0^t a(\xi) \,d\xi - \int_\tau^{T} \theta(t)\,dt \end{eqnarray} with the boundary conditions: \begin{eqnarray} \label{eq11} a(0)=a_0^*, \quad a(T)=a_T^*, \quad a_0^*, \;a_T^* \quad \mbox{are given.} \end{eqnarray} Under conditions (\ref{eq11}), the associated homogeneous equation \begin{eqnarray*} \alpha \left( a'' -a \right) = \int_\tau^{T}\,dt \int_0^t a(\xi) \,d\xi \end{eqnarray*} can not possess a non-trivial solution. In fact, if $a(\tau)$ were such a solution, then multiplying both sides above by $a(\tau)$ and integrating with respect to $\tau$ from $0$ to $T$, one should get the following equality \begin{eqnarray*} -\alpha ||a ||_{W_2^1[0,T]}^2 = || \, A[a] \, ||_{L^2 [0,T]}^2, \end{eqnarray*} which would contradict the hypothesis that $\alpha>0$. Therefore, the inhomogeneous equation (\ref{eq10}) has one and only one solution $a_{\alpha}$. Thus the theorem is proved. The above theorem means that an operator $R(\theta, \alpha)$ into $C[0,T]$ is defined on the set of pairs $(\theta, \alpha)$, where $\theta\in L^2 [0,T]$ and $\alpha>0$, so that the element $a_{\alpha}=R(\theta, \alpha)$ minimizes the functional $M^\alpha[a,\theta]$. We also need to show that the operator $R(\theta, \alpha)$ is a regularizing one for equation (\ref{eq6}) by selecting an appropriate parameter $\alpha$. \begin{theorem} Let $a_{T}$ be the solution of equation (\ref{eq6}) corresponding to a given $\theta=\theta_{T}$, that is, $A[a_{T}]=\theta_{T}$. Then for any positive number $\epsilon$, there exists a number $\delta(\epsilon)>0$ such that the inequality \begin{eqnarray*} ||\bar{\theta}-\theta_{T}||_{L^2} \leq \delta \leq \delta(\epsilon) \end{eqnarray*} implies the inequality \begin{eqnarray*} ||a_{\alpha}-a_{T}||_C \leq \epsilon, \end{eqnarray*} where \begin{eqnarray*} a_{\alpha}=R(\bar{\theta}, \alpha), \quad \mbox{with} \quad \alpha=\alpha(\delta)=\delta^\lambda, \;\; 0 < \lambda \leq 2. \end{eqnarray*} \end{theorem} Proof: Since the functional $M^\alpha[a,\bar{\theta}]$ attains its minimum when $a=a_{\alpha}$, we have \begin{eqnarray*} M^\alpha[a_{\alpha},\bar{\theta}] \leq M^\alpha[a_{T},\bar{\theta}]. \end{eqnarray*} Therefore \begin{eqnarray*} \alpha ||a_{\alpha} ||_{W_2^1[0,T]}^2 \leq M^\alpha[a_{\alpha},\bar{\theta}] \leq M^\alpha[a_{T},\bar{\theta}] \leq \delta^2 + \alpha ||a_{T} ||_{W_2^1[0,T]}^2 \leq \delta^{\lambda} d, \end{eqnarray*} where \begin{eqnarray*} d = 1 + ||a_{T} ||_{W_2^1[0,T]}^2. \end{eqnarray*} Thus \begin{eqnarray*} ||a_{\alpha} ||_{W_2^1[0,T]}^2 \leq d, \quad \mbox{and} \quad ||a_{T} ||_{W_2^1[0,T]}^2 \leq d. \end{eqnarray*} Consequently, both $a_{\alpha}$ and $a_{T}$ belong to the following compact subset of space $C[0,T]$ \begin{eqnarray*} E = \left \{ a(t): \;\; ||a||_{W_2^1[0,T]}^2 \leq d \right \}. \end{eqnarray*} By virtue of the continuity of the inverse $A^{-1}$ defined on $AE$, for arbitrary $\epsilon>0$, there exists a number $\eta(\epsilon)>0$ such that the inequality \begin{eqnarray*} &&||\theta_{\alpha} - \theta_T ||_{L^2[0,T]} \leq \eta(\epsilon), \quad \mbox{for} \quad \theta_{\alpha} = A[a_{\alpha}], \quad \theta_{T} = A[a_{T}] \in AE \end{eqnarray*} implies the inequality \begin{eqnarray*} ||a_{\alpha} - a_{T}||_{C[0,T]} \leq \epsilon. \end{eqnarray*} Now for $\bar{\theta}$ and $\theta_{\alpha}$, we have \begin{eqnarray*} ||\theta_{\alpha} - \bar{\theta}||_{L^2[0,T]} = ||A[a_{\alpha}]-\bar{\theta}||_{L^2[0,T]} \leq M^\alpha[a_{\alpha},\bar{\theta}] \leq M^\alpha[a_{T},\bar{\theta}] \leq \delta^{\lambda} d, \end{eqnarray*} and thus \begin{eqnarray*} ||\theta_{\alpha} -\theta_{T}||_{L^2[0,T]} &\leq& ||\theta_{\alpha} - \bar{\theta}||_{L^2[0,T]} + ||\bar{\theta} - \theta_{T}||_{L^2[0,T]} \\ [5pt] &\leq& \delta^{\lambda/2} \sqrt{d} + \delta \leq \ \delta^{\lambda/2}( 1+ \sqrt{d}). \end{eqnarray*} If we set \begin{eqnarray*} \delta(\epsilon) = \left( \frac{ \eta(\epsilon)}{1 + \sqrt{d}} \right)^{2/\lambda}, \end{eqnarray*} then the conclusion of the theorem follows. Therefore it is justified to take $a_{\alpha}$ as an approximate solution of equation (\ref{eq6}) with an approximate left hand side $\theta=\bar{\theta}$. Finally, the continuous dependence of the $\theta_T$ on $y$ is almost clear from the following. If $||y_{\delta}||_C=||\bar{y}-y_{\eta_{\tau}}||_C\leq \delta$, from (\ref{eq8}), we can conclude: \begin{eqnarray} ||\bar{\theta} - \theta_{T}||_{L^2[0,T]}^2 = \int_{0}^{T} \left( \ln\bar{y}(t) - \ln y_{T}(t) \right)^2\,dt \leq \int_{0}^{T} \frac{ y_{\delta}^2 }{ y_{T}^2 }\,dt \leq c^2 \delta^2, \end{eqnarray} where \begin{eqnarray*} c^2 = \int_{0}^{T} \frac{1}{ y_{T}^{2}(t) } dt. \end{eqnarray*} The following theorem shows that $y(t)$ depends on the initial data $f(t)$ and $g(t)$ continuously. \begin{theorem} Suppose \begin{eqnarray*} ||f_{\delta}||_C = ||\bar{f}-f_{T}||_C\leq \delta, \quad \mbox{and} \quad ||g_{\delta}||_C = ||\bar{g}-g_{T}||_C\leq \delta. \end{eqnarray*} Define \begin{eqnarray} D=\frac{ \sqrt{\pi} \int_{0}^{T} y_{T}(t)\,dt + \int_{0}^{T} \frac{y_T(t)}{\sqrt{T-t}}\,dt }{ \sqrt{\pi} \int_{0}^{T} g_{T}(t)\,dt + \int_{0}^{T} \frac{f_T(t)}{\sqrt{T-t} }\,dt }, \end{eqnarray} then \begin{eqnarray} \label{eq14} ||y_{\delta}||_C = ||\bar{y}-y_{T}||_C\leq 2D\delta. \end{eqnarray} \end{theorem} Proof: From (\ref{eq7}) we can write \begin{eqnarray*} \int_{0}^{t} \frac{ \bar{g}(\tau) }{ \sqrt{t-\tau} } \bar{y}(\tau) d \tau &=& \sqrt{\pi} \bar{f}(t) \bar{y}(t), \\ \mbox{and} \quad \int_{0}^{t} \frac{ g_T(\tau) }{ \sqrt{t-\tau} } y_T(\tau) d \tau &=& \sqrt{\pi} f_T(t) y_T(t). \end{eqnarray*} This implies \begin{eqnarray*} \sqrt{\pi} \left( \bar{f}(t)y_{\delta}(t) + f_{\delta}(t) y_T(t) \right) = \int_{0}^{t} \frac{\bar{g}(\tau)y_{\delta}(\tau) + g_{\delta}(\tau) y_T(\tau) }{ \sqrt{t-\tau} } d \tau. \end{eqnarray*} Multiplying both sides by $1/\sqrt{T-t}$ and integrating with respect to $t$ over $[0,T]$, we can obtain \begin{eqnarray*} \int_{0}^{T} \frac{\bar{f}(t)y_{\delta}(t) + f_{\delta}(t) y_T(t) } { \sqrt{T-t} } d t = \sqrt{\pi} \int_{0}^{T} \left( \bar{g}(t)y_{\delta}(t) + g_{\delta}(t) y_T(t) \right) dt. \end{eqnarray*} Thus \begin{eqnarray*} \left | \int_{0}^{T} \left( \frac{\bar{f}(t)}{\sqrt{T-t} } - \sqrt{\pi} \bar{g}(t) \right) y_{\delta}(t)\,dt \right | \leq \delta \int_{0}^{T} \left( \frac{y_T(t) }{ \sqrt{T-t} } + \sqrt{\pi} y_T (t) \right) dt, \end{eqnarray*} from which (\ref{eq14}) follows. This completes the proof of the theorem. \section{A numerical example} In this section, we provide an example with exact solution to see how the regularization method proposed in this paper works. Take \begin{eqnarray} f_T(t) = 2 (t+1) \sqrt{t/\pi}, \quad \mbox{and} \quad g_T(t) = t+1. \end{eqnarray} From (\ref{eq6}) and (\ref{eq7}), it is easy to verify that \begin{eqnarray} y_T= \frac{1}{t+1}, \quad \mbox{and} \quad a_T(t) = \frac{1}{t+1}. \end{eqnarray} For simplicity, we use a uniform grid with step size $h=T/(n+1)$. The first step in the numerics is to replace (\ref{eq7}) and (\ref{eq10}) with finite difference approximation on the grid to get the following recursive relations \begin{eqnarray*} y_1 &=& 2 g_0 \frac{ \sqrt{h/\pi} }{ f_1} \\ y_i &=& \frac{ 2 \left( (\sqrt{i} - \sqrt{i-1} ) g_0 + \sum_{j=2}^{i} ( \sqrt{i-j+1} - \sqrt{i-j} ) g_{j-1} y_{j-1} \right) \sqrt{h/\pi}} {f_i}, \\ && i=2, \cdots, n\,, \end{eqnarray*} and the system of linear equations \begin{eqnarray} \label{eq15} \alpha \left( \frac{ a_{j-1} - 2 a_{j} + a_{j+1}}{h^2} - a_j \right) = h^2 \sum_{i=j}^n \sum_{k=1}^{i} a_k - h \sum_{i=j}^n \theta_i, \quad j=1, \cdots, n, \end{eqnarray} with $a_0^*=1$ and $a_T^*=0.5$ corresponding to $T=1$, see (\ref{eq11}). Next, we take the regularization parameter $\alpha$ in the form \begin{eqnarray} \alpha = \alpha(\delta) = (2CD\delta)^{\lambda}, \quad 0 < \lambda \leq 2. \end{eqnarray} Table~\ref{tab1} shows the exact solution and the solution using the regularization approach with the perturbations $f_{\delta}(t)=g_{\delta}(t)=\delta \sin(2 \pi t)$, $n=79$, $\delta=10^{-6}$ and $\lambda=0.6$. The results agree with each other pretty well. \begin{table}[hbtp] \caption{Reconstruction of $a(t)$ using the regularization method. $a_T(t)$ is the exact solution. $a_{\alpha}(t)$ is the solution of the regularization method. The parameters are: $n=79$, $T=1.0$, $\delta=10^{-6}$, and $\lambda=0.6$.