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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 130, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2008/130\hfil Regularization of the backward heat equation]
{Regularization of the backward heat equation via heatlets}
\author[B. M. Campbell H., R. Hughes, E. McNabb, \hfil EJDE-2008/130\hfilneg]
{Beth Marie Campbell Hetrick, Rhonda Hughes, Emily McNabb} % in alphabetical order
\address{Beth Marie Campbell Hetrick \newline
Gettysburg College, Gettysburg, PA 17325, USA}
\email{bcampbel@gettysburg.edu}
\address{Rhonda Hughes \newline
Bryn Mawr College, Bryn Mawr, PA 19010, USA}
\email{rhughes@brynmawr.edu}
\address{Emily McNabb \newline
Bryn Mawr College, Bryn Mawr, PA 19010, USA}
\email{emily.a.mcnabb@accenture.com}
\thanks{Submitted March 27, 2008. Published September 18, 2008.}
\subjclass[2000]{47A52, 42C40}
\keywords{Ill-posed problems; backward heat equation; wavelets;\hfill\break\indent
quasireversibility}
\begin{abstract}
Shen and Strang \cite{Shen2} introduced heatlets in order to solve
the heat equation using wavelet expansions of the initial data.
The advantage of this approach is that heatlets, or the heat evolution
of the wavelet basis functions, can be easily computed and stored.
In this paper, we use heatlets to regularize the {\it backward}
heat equation and, more generally, ill-posed Cauchy problems.
Continuous dependence results obtained by Ames and Hughes \cite{AHb}
are applied to approximate stabilized solutions to ill-posed problems
that arise from the method of quasi-reversibility.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\section{Introduction} \label{Introduction}
Shen and Strang \cite{Shen2} introduced heatlets in order to solve
the heat equation using wavelet expansions of the initial data.
The advantage of this approach is that heatlets, or the heat
evolution of the wavelet basis functions, can be computed easily
and stored. When the initial data is expanded in terms of the
wavelet basis, the solution to the heat equation is then obtained
from an expansion using the heatlets and the corresponding wavelet
coefficients of the data. In this paper, we turn our attention to
ill-posed problems, using heatlets, and the method of
quasi-reversibility \cite{LL}, to regularize the {\it
backward} heat equation \cite{Miller1, Payne1,Showalter} as
well as more general ill-posed problems.
Given an ill-posed problem, it is often convenient to define an
approximate problem that is well-posed. Generally, we seek to
ensure that a solution to the original problem, if it exists, will
be appropriately close to the solution to the approximate problem.
In our main results, we show that for a wide range of ill-posed
problems, heatlets may be used to obtain such approximate
solutions. In addition, applying the results of \cite{AHb, AH2}, we
obtain H\"{o}lder-continuous dependence results for the difference
between solutions of the ill-posed and approximate well-posed
problems. Previously, wavelets have been used by Liu et al. to
decompose the regularized solution of inverse heat conduction
problems using a sensitivity decomposition \cite{Liu}, but
heatlets do not play a role in that work.
We consider the backward heat equation
\begin{equation} \label{e1}
\begin{gathered}
\frac{\partial u}{\partial t}
= - \frac{\partial ^2 u}{\partial x^2} \quad \text{where } 0 0$, such that for $0 \le t 0$, and $c_{j,n}$ is the wavelet coefficient of $f(x)$
attached to $\psi_{j,n}=2^{j/2}\psi(2^{j}x-n)$. Thus,
for small values of $\epsilon > 0$,
\[
\sum_{j,n \in \mathbb{Z}}
c_{j,n} e^{T(A-\epsilon A^{2})}\Psi^{h}_{j,n} (x,T-t)
\]
is close to $u(t)$ in $L^{2}(\mathbb{R})$, for $0 \leq t < T$.
