\documentclass[reqno]{amsart}
\usepackage[notref,notcite]{showkeys}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 219, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/219\hfil Backward heat conduction problems]
{A wavelet method for solving backward heat
conduction problems}

\author[C. Qiu,  X. Feng \hfil EJDE-2017/219\hfilneg]
{Chunyu Qiu, Xiaoli Feng}

\address{Chunyu Qiu \newline
School of Mathematics and Statistics,
Lanzhou University, Lanzhou 730000, China}
\email{qcy@lzu.edu.cn}

\address{Xiaoli Feng \newline
School of Mathematics and Statistics,
Xidian University, Xi'an 710071, China}
\email{xiaolifeng@xidian.edu.cn}

\thanks{Submitted March 29, 2017. Published September 14, 2017.}
\subjclass[2010]{65T60, 65M30, 35R25}
\keywords{Backward heat equation; Ill-posed problem; regularization;
\hfill\break\indent  Meyer wavelet; error estimate}

\begin{abstract}
 In this article, we consider the backward heat conduction problem (BHCP).
 This classical problem is more  severely ill-posed than some other problems,
 since the error of the data will be exponentially amplified at high frequency
 components. The Meyer wavelet method can eliminate the influence of the
 high frequency components of the noisy data. The known works on this method
 are limited to the \emph{a priori} choice of the regularization parameter.
 In this paper, we consider also  \emph{a posteriori} choice of the
 regularization parameter. The H\"older type stability estimates for
 both \emph{a priori} and \emph{a posteriori} choice rules are established.
 Moreover several numerical examples are also provided.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

Wavelet theory has been widely developed in the late of the previous century.
The multiscale analysis and wavelet decomposition are now still subjects of
intensive development. At the same time, the investigation of mutual interactions
between wavelet theory and ill-posed problems has never stopped.
Some results have been applied to the analysis of some inverse problems.
Wavelet methods have advantages for use in certain inverse problems:
1. They allow for the decomposition of an object into multiple resolutions
 (or scales). This is a particular advantage for high precision inversion
of the objects;
2. The localization of wavelets in both time and frequency makes them quite
useful for analyzing local features;
3. Wavelets have excellent data compression capabilities for spatially
variable objects, such as signals characterized by singularities or images
determined primarily by a set of edges;
4. Some methods of denoising, based on thresholding of the wavelet coefficients,
have been proven to be nearly optimal for a number of tasks across a wide range
of function classes \cite{Kolaczyk}.

We emphasized that Meyer wavelets possess specially important significance
for solving many ill-posed problems. Meyer wavelets have the property that
their Fourier transforms have compact supports. This means that they can
be used to prevent high frequency
noise from destroying the solution, i.e., by expanding the data and the
solution in a basis of Meyer wavelets, high-frequency components can be
filtered away. Within $V_j$, which is generated by the father wavelet of Meyer,
the original ill-posed problem is well-posed, and we can find a regularization
parameter $J$ depending on the noise level of the data such that the solution
in $V_j$ is a good approximation of the original problem.
Moreover, the combination of Meyer wavelets method with the fast Fourier
transform (FFT) can build high-speed algorithm.

Meyer wavelet techniques have been used by Regi\'{n}ska et al
\cite{s9,Reginska+Elden-1997}, H\`{a}o et al \cite{s11}, Eld\'{e}n \cite{s10},
and Wang \cite{Wang+2010} to solve the inverse heat conduction problems (IHCP),
 and by Vani \cite{vani+2002}, Qiu et al \cite{Qiu+2008} to solve the
Cauchy problem for the Laplace equation. However, they all used
the \emph{a priori} wavelet method, and did not consider the \emph{a posteriori}
 error estimate. Since the numerical results for the \emph{a posteriori}
method do not depend on the \emph{a priori} information,
the \emph{a posteriori} wavelet method is more effective to solve practical
problems than the \emph{a priori} method. In this paper, we continue
to study the \emph{a priori} wavelet method, and then focus on the
\emph{a posteriori} wavelet method and its numerical solution.

Although the application of wavelet theory in differential equation has been
mostly focused on numerical computation, the connection between the wavelet
theory and differential equation is also
searched all the time. Shen and Strang in \cite{Shen+Strong-2000}
have introduced the concept of heatlets in order to solve the heat equation
using wavelet expansions of the initial data. The heatlet is a ``fundamental''
solution to the heat equation, 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 \cite{Marie-2008} the authors combined heatlets with quasi-reversibility
 method to regularize the backward heat equation, and obtained some theoretical
 error estimates. However, there are no numerical results.

In the present paper, we  consider the following backward heat equation in
a strip domain by a Meyer wavelet method.
\begin{align}\label{e11}
\begin{gathered}
u_t=u_{xx}, \quad -\infty<x<\infty, \; 0\leq t<T,\\
u(x,T)=\varphi_T(x), \quad -\infty<x<\infty,
\end{gathered}
\end{align}
where we want to determine the temperature distribution $u(\cdot,t)$
on the interval $t\in[0,T)$ from the data $\varphi_T(x)$.
Backward heat conduction problem is a classical ill-posed
problem \cite{s1,s2}, and is known as the most severely ill-posed problem,
which has been studied by many authors by different
methods \cite{s3,s4,s5,s6,s7,s8}.

Let $\widehat{g}(\xi)$ denote the Fourier transform of $g(x)$ defined by
\begin{equation}\label{e12}
\widehat{g}(\xi):=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}e^{-ix\xi}g(x)dx,
\end{equation}
and $\|f\|_{H^s}$ denote the norm on the Sobolev space $H^s$ defined by
\begin{equation}\label{e13}
   \|f\|_{H^s}:=\Big(\int_{-\infty}^{\infty}|\widehat{f}(\xi)|^2(1+\xi^2)^sd\xi
\Big)^{1/2}.
\end{equation}
When $s=0$, $ \|\cdot\|_{H^0}:=\|\cdot\|$ denotes the
$L^2(\mathbb{R})$-norm, and $(\cdot,\cdot)$ denotes the
$L^2(\mathbb{R})$-inner product. It is easy to know that
$L^2(\mathbb{R})\subset H^s(\mathbb{R})$ for $s\leq0$.