} \label{tab1} \vspace{0.5cm} \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline $t$ & $0.05$ & $0.1$ & $0.15$ & $0.2$ & $0.25$ \\ \hline $a_T(t) $ & $0.95238 $ & $0.90909 $ & $0.86956 $ & $0.83333 $ & $0.8 $\\ $a_{\alpha}(t)$ &$0.95614$& $0.91484$ & $0.87606$ & $0.83978$&$0.80595$\\ \hline \hline $t $ & $0.3 $ & $0.35 $ & $0.4 $ & $0.45 $ & $0.5 $ \\ \hline $a_T(t)$ & $0.76923$ & $0.74074$ & $0.71429$ & $0.68966$ & $0.66667$ \\ $a_{\alpha}(t)$ & $0.77446$ & $0.74520$ & $0.71802$ &$0.69274$&$0.66919$\\ \hline \hline $t $ & $0.55 $ & $0.6 $ & $0.65 $ & $0.7 $ & $0.75 $ \\ \hline $a_T(t)$ & $0.64516$ & $0.62500$ & $0.60606$ & $0.58824$ & $0.57143 $ \\ $a_{\alpha}(t)$ & $0.64720$ & $0.62661$ & $0.60726$& $0.58903$&$0.57181$\\ \hline \hline $t $ & $0.8 $ & $0.85 $ & $0.9 $ & $0.95 $ & $ $ \\ \hline $a_T(t)$ & $0.55556 $ & $0.54054 $ & $0.52632 $ & $0.51282 $ & $ $ \\ $a_{\alpha}(t)$ & $0.55555$ & $0.54020 $ & $0.52578 $ & $0.51234$ & $$ \\ \hline \hline \end{tabular} \end{center} \end{table} In practice, the exact solution $a_T$ is unknown. More work remains to be done to study the behavior, such as accuracy and stability, of a numerical method applied to the equation (\ref{eq15}). In a summary, the regularization method proposed in this paper seems to be an effcient way for solving the inverse problem described at the begining of this paper. We have proved some good theoretical results about this method. Naive numerical discretization gives reasonably accurate result for the example provided in the paper. More work needs to be done on numerical study of such problems especially for two or higher dimensional problems. \begin{thebibliography}{10} \bibitem{aless} G. Alessandrini. \newblock Stable determination of conductivity by boundary measurements. \newblock {\em Appl. Anal.},27:153--218, 1988. \bibitem{cannon} J. Cannon. \newblock Determination of an unknown coefficient in a parabolic differential equation. \newblock {\em Duke Math. J.}, 30:313--323, 1963. \bibitem{cannon1} J. Cannon. \newblock Determination of certain parameters in heat conduction problems. \newblock {\em J. Math. Anal. Appl.}, 8:188--321, 1964 \bibitem{cannon2} J. Cannon and D. Zachmann. \newblock Parameter determination in parabolic partial differential equations from over-specified boundary data. \newblock {\em Int. J. Eng. Sci.}, 20:779-788, 1982. \bibitem{kohn} R. Kohn and G. Strang. \newblock Determining conductivity by boundary measurements. \newblock {\em Comm. Pure Appl. Math.}, 37:289--298, 1984. \bibitem{kohn1} R. Kohn and G. Strang. \newblock Determining conductivity by boundary measurements, II. Interior results. \newblock {\em Comm. Pure Appl. Math.}, 38:644-667, 1985. \bibitem{mclaughlin} D. Mclaughlin. \newblock Inverse Problems, editor. \newblock {\em SIAM-AMS Proc.}, 14, AMS, Providence, 1984. \bibitem{tikhonov} A. Tikhonov and V. Arsenin. \newblock Solutions of Ill-posed problems. \newblock {\em John Wiley \& Sons}, 1977. \end{thebibliography} \bigskip {\sc Zhilin Li} \newline Department of Mathematics \& Center For Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8205.\newline E-mail: zhilin@math.ncsu.edu \medskip {\sc Kewang Zheng}\newline Department of Mathematics, Hebei University of Science \& Technology, Hebei 050018, China \end{document}