\end{theorem}
The value of the above theorem lies in the fact that,
as in the case of the well-posed heat equation, the heatlets may be
computed and stored, and the approximation $w(t)$ will require
evaluation of $e^{T(A-\epsilon A^{2})}\Psi^{h}_{j,n} (x,T-t)$,
rather than $e^{T(A- \epsilon A^{2})}e^{(t-T)A}f$. Finally, in
Section 4, we show that Theorem 1.3 may be framed in a more general
setting, with other choices of the approximating operators. To
pursue this generalization, we introduce the terminology of
\cite{AHb},
and define {\it generalized heatlets}, that is, solutions of the
{\it abstract} Cauchy problem with initial data consisting of
elements of a wavelet basis. We then approximate the solution to
the ill-posed problem using the wavelet coefficients in a manner
analagous to that in Theorem 1.3 (Theorem 4.2).
\section{Wavelets and Heatlets}
In $L^{2}(\mathbb{R})$ we define the \emph{mother
wavelet} of the Haar basis as
\[
\psi(x) = \begin{cases}
1 & 0 \le x < \frac{1}{2}\\
-1 & \frac{1}{2} \le x < 1\\
0 & \text{otherwise.}
\end{cases}
\]
For positive integers $n,j$ define
$\psi_{n}^{j}(x)=2^{j/2}\psi(2^{j}x-n)$. Then according to a
theorem of Haar, $\{\psi_{n}^{j}\}$ is an orthonormal basis for
$L^{2}(\mathbb{R})$ (cf. \cite{Daub}).
\noindent \textbf{Definition.}
A \emph {multiresolution analysis}
of $L^{2}( \mathbb{R})$ is a chain of approximate spaces $V_{j}$
such that $-\infty \le j \le \infty$. These closed subspaces
satisfy the following properties:
\begin{itemize}
\item[(i)] The $V_{j}$ spaces are nested:
$\dots V_{-1} \subset V_{0} \subset V_{1} \subset V_{2} \subset
\dots$
\item[(ii)] These spaces are complete; that is,
\begin{gather*}
\overline{\cup_{j \in \mathbb{Z}} V_{j}}
= L^{2}(\mathbb{R}) \quad
(\text{i.e. } \lim_{j \to \infty} V_{j} = L^{2}(\mathbb{R}) ),\\
\cap_{j \in \mathbb{Z}} V_{j}= {0} \quad
(\text{i.e. } \lim_{j \to -\infty} V_{j} = {0}).
\end{gather*}
\item[(iii)] $f (x) \in V_{j}$ if and only if $f(2x) \in V_{j+1}$.
\item[(iv)] $f (x) \in V_{0}$ if and only if $f(x-k) \in V_{0}$.
\item[(v)] There exists a scaling function $\phi (x) \in V_{0}$
such that $\{ \phi(x-k): k \in \mathbb{Z} \}$ is an orthonormal
basis of $V_{0}$ (cf. \cite{Daub}).
\end{itemize}
To create a multiresolution, one needs to construct a
scaling function $\phi(x)$. Then, using the properties of a
multiresolution analysis, the entire chain can be constructed from
$\phi(x)$. For example, we can let $V_{0}=\{\phi(x-n)|n \in \mathbb{Z}\}$.
Then
\begin{gather*}
V_{1}=\{\phi(2x-n):n \in \mathbb{Z}\}, \\
V_{2}=\{\phi(2^{2}x-n):n \in \mathbb{Z}\}, \\
V_{-1}=\{\phi(\frac{x}{2}-n):n \in \mathbb{Z}\}.
\end{gather*}
This chain of approximate spaces $V_{j}$ forms a multiresultion analysis of
$L^{2}(\mathbb{R})$ \cite{Daub}.
The multiresolution analysis associated with the Haar basis is
provided by
\[
V_{j} = \{ f \in L^{2}( \mathbb{R}) :
f | _ {[\frac{k}{2^{j}}, \frac{(k+1)}{2^{j}}]} = \text{constant, }
k \in \mathbb{Z} \}.
\]
Next, we summarize the definitions and results from
\cite[Section 3]{Shen2}.