We assume that there exists a solution $u(x,t)$ satisfying  \eqref{e11}
 in the classical sense and $u(\cdot, t)\in L^2(\mathbb{R})$ for $0<t<T$.
Using the Fourier transform technique to problem \eqref{e11} with respect
to variable $x$, we can get the Fourier transform $\widehat{u}(\xi,t)$
of the exact solution $u(x,t)$ of problem \eqref{e11}:
\begin{equation}\label{e14}
\widehat{u}(\xi,t)=e^{\xi^2(T-t)}\widehat{\varphi}_T(\xi),
\end{equation}
or equivalently,
\begin{equation}\label{e15}
u(x,t)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}e^{i\xi
x}e^{\xi^2(T-t)}\widehat{\varphi}_T(\xi)d\xi.
\end{equation}

For any ill-posed problem some \emph{a priori} assumptions on the exact solution
are needed, otherwise the convergence of the regularization approximate
solution will not be obtained or the
convergence rate can be arbitrarily slow \cite{Engl-1996}.
When we consider problem \eqref{e11} in $L^2(\mathbb{R})$ for the variable $x$,
we assume there exists an \emph{a priori} bound for $\varphi_0(x):=u(x,0)$:
\begin{equation}\label{e17}
    \|\varphi_0\|=\|u(\cdot,0)\|\leq E.
\end{equation}
From the formula \eqref{e15} and the Parseval formula, we know
\begin{equation}\label{e18}
\|\varphi_0\|^2=\int^{\infty}_{-\infty}|e^{\xi^2T}\widehat{\varphi}_T(\xi)|^2d\xi.
\end{equation}

For a concrete ill-posed problem, not all regularization methods are effective.
For example, for the severely ill-posed problems with growth rate of magnitude
factor reaching or exceeding
$O(e^{\gamma\xi^2}), \gamma>0,\xi\to \infty$, the Mollification method with
 Gauss kernel suggested by
Murio \cite{murio-1993} cannot deal with them.
Problem \eqref{e11} considered in the present paper just is the case
as the magnitude factor $e^{\xi^2(T-t)}$. Therefore, the Mollification
method is not effective both in theory and numerical computation.
In addition, the convergence rate and numerical results are also not completely
the same for different regularization methods. For example,
for the Modified method suggested by \cite{Qian+Fu+Shi+2007},
the theoretical convergence rate for problem \eqref{e11} is  only
logarithm type not H\"{o}lder type. So construction of specific regularization
methods for different ill-posed problems is  significant.

The main goal of this paper is to provide a Meyer wavelet method for
solving the backward heat equation \eqref{e11}. Our method of proving
stability estimates is constructive: We construct a stable solution
to the problem for both \emph{a priori} and \emph{a posteriori} choice rules.

The outline of this paper is as follows.
In Section 2, a brief survey on some fundamental properties of Meyer wavelet
is presented. On this basis, we give the Meyer wavelet regularization method
to solve the problem \eqref{e11} for both \emph{a priori}
and \emph{a posteriori} parameter choice rules, and obtain the error
estimates of the \emph{a priori} and \emph{a posteriori} situations respectively.
In Section 3, four numerical examples are provided, and the comparison of
numerical effects for \emph{a posteriori} wavelet method with other methods
for Examples \ref{examp1} and \ref{examp2} are also taken into account.

\section{Wavelet regularization and error estimates}

Let $\varphi(x),\psi(x)$ be Meyer scaling and wavelet functions respectively.
Then from \cite{s12} we know
$$
\operatorname{supp}\widehat{\varphi}=[-\frac{4}{3}\pi,\frac{4}{3}\pi],\quad
\operatorname{supp}\widehat{\psi}
=[-\frac{8}{3}\pi,-\frac{2}{3}\pi]\cup[\frac{2}{3}\pi,\frac{8}{3}\pi],
$$
and $\psi_{jk}(x)=2^{\frac{j}{2}}\psi(2^jx-k),j,k\in\mathbb{Z}$
constitute an orthonormal basis of $L^2(\mathbb{R})$ and
\begin{equation}\label{e21}
\operatorname{supp}\widehat{\psi}_{jk}(\xi)=[-\frac{8}{3}\pi2^j,
-\frac{2}{3}\pi2^j]\cup[\frac{2}{3}\pi2^j,\frac{8}{3}\pi2^j],\quad
k\in\mathbb{Z}.
\end{equation}
The multiresolution analysis (MRA) $\{V_j\}_{j\in\mathbb{Z}}$
of Meyer wavelet is generated by
\begin{equation} \label{e22}
\begin{gathered}
  V_j=\overline{\{\varphi_{jk}:k\in\mathbb{Z}\}},\quad
  \varphi_{jk}:=2^{\frac{j}{2}}\varphi(2^jx-k),\quad
  j,k\in\mathbb{Z}, \\
\operatorname{supp}\widehat{\varphi}_{jk}(\xi)
=[-\frac{4}{3}\pi2^j,\frac{4}{3}\pi2^j],\quad
k\in\mathbb{Z}.
\end{gathered}
\end{equation}
The orthogonal projection of a function $g\in L^2(\mathbb{R})$ on space
$V_J$ is given by
$P_Jg:=\sum_{k\in\mathbb{Z}}(g,\varphi_{Jk})\varphi_{Jk}$, while
$Q_Jg:=\sum_{k\in\mathbb{Z}}(g,\psi_{Jk})\psi_{Jk}$ denotes the orthogonal
projection on wavelet space $W_J$ with $V_{J+1}=V_J\bigoplus W_J$.
It is easy to see from \eqref{e22} and \eqref{e21} that
\begin{gather}
\widehat{P_Jg}(\xi)=0,\quad\text{for } |\xi|\geq\frac{4}{3}\pi
2^J,\label{e23}\\
\widehat{Q_jg}(\xi)=0,\quad\text{for } j>J\,\,
\mbox{and}\,\,|\xi|<\frac{4}{3}\pi 2^J.\label{e24}
\end{gather}
Since $(I-P_J)g=\sum_{j\geq J}Q_jg$ and from \eqref{e24}, we know
\begin{equation}\label{e25}
((I-P_J)g)^{\widehat{}}(\xi)=\widehat{Q_Jg}(\xi),\quad
\text{for } |\xi|<\frac{4}{3}\pi2^J.
\end{equation}

\begin{lemma}[\cite{s11}] \label{lem2.1}
  Let $\{V_j\}_{j\in\mathbb{Z}}$ be Meyer's
MRA and suppose $J\in\mathbb{N}, r\in\mathbb{R}$. Then for all
$g\in V_J$, it holds the estimate
\begin{equation}\label{e26}
    \|D^kg\|_{H^r}\leq C2^{(J-1)k}\|g\|_{H^r},\quad
    k\in\mathbb{N},
\end{equation}
where $C$ is a positive constant and $D^k=\frac{d^k}{dx^k}$.
\end{lemma}