\noindent \textbf{Definition.}
Let $\phi(x)$ be the scaling function and $\psi(x)$ be the wavelet
associated to a multiresolution analysis. Define the
\emph{heat evolutions} of $\phi(x)$ and $\psi(x)$ to be $\Phi^{h}(x,t)$
and $\Psi^{h}(x,t)$, where
\[
\Phi^{h}_{t}= \Phi^{h}_{xx}, \quad \Phi^{h}(x,0)=\phi(x), \quad
\text{for } t>0, \; x \in \mathbb{R}.
\]
Similarly,
\[
\Psi^{h}_{t}= \Psi^{h}_{xx}, quad \Psi^{h}(x,0)=\psi(x),\quad
\text{for } t>0, \; x \in \mathbb{R}.
\]
The function $\Psi^{h}$ is called a \emph{heatlet} and
$\Phi^{h}$ is a \emph{refinable heat}.
\begin{proposition} \label{prop1}
Assume that $\phi(x)$ and $\psi(x)$ satisfy the equations
\begin{gather*}
\phi(x)=2 \sum_{n \in Z} h_{n}\phi(2x-n), \\
\psi(x)=2 \sum_{n \in Z} g_{n}\phi(2x-n),
\end{gather*}
where $(h_{n}),(g_{n}) \in l^{2}$. Then, the refinable heat and
heatlet will satisfy
\begin{gather*}
\Phi^{h}(x,t)=2 \sum_{n \in Z} h_{n}\Phi^{h}(2x-n,4t), \\
\Psi^{h}(x,t)=2 \sum_{n \in Z} g_{n}\Phi^{h}(2x-n,4t).
\end{gather*}
\end{proposition}
\begin{proposition} \label{prop2}
Define $\Psi^{h}_{j,n} (x,t)$ to be the solution of \eqref{e5}
with initial data $\psi_{j,n}$. Then
\[
\Psi^{h}_{j,n} (x,t)= 2^{j/2}\Psi^{h}(2^{j}x-n,4^{j}t).
\]
\end{proposition}
The main theorem of Shen and Strang \cite{Shen2} is as follows.
\begin{theorem}[\cite{Shen2}] \label{thmc}
Let $f \in L^{2}(\mathbb{R})$.
Then the corresponding heat evolution in $L^{2}(\mathbb{R})$ is
given by
\[
u(x,t)= \sum_{j,n \in \mathbb{Z}} c_{j,n}\Psi^{h}_{j,n} (x,t),
\]
where $c_{j,n}$ is the wavelet coefficient of $f(x)$ attached to
$\psi_{j,n}=2^{j/2}\psi(2^{j}x-n)$. Moreover, the
infinite series converges in $L^{2}(\mathbb{R})$ uniformly with
respect to $t$ .
\end{theorem}
\section{Regularization of the Backward Heat Equation}
\label{Regularization of the Backward Heat Equation}
Consider the \emph{final value} problem
\begin{gather*}
\frac{\partial u}{\partial t} = \frac {\partial ^{2} u} {\partial x^{2}}
\quad \text{for } 0 0$, such that
\[
\| u(t)- \sum_{j,n \in \mathbb{Z}} c_{j,n}
e^{T(A-\epsilon A^{2})}\Psi^{h}_{j,n} (x,T-t) \|
\le C \epsilon^{1- \frac{t}{T}}M^{t/T}.
\]
Thus, for small values of $\epsilon > 0$,
$\sum_{j,n \in \mathbb{Z}} c_{j,n} e^{T(A-\epsilon A^{2})}\Psi^{h}_{j,n}
(x,T-t)$ is close to $u(t)$ in $L^{2}(\mathbb{R})$, for
$0 \leq t < T$.
\end{theorem}
\begin{proof}
Recall that the solution to \eqref{e5} is
\begin{align*}
w(t) & = e^{(t-T)A}e^{T(A-\epsilon A^{2})}f \\
& = \sum_{j,n \in \mathbb{Z}} c_{j,n}e^{(t-T)A} e^{T(A-\epsilon A^{2})}
\psi_{j,n} \\
& = \sum_{j,n \in \mathbb{Z}} c_{j,n}e^{T(A-\epsilon A^{2})}
\Psi^{h}_{j,n} (x,T-t),
\end{align*}
where for each $j,n$, $e^{(t-T)A} \psi_{j,n}$ is the heatlet
$\Psi^{h}_{j,n} (x,T-t)$.