Define an operator $A_t:\varphi_T(x)\mapsto u(x,t)$ by
\eqref{e14}, i.e.,
$$
A_t\varphi_T=u(x,t),\quad 0\leq t<T,
$$
or
\begin{equation}\label{e27}
    \widehat{A_t\varphi_T}(\xi)=e^{\xi^2(T-t)}\widehat{\varphi}_T(\xi),\quad 0\leq
    t<T.
\end{equation}
Then we have the following lemma.

\begin{lemma} \label{lem2.2}
Let $\{V_j\}_{j\in\mathbb{Z}}$ be Meyer's MRA and suppose
$J\in\mathbb{N}, r\in\mathbb{R},0\leq t<T$. Then for all $g\in V_J$ we have
\begin{equation}\label{e28}
    \|A_tg\|_{H^r}\leq2C\exp\big\{2^{2(J-1)}(T-t)\big\}\|g\|_{H^r},
\end{equation}
where constant $C$ is the same as in \eqref{e26}.
\end{lemma}

\begin{proof}
From \eqref{e26} we know that
\begin{align*}
\|A_tg\|_{H^r}
&=\Big(\int^{\infty}_{-\infty}|\widehat{A_tg}(\xi)|^2(1+\xi^2)^rd\xi\Big)^{1/2}\\
&=\Big(\int^{\infty}_{-\infty}|e^{\xi^2(T-t)}\widehat{g}(\xi)|^2(1+\xi^2)^rd\xi
 \Big)^{1/2}\\
&\leq\Big(\int^{\infty}_{-\infty}|2\cosh({\xi^2(T-t)})\widehat{g}(\xi)|^2
 (1+\xi^2)^rd\xi\Big)^{1/2}\\
&=2\Big(\int^{\infty}_{-\infty}|\sum^{\infty}_{k=0}\frac{(T-t)^{2k}}{(2k)!}
 \xi^{4k}\widehat{g}(\xi)|^2(1+\xi^2)^rd\xi\Big)^{1/2}\\
&=2\Big(\int^{\infty}_{-\infty}|\sum^{\infty}_{k=0}\frac{(T-t)^{2k}}{(2k)!}
 (i\xi)^{4k}\widehat{g}(\xi)|^2(1+\xi^2)^rd\xi\Big)^{1/2}\\
&\leq2\sum^{\infty}_{k=0}\frac{(T-t)^{2k}}{(2k)!}\|D^{4k}g\|_{H^r}\\
&\leq2C\sum^{\infty}_{k=0}\frac{(T-t)^{2k}}{(2k)!}2^{(J-1)4k}\|g\|_{H^r}\\
&=2C\cosh\big(2^{2(J-1)}(T-t)\big)\|g\|_{H^r}\\
&\leq2C\exp\big\{2^{2(J-1)}(T-t)\big\}\|g\|_{H^r}.
\end{align*}
\end{proof}

Let $\varphi_T(x),\varphi^{\delta}_{T}(x)$ be exact and measured data,
respectively, which satisfy
\begin{equation}\label{e29}
\|\varphi_T-\varphi^{\delta}_{T}\|_{H^r}<\delta,\quad \text{for some } r\leq0.
\end{equation}
Since $\varphi^{\delta}_{T}(x)$ belongs, in general, to
$L^2(\mathbb{R})\subset H^r(\mathbb{R})$ for $r\leq0$, $r$ should not be positive.
In general, $L^2$ \emph{a priori} bound for exact solution as \eqref{e17} can only
lead to a H\"{o}lder stability estimate for the regularization solution,
but this \emph{a priori} assumption can not ensure the convergence of the
regularization solution at $t=0$ for problem \eqref{e11}.
To obtain a more sharp convergence for the regularization solution,
here we assume, for some $s\ge r$, there exists an \emph{a priori} bound:
\begin{equation}\label{e210}
    \|\varphi_0\|_{H^s}\leq E.
\end{equation}
Denote operator $A_{t,J}:=A_tP_J$, we can show it approximates $A_t$ in a
stable way for an appropriate choice of $J\in\mathbb{N}$ depending on
$\delta$ and $E$. In fact, we have
\begin{equation} \label{e211}
\|A_t\varphi_T-A_{t,J}\varphi^{\delta}_T\|_{H^r}
\leq\|A_t\varphi_T-A_{t,J}\varphi_T\|_{H^r}+\|A_{t,J}
\varphi_T-A_{t,J}\varphi^{\delta}_T\|_{H^r}.
\end{equation}
From Lemma \ref{lem2.2} and condition \eqref{e29}, we can see that the second term
of the right-hand side of \eqref{e211} satisfies
\begin{equation} \label{e212}
\begin{aligned}
\|A_{t,J}\varphi_T-A_{t,J}\varphi^{\delta}_T\|_{H^r}
&=\|A_tP_J(\varphi_T-\varphi_T^{\delta})\|_{H^r} \\
&\leq2C\exp\big\{2^{2(J-1)}(T-t)\big\}\|P_J(\varphi_T-\varphi_T^{\delta})\|_{H^r} \\
&\leq2C\exp\big\{2^{2(J-1)}(T-t)\big\}\delta.
\end{aligned}
\end{equation}
For the first term of the right-hand side of \eqref{e211}, from \eqref{e23} we have
\begin{equation} \label{e213}
\begin{aligned}
&\|A_t\varphi_T-A_{t,J}\varphi_T\|_{H^r} \\
&=\|A_t(I-P_J)\varphi_T\|_{H^r} \\
&=\Big(\int^{\infty}_{-\infty}|e^{\xi^2(T-t)}
 \big((I-P_J)\varphi_T\big)^{\widehat{}}(\xi)|^2(1+\xi^2)^rd\xi\Big)^{1/2} \\
&=\Big(\int_{|\xi|\geq\frac{4}{3}\pi2^J}|e^{\xi^2(T-t)}\widehat{\varphi}_T(\xi)|^2
 (1+\xi^2)^rd\xi\Big)^{1/2} \\
&\quad +\Big(\int_{|\xi|<\frac{4}{3}\pi2^J}|e^{\xi^2(T-t)}
 \big((I-P_J)\varphi_T\big)^{\widehat{}}(\xi)|^2(1+\xi^2)^rd\xi\Big)^{1/2}
:=I_1+I_2.
\end{aligned}
\end{equation}
Noting \eqref{e14} and \eqref{e210}, we have
\begin{equation} \label{e214}
\begin{aligned}
I_1
&=\Big(\int_{|\xi|\geq\frac{4}{3}\pi2^J}|e^{\xi^2(T-t)}
 \widehat{\varphi}_T(\xi)|^2(1+\xi^2)^rd\xi\Big)^{1/2} \\
&=\Big(\int_{|\xi|\geq\frac{4}{3}\pi2^J}|e^{-t\xi^2}
 \widehat{\varphi}_0(\xi)|^2(1+\xi^2)^rd\xi\Big)^{1/2} \\
&\leq\sup_{|\xi|\geq\frac{4}{3}\pi2^J}e^{-t\xi^2}
 \frac{1}{(1+\xi^2)^{\frac{s-r}{2}}}
 \Big(\int_{|\xi|\geq\frac{4}{3}\pi2^J}|\widehat{\varphi}_0(\xi)|^2
 (1+\xi^2)^sd\xi\Big)^{1/2} \\
&\leq\sup_{|\xi|\geq\frac{4}{3}\pi2^J}e^{-t\xi^2}
 \frac{1}{(1+\xi^2)^{\frac{s-r}{2}}}\|\varphi_0\|_{H^s} \\
&\leq e^{-t(\frac{4}{3}\pi2^J)^2}\frac{E}{(\frac{4}{3}\pi2^J)^{s-r}}
 \leq e^{-t2^{2(J+2)}}\frac{E}{2^{(J+2)(s-r)}} \\
&=2^{-(J+2)(s-r)}\exp\{-t2^{2(J+2)}\}E.
\end{aligned}
\end{equation}
From \eqref{e25}, Lemma \ref{lem2.2}, and noting that $Q_J\varphi_T\in W_J\subset V_{J+1}$,
 it is easy to see that $I_2$ satisfies:
\begin{align*}
I_2
&=\Big(\int_{|\xi|<\frac{4}{3}\pi2^J}|e^{\xi^2(T-t)}
\left((I-P_J)\varphi_T\right)^{\widehat{}}(\xi)|^2(1+\xi^2)^rd\xi\Big)^{1/2}\\
&=\Big(\int_{|\xi|<\frac{4}{3}\pi2^J}|e^{\xi^2(T-t)}\widehat{Q_J\varphi_T}
 (\xi)|^2(1+\xi^2)^rd\xi\Big)^{1/2}\\
&\leq\|A_tQ_J\varphi_T\|_{H^r}\\
&\leq2C\exp\{2^{\,2J}(T-t)\}\|Q_J\varphi_T\|_{H^r}.
\end{align*}
Denote $\chi_J$ as the characteristic function of the interval
$[-\frac{2}{3}\pi2^J,\frac{2}{3}\pi2^J]$. We introduce an operator
$M_J$ defined by
$$
\widehat{M_Jg}=(1-\chi_J)\widehat{g},\quad g\in L^2(\mathbb{R}).
$$
Noting that $\varphi_T\in L^2(\mathbb{R})$, so from Parseval formula
and \eqref{e21} we have
\begin{align*}
Q_J\varphi_T
&=\sum_{k\in \mathbb{Z}}(\varphi_T,\psi_{Jk})\psi_{Jk}
=\sum_{k\in \mathbb{Z}}(\widehat{\varphi}_T,\widehat{\psi}_{Jk})\psi_{Jk}\\
&=\sum_{k\in \mathbb{Z}}((1-\chi_J)\widehat{\varphi}_T,\widehat{\psi}_{Jk})\psi_{Jk}\\
&=\sum_{k\in \mathbb{Z}}(M_J\varphi_T,\psi_{Jk})\psi_{Jk}\\
&=Q_JM_J\varphi_T.
\end{align*}
So, from \eqref{e14},
\begin{equation} \label{e2*}
\begin{aligned}
\|Q_J\varphi_T\|_{H^r}
&=\|Q_JM_J\varphi_T\|_{H^r}
 \leq\|M_J\varphi_T\|_{H^r} \\
&=\Big(\int_{|\xi|\geq\frac{2}{3}\pi2^J}|\widehat{\varphi}_T(\xi)|^2
 (1+\xi^2)^rd\xi\Big)^{1/2} \\
&=\Big(\int_{|\xi|\geq\frac{2}{3}\pi2^J}|e^{-\xi^2T}\widehat{\varphi}_0(\xi)|^2
 (1+\xi^2)^rd\xi\Big)^{1/2} \\
&=\Big(\int_{|\xi|\geq\frac{2}{3}\pi2^J}|e^{-\xi^2T}
 \frac{1}{(1+\xi^2)^{\frac{s-r}{2}}}|^2|\widehat{\varphi}_0
 (\xi)|^2(1+\xi^2)^sd\xi\Big)^{1/2} \\
&\leq\sup_{|\xi|\geq\frac{2}{3}\pi2^J}
 \frac{e^{-\xi^2T}}{|\xi|^{s-r}}\|\varphi_0\|_{H^s} \\
&\leq 2^{-(J+1)(s-r)}\exp\big\{-T2^{2(J+1)}\big\}E.
\end{aligned}
\end{equation}
Therefore,
\begin{equation} \label{e215}
\begin{aligned}
I_2&\leq2C\exp\left\{2^{2J}(T-t)\right\}2^{-(J+1)(s-r)}
\exp\big\{-T2^{2(J+1)}\big\}E \\
&\leq2C2^{-(J+1)(s-r)}\exp\big\{2^{2(J+1)}(T-t)-T2^{2(J+1)}\big\}E \\
&=2C2^{-(J+1)(s-r)}\exp\big\{-t2^{2(J+1)}\big\}E.
\end{aligned}
\end{equation}
From \eqref{e214}, \eqref{e215} and \eqref{e213}, we obtain
\begin{equation} \label{e216}
\begin{aligned}
&\|A_t\varphi_T-A_{t,J}\varphi_T\|_{H^r} \\
&\leq2^{-(J+2)(s-r)}\exp\{-t2^{2(J+2)}\}E+2C2^{-(J+1)(s-r)}
\exp\big\{-t2^{2(J+1)}\big\}E \\
&\leq(1+2C)2^{-(J+1)(s-r)}\exp\big\{-t2^{2(J+1)}\big\}E.
\end{aligned}
\end{equation}
Combining \eqref{e216}, \eqref{e212} with \eqref{e211}, we have
\begin{equation} \label{e217}
\begin{aligned}
&\|A_t\varphi_T-A_{t,J}\varphi^{\delta}_T\|_{H^r} \\
&\leq2C\exp\big\{2^{2(J-1)}(T-t)\big\}\delta+(1+2C)2^{-(J+1)(s-r)}
\exp\big\{-t2^{2(J+1)}\big\}E \\
&\leq2C\exp\left\{2^{2J}(T-t)\right\}\delta+(1+2C)2^{-(J+1)(s-r)}
\exp\big\{-t2^{2(J+1)}\big\}E.
\end{aligned}
\end{equation}
Based on the above results, we will give the estimates for the
\emph{a priori} and \emph{a posteriori} parameter choice rules, respectively.

\subsection{\emph{A-priori} parameter choice}

Now we can firstly give an error estimate between the wavelet regularization
solution $A_{t,J}\varphi^{\delta}_T$ and the exact solution $u(x,t)=A_t\varphi_T$
in $L^2(\mathbb{R})$.


\begin{theorem} \label{thm2.1}
Suppose that $\varphi_0\in L^2(\mathbb{R})$ and \eqref{e29}, \eqref{e210} hold
for $r=s=0$. The problem of calculating  $A_{t,J}\varphi^{\delta}_T$ is stable.
Furthermore, taking
\begin{equation}\label{e218}
   J^*:=\Big[\frac{1}{2}\log_2\Big(\ln\big(\frac{E}{\delta}\Big)^{1/T}\big)\Big],
\end{equation}
where $[a]$ denotes the largest integer less than or equal to $a\in \mathbb{R}$,
 then the following stability estimate holds:
\begin{equation}\label{e219}
   \|A_t\varphi_T-A_{t,J^*}\varphi^{\delta}_T\|
\leq(4C+1)E^{1-\frac{t}{T}}\delta^{t/T},
\end{equation}
where $C$ is the same as in \eqref{e26}.
\end{theorem}

\begin{proof}
 Note that
\begin{gather*}
\begin{aligned}
&\exp\big\{2^{2J^*}(T-t)\big\}\delta\\
&\leq\exp\big\{(T-t)\ln\big(\frac{E}{\delta}\big)^{1/T}\big\}\delta
 =\big(\frac{E}{\delta}\big)^{\frac{T-t}{T}}\delta
=E^{1-\frac{t}{T}}\delta^{t/T},
\end{aligned}\\
\exp\big\{-t2^{2(J^*+1)}\big\}E
\leq\exp\big\{-t\ln\big(\frac{E}{\delta}\big)^{1/T}\big\}E
=E^{1-\frac{t}{T}}\delta^{t/T}.
\end{gather*}
and from estimate \eqref{e217} we have
\begin{align*}
\|A_t\varphi_T-A_{t,J^*}\varphi^{\delta}_T\|_{H^0}
&=\|A_t\varphi_T-A_{t,J^*}\varphi^{\delta}_T\|\\
&\leq 2CE^{1-\frac{t}{T}}\delta^{t/T}+(1+2C)E^{1-\frac{t}{T}}\delta^{t/T} \\
&=(4C+1)E^{1-\frac{t}{T}}\delta^{t/T}.
\end{align*}
The proof is complete.
\end{proof}

\begin{remark} \label{rmk2.1} \rm
From the result of reference \cite{s7} we know estimate \eqref{e219}
is an order optimal H\"{o}lder stability estimate in $L^2(\mathbb{R})$.
This suggests that wavelet method must be useful for solving the considered
ill-posed problem. However, from \eqref{e219} we know when $t\to0^+$,
the accuracy of the regularized solution becomes progressively lower.
At $t=0$, it merely implies that the error is bounded by $4C+1$, i.e., the
convergence of the regularized solution at $t=0$ is not proved theoretically.
This defect is remedied by the following result.
\end{remark}

\begin{theorem} \label{thm2.2}
Suppose that $\varphi_0\in H^s(\mathbb{R})$ for some $s\in \mathbb{R}$
and \eqref{e29} holds for $r\leq\min\{0,s\}$. Take
\begin{equation}\label{e220}
J^{**}:=\Big[\frac{1}{2}\log_2\Big(\ln\Big(\big(\frac{E}{\delta}\big)^{1/T}
\big(\ln\frac{E}{\delta}\big)^{-\frac{(s-r)}{2T}}\Big)\Big)\Big],
\end{equation}
where the bracket $[a]$ is the same as in \eqref{e218}.
Then
\begin{equation} \label{e221}
\begin{aligned}
&\|A_t\varphi_T-A_{t,J^{**}}\varphi^{\delta}_T\|_{H^r}\\ 
&\leq\Big(2C+(1+2C)\Big(\frac{\ln\frac{E}{\delta}}
{\frac{1}{T}\ln\frac{E}{\delta}+\ln\big(\ln\frac{E}{\delta}
\big)^{-(s-r)/(2T)}} \Big)^{\frac{s-r}{2}}\Big) \\
&\quad\times E^{1-\frac{t}{T}}\delta^{t/T}\big(\ln\frac{E}{\delta}
\big)^{-\frac{(T-t)(s-r)}{2T}}.
\end{aligned}
\end{equation}
\end{theorem}

\begin{proof} Note that
\begin{align*}
\exp\big\{2^{2J^{**}}(T-t)\big\}\delta 
&\leq\exp\Big\{(T-t)\ln\Big(\big(\frac{E}{\delta}\big)^{1/T}
\big(\ln\frac{E}{\delta}\big)^{-\frac{(s-r)}{2T}}\Big)\Big\}\delta\\
&=\big(\frac{E}{\delta}\big)^{\frac{T-t}{T}}\big(\ln\frac{E}{\delta}
\big)^{-\frac{(T-t)(s-r)}{2T}}\delta\\
&=E^{1-\frac{t}{T}}\delta^{t/T}\big(\ln\frac{E}{\delta}
 \big)^{-\frac{(T-t)(s-r)}{2T}},
\end{align*}
\begin{align*}
\exp\big\{-t2^{2(J^{**}+1)}\big\}E
&\leq\exp\Big\{-t\ln\Big(\big(\frac{E}{\delta}\big)^{1/T}
 \big(\ln\frac{E}{\delta}\big)^{-\frac{(s-r)}{2T}}\Big)\Big\}E\\
&=\big(\frac{E}{\delta}\big)^{-\frac{t}{T}}
 \big(\ln\frac{E}{\delta}\big)^{\frac{t(s-r)}{2T}}E \\
&=E^{1-\frac{t}{T}}\delta^{t/T}\big(\ln\frac{E}{\delta}\big)^{\frac{t(s-r)}{2T}},
\end{align*}
\begin{align*}
2^{-(J^{**}+1)(s-r)}
&=2^{2(J^{**}+1)(-\frac{s-r}{2})}\\
&\leq\Big(\ln\Big(\big(\frac{E}{\delta}\big)^{1/T}
 \big(\ln\frac{E}{\delta}\big)^{-\frac{(s-r)}{2T}}\Big)\Big)^{-\frac{s-r}{2}}\\
&=\Big(\frac{1}{\frac{1}{T}\ln\frac{E}{\delta}
 +\ln\big(\ln\frac{E}{\delta}\big)^{-\frac{(s-r)}{2T}}}\Big)^{\frac{s-r}{2}}\\
&=\Big(\frac{\ln\frac{E}{\delta}}{\frac{1}{T}\ln\frac{E}{\delta}
+\ln\big(\ln\frac{E}{\delta}\big)^{-\frac{(s-r)}{2T}}}\Big)^{\frac{s-r}{2}}
\big(\ln\frac{E}{\delta}\big)^{-\frac{(s-r)}{2}},
\end{align*}
and from \eqref{e217} we know
\begin{align*}
& \|A_t\varphi_T-A_{t,J^{**}}\varphi^{\delta}_T\|_{H^r} \\
&\leq 2CE^{1-\frac{t}{T}}\delta^{t/T}
 \big(\ln\frac{E}{\delta}\big)^{-\frac{(T-t)(s-r)}{2T}}
 +(1+2C)\Big(\frac{\ln\frac{E}{\delta}}{\frac{1}{T}\ln\frac{E}{\delta}
+\ln\big(\ln\frac{E}{\delta}\big)^{-\frac{(s-r)}{2T}}}\Big)^{\frac{s-r}{2}} \\
&\quad\times \big(\ln\frac{E}{\delta}\big)^{-\frac{T(s-r)}{2T}}
E^{1-\frac{t}{T}}\delta^{t/T}\big(\ln\frac{E}{\delta}\big)^{\frac{t(s-r)}{2T}}\\
&\leq\Big(2C+(1+2C)\Big(\frac{\ln\frac{E}{\delta}}{\frac{1}{T}
 \ln\frac{E}{\delta}+\ln\big(\ln\frac{E}{\delta}\big)^{-\frac{s-r}{2T}}}
 \Big)^{\frac{s-r}{2}}\Big)E^{1-\frac{t}{T}}\delta^{t/T}
 \big(\ln\frac{E}{\delta}\big)^{-\frac{(T-t)(s-r)}{2T}},
\end{align*}
where the factor
\[
\frac{\ln\frac{E}{\delta}}{\frac{1}{T}\ln\frac{E}{\delta}
+\ln\big(\ln\frac{E}{\delta}\big)^{-\frac{s-r}{2T}}}
\]
is bounded as $\delta\to0^+$. So, the proof of estimate \eqref{e221} is complete.
\end{proof}

\begin{remark} \label{rmk2.2} \rm
When $s=r=0$, estimate \eqref{e220} becomes estimate \eqref{e218}
 and the convergence speed given by \eqref{e221} is faster than the one
 given by \eqref{e219} for $s>r$. Especially, when
$t=0$, estimate \eqref{e221} becomes
\begin{align*}
&\|A_0\varphi_T-A_{0,J^{**}}\varphi^{\delta}_T\|_{H^r}\\
&\leq \Big(2C+(1+2C)\Big(\frac{\ln\frac{E}{\delta}}{\frac{1}{T}
\ln\frac{E}{\delta}+\ln\big(\ln\frac{E}{\delta}\big)^{-\frac{s-r}{2T}}}
\Big)^{\frac{s-r}{2}}\Big)\big(\ln\frac{E}{\delta}\big)^{-\frac{(s-r)}{2}}E\to0,
\end{align*}
when $\delta\to0^+$ and $s>r$. This is a logarithmical stability estimate 
and  an important improvement to estimate \eqref{e219}.
\end{remark}

\begin{remark} \label{rmk2.3} \rm
Noting that
\[
\lim_{\delta\to0}\frac{\ln\frac{E}{\delta}}{\frac{1}{T}\ln\frac{E}{\delta}
+\ln\big(\ln\frac{E}{\delta}\big)^{-\frac{s-r}{2T}}}=T,
\]
 estimate \eqref{e221} also can be rewritten in the  asymptotic form
\begin{align*}
& \|A_t\varphi_T-A_{t,J^{**}}\varphi^{\delta}_T\|_{H^r} \\
&\leq\left(2C+(1+2C)T^{\frac{s-r}{2}}+o(1)\right)E^{1-\frac{t}{T}}
\delta^{t/T}\big(\ln\frac{E}{\delta}\big)^{-\frac{(T-t)(s-r)}{2T}},
\end{align*}
as $\delta\to 0$.
\end{remark}

\subsection{\emph{A-posteriori} parameter choice}

In this subsection, we consider the \emph{a posteriori} regularization 
parameter choice in the Morozov's discrepancy principle. 
This principle has been used by Feng et al \cite{Feng+Ning+2015} 
to solve the numerical analytic continuation, however the backward heat
 conduction problem is more severely ill-posed than the numerical analytic 
continuation.

\begin{lemma} \label{lem2.3} 
Assume conditions \eqref{e29} for $r=0$ and \eqref{e210} for $s=0$ hold. 
If $J$ is chosen as the solution of the inequalities
\begin{equation}\label{e227}
\|P_J\varphi_T^\delta-\varphi_T^\delta\|\leq\tau\delta
\leq\|P_{J-1}\varphi_T^\delta-\varphi_T^\delta\|,\,\tau>1 ,
\end{equation}
then it holds
\begin{equation}\label{e228}
\exp(2^{2J}T)\leq\frac{2E}{(\tau-1)\delta}.
\end{equation}
\end{lemma}

\begin{proof} 
From Equations \eqref{e23} and \eqref{e25}, we know
\begin{equation} \label{e229}
\begin{aligned}
 & \|P_{J-1}\varphi_T-\varphi_T\| \\
&=\Big(\int_{-\infty}^{\infty}|((I-P_{J-1})\varphi_T)^{\widehat{}}(\xi)|^2d\xi
 \Big)^{1/2} \\
&\leq \Big(\int_{|\xi|\geq\frac{4}{3}\pi2^{J-1}}|\hat{\varphi_T}(\xi)|^2d\xi
 \Big)^{1/2}+\Big(\int_{|\xi|<\frac{4}{3}\pi2^{J-1}}|(Q_{J-1}\varphi_T)^{\widehat{}}
 (\xi)|^2d\xi\Big)^{1/2} \\
&=:  I_3+I_4.
\end{aligned}
\end{equation}
From \eqref{e14},
\begin{equation}\label{e230}
I_3\leq\exp(-(\frac{4}{3}\pi2^{J-1})^2T)E\leq\exp(-2^{2J}T)E.
\end{equation}
From \eqref{e2*},
\begin{equation} \label{e231}
I_4 \leq\|(Q_{J-1}\varphi_T)^{\widehat{}}(\xi)\|
\leq 2^{-Js}\exp(-T2^{2J})E
\leq\exp(-2^{2J}T)E,\quad s\geq0
\end{equation}
Combining  \eqref{e229} with \eqref{e230} and \eqref{e231}, we obtain
\begin{equation}\label{e232}
\|P_{J-1}\varphi_T-\varphi_T\|\leq2\exp(-2^{2J}T)E.
\end{equation}
On the other hand, by \eqref{e29} for $r=0$, \eqref{e227},
and the triangle inequality give
\begin{equation}\label{e233}
\|P_{J-1}\varphi_T-\varphi_T\|\geq\|(I-P_{J-1})\varphi^\delta_T\|
-\|(I-P_{J-1})(\varphi_T-\varphi^\delta_T)\|\geq(\tau-1)\delta.
\end{equation}
From \eqref{e232} and \eqref{e233}, estimate \eqref{e228} is proved.
\end{proof}

\begin{theorem} \label{thm2.3} 
Assume that conditions \eqref{e29} for $r=0$ and \eqref{e210} for $s=0$ hold. 
If the regularization parameter $J$ is chosen as the solution of inequalities 
\eqref{e227}, then 
\begin{equation}\label{e234}
\|A_t\varphi_T-A_{t,J}\varphi_T^\delta\|
\leq \widetilde{C}E^{1-\frac{t}{T}}\delta^{t/T},
\end{equation}
where $\widetilde{C}=(2C(\frac{2}{\tau-1})^{1-\frac{t}{T}}+(\tau+1)^{t/T})$, 
and the constant $C$ is the same as in \eqref{e26}.
\end{theorem}


\begin{proof} 
By the Parseval formula and the triangle inequality, 
\begin{equation}\label{e235}
\|A_t\varphi_T-A_{t,J}\varphi_T^\delta\|
\leq \|\widehat{A_{t,J}\varphi_T^\delta}-\widehat{A_{t,J}\varphi_T}\|
+\|\widehat{A_{t,J}\varphi_T}-\widehat{A_{t}\varphi_T}\|=:I_5+I_6.
\end{equation}
From Equation \eqref{e212} and Lemma \ref{lem2.3},
\begin{equation} \label{e236}
\begin{aligned}
I_5
&\leq2C\exp(2^{2(J-1)}(T-t))\delta \\
&=2C(\exp(2^{2(J-1)}T))^{\frac{T-t}{T}}\delta \\
&\leq2C\Big(\frac{2E}{(\tau-1)\delta}\Big)^{\frac{T-t}{T}}\delta.
\end{aligned}
\end{equation}
Moreover,
\begin{equation} \label{e237}
\begin{aligned}
&I_6^2\\
& = \int_{-\infty}^{\infty}|e^{\xi^2(T-t)}((I-P_{J})\varphi_T)^{\widehat{}}
 (\xi)|^2d\xi \\
& =\int_{-\infty}^{\infty}|e^{\xi^2T}((I-P_{J})\varphi_T)^{\widehat{}}
 (\xi)|^{2\frac{T-t}{T}}|((I-P_{J})\varphi_T)^{\widehat{}}(\xi)|^{2\frac{t}{T}}d\xi \\
&\leq\Big(\int_{-\infty}^{\infty}|e^{\xi^2T}((I-P_{J})\varphi_T)^{\widehat{}}
 (\xi)|^2d\xi\Big)^{\frac{T-t}{T}}
\Big(\int_{-\infty}^{\infty}|((I-P_{J})\varphi_T)^{\widehat{}}(\xi)|^2d\xi\Big)^{t/T}.
\end{aligned}
\end{equation}
By combining  \eqref{e17} with  \eqref{e227} and \eqref{e237}, it holds
\begin{align}\label{e238}
I_6^2 \leq E^{2(1-\frac{t}{T})}((\tau+1)\delta)^{2\frac{t}{T}}.
\end{align}
From \eqref{e235}, \eqref{e236} and \eqref{e238}, we complete the proof.
\end{proof}

\section{Numerical aspects}

In this section, we want to discuss some numerical aspects of the proposed method.

\subsection{Numerical implementation}
Suppose that the vector $\{\Phi(x_i)\}_{i=1}^N$ represents samples from 
the function $\varphi_{T}(x)$, and $N$ is even, then we add a random 
normal distribution to each data and obtain
the perturbation data
\begin{equation}
\Phi^{\delta}=\Phi+\epsilon \operatorname{randn}(\operatorname{size}(\Phi)),
\end{equation}
where the function ``$\operatorname{randn}(\cdot)$" 
generates arrays of random numbers whose elements are normally
 distributed with mean 0, variance $\sigma^2=1$, and standard deviation 
$\sigma=1$.

The total noise $\delta$ can be measured in the sense of root mean square
 error according to
\begin{equation}
\delta=\|\Phi^{\delta}-\Phi\|_{l^2}
:=\Big(\frac{1}{N}\sum_{i=1}^N(\Phi^{\delta}(x_i)-\Phi(x_i))^2\Big)^{1/2}.
\end{equation}
For a given measured function $\varphi_{T}^{\delta}$, from Section 2, we have
\begin{equation} \label{e3.3}
\widehat{A_{t,J}\varphi_{T}^{\delta}}(\xi)=\sum_{k\in
Z}(\widehat{\varphi_{T}^{\delta}} ,
\widehat{\varphi}_{Jk})e^{\xi^2(T-t)}\widehat{\varphi}_{Jk}=e^{\xi^2(T-t)}\sum_{k\in
Z}(\widehat{\varphi_{T}^{\delta}} ,
\widehat{\varphi}_{Jk})\widehat{\varphi}_{Jk}.
\end{equation}

If no specific instructions are assumed, we will compute the regularization 
parameter $J$ according to inequalities \eqref{e227} with $\tau=1.1$. 
By using the Discrete Meyer wavelet Transform (DMT) and the Fast Fourier 
Transform (FFT), we can easily compute the regularized solution according 
to formula \eqref{e3.3}. Algorithms for DMT are described in \cite{Kolaczyk}. 
These algorithms are based on the FFT, and computing the DMT of a vector in
 $\mathbb{R}$ requires $\mathcal{O}(N\log_2^2N)$ operations.


\subsection{Numerical tests}
In this subsection some numerical tests are presented to demonstrate the 
usefulness of our method. The tests are performed using Matlab 7.0 and the 
wavelet package WaveLab 850, which is downloaded from
http://www-stat.stanford.edu/$\sim$wavelab/.

Examples \ref{examp1} and \ref{examp2} are from \cite{Fu+Xiong+Qian+2007} and 
\cite{Qian+Fu+Shi+2007}, respectively. In theoretical aspect, the error 
estimates of regularization solutions for the wavelet method (WM) and 
Fourier method (FM) both are sharper H\"{o}lder-logarithm type, but it is 
only weaker logarithm type for the Modified method (MM). Here some
comparison of numerical result for these three methods are considered. 
The case with no explicit solutions is considered in Examples \ref{examp3} 
and \ref{examp4}.

In the following tests, the initial time is chosen as $t=0$, and the number 
of the discrete points $N$ is $128$. The selection of regularization parameters 
for Fourier method and Modified method are chosen according to
\cite[(2.8) with $s=0$]{Fu+Xiong+Qian+2007} and 
\cite[(3.24)]{Qian+Fu+Shi+2007} in Examples \ref{examp1} and \ref{examp2}, 
respectively.

Let $u$ be the exact solution and $v$ be the approximation of some 
regularization method. The absolute error $e_a(u)$ is defined as
$$
e_a(u):=\|u-v\|_{l^2}=\Big(\frac{1}{N}\sum_{n=1}^{N}|u(n)-v(n)|^2\Big)^{1/2},
$$


\begin{example}[\cite{Fu+Xiong+Qian+2007}] \label{examp1} \rm
  It is easy to verify that the function
\begin{equation}
u(x,t)=\frac{1}{\sqrt{1+4t}}e^{-\frac{x^2}{1+4t}}
\end{equation}
is the unique solution of the  problem
\begin{equation} \label{eq:1}
\begin{gathered}
u_t= u_{xx}, \quad x\in \mathbb{R}, \quad t>0, \\
u(x,T)=\varphi_T(x):=\frac{1}{\sqrt{1+4T}}e^{-\frac{x^2}{1+4T}},
\quad x\in \mathbb{R}.
\end{gathered}
\end{equation}
\end{example}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig1a}\\
\includegraphics[width=0.6\textwidth]{fig1b}\\
\includegraphics[width=0.6\textwidth]{fig1c}
\end{center}
\caption{Example \ref{examp1}. Exact solution and approximation at $t=0$ 
from $T=0.01$ (top), $T=0.04$ (middle), 
$T=0.09$ (bottom) with $\epsilon=10^{-2}$.}
\label{fig:1}
\end{figure}

\begin{table}[htb]
\caption{Regularization parameters for different methods with $\epsilon=10^{-2}$}
\label{tab:1}
\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|cccc|}
\hline
$T$ & $\xi_{\max}$(FM) & $\mu$(MM) & $J$(WM)  \\
\hline
 $0.01$&  21.4597  &   0.0046  &  4 \\
 $0.04$&  10.7298  &   0.0185  &  4\\
 $0.09$&   7.1532  &   0.0416  &  4 \\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2}
\end{center}
  \caption{Example \ref{examp1}. Reconstruction error obtained by the different methods for
 different numbers of discrete points with $T=0.01$, $\epsilon=10^{-3}$.}
\label{fig:2}
\end{figure}

Figure \ref{fig:1} illustrates the exact solution and the approximation
 corresponding to the three methods at different times $t=0$ from 
$T=0.01, T=0.04, T=0.09$  with $\epsilon=10^{-2}$ for 
Example \ref{examp1} in the interval $x\in[-10,10]$. 
The regularization parameters are