We consider
\[
\|u(t) - w(t) \| = \|e^{tA} \chi - e^{tA} e^{-\epsilon T A^2} f\| =
\|(I - e^{-\epsilon TA^2}) e^{tA} f\|.
\]
In order to obtain a convexity result, we set
\[
\phi_n(\alpha) = (e^{ {\alpha}^2}[e^{\alpha A} - e^{\alpha A}
e^{-\epsilon T A^2}]f_n,\,h),
\]
where $f_n = E(e_n)$, $E(\cdot)$ is the resolution of the identity
for $A$, $e_n$ is a bounded Borel function, and $h$ is an arbitrary
element of $\mathcal{H}$. Then
\begin{align*}
|\phi_n(\alpha)|
&\leq e^{t^2 - \eta^2} \|e^{(t+i\eta)A} f_n -
e^{(t+i\eta)A}e^{-\epsilon TA^2} f_n\| \, \|h\|\\
&\leq e^{t^2 - \eta^2}\|(I - e^{-\epsilon TA^2})e^{tA} f_n\|\,\|h\|\\
&\leq C_1 \, e^{t^2 - \eta^2} \epsilon \|A^{2}e^{tA}f_n\|\,\|h\|.
\end{align*}
Thus $\phi_n(\alpha)$ is
bounded in the strip $0 \leq \Re \alpha \leq T$, and so by the Three
Lines Theorem, we obtain
\[
|\phi_n(t)| \leq M(0)^{1-t/T}M(T)^{t/T} ,
\]
where $M(t) = \max_{\eta \in \mathbb{R}}|\phi(t +i\eta)|$.
Since $M(0) \leq C_1 \epsilon \|A^{2} f_n\|\,\|h\|$,
and
\[
M(T) \leq e^{T^2} \|(I - e^{-\epsilon TA^2})
e^{TA}f_n\|\,\|h\| \leq C_2 \, e^{T^2}\|e^{TA}f_n\|\,\|h\|,
\]
we obtain, taking the supremum over all $h \in \mathcal{H}$,
with $\|h\| \leq 1$,
\[
\|u(t) - w(t)\| \leq C\{\epsilon \|A^{2} f_n\|\}^{1-t/T} \{
\|e^{TA}f_n\|\}^{t/T}.
\]
for a suitable constant $C$. If we take the limit as $n
\to \infty$, and assume in addition that
$\|e^{TA}f\| \leq
\tilde{M}$, from which it follows that
$\|A^{2}f\| \leq \tilde{M}$, for a possibly different
constant, we have
\[
\|u(t) - \sum_{j,n \in \mathbb{Z}} c_{j,n}e^{T(A-\epsilon A^{2})}
\Psi^{h}_{j,n} (x,T-t)\|
= \|u(t) - w(t)\|
\leq C\epsilon^{1-t/T}M^{t/T}.
\]
Thus, for small values of $\epsilon > 0$,
$\sum_{j,n \in \mathbb{Z}} c_{j,n} e^{T(A-\epsilon A^{2})}\Psi^{h}_{j,n}
(x,T-t)$ is close to $u(t)$ in $L^{2}(\mathbb{R})$, for
$0 \leq t 0$, such that
\[
\| u(t)- \sum_{j,n \in \mathbb{Z}} c_{j,n} e^{Tf(A)}\Psi^{h}_{j,n} (x,T-t)\|
\le C \epsilon^{1- \frac{t}{T}}M^{t/T}.
\]
Thus, for small values of $\epsilon > 0$,
$\sum_{j,n \in \mathbb{Z}} c_{j,n} e^{Tf(A)}\Psi^{h}_{j,n} (x,T-t)$
is close to
$u(t)$ in $L^{2}(\mathbb{R})$, for $0 \leq t < T$.
\end{theorem}
\subsection*{Acknowledgements}
The authors gratefully
acknowledge the contributions of Professor Walter Huddell (Eastern
University) and Ayako Fukui (Bryn Mawr College) to this work.
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\end{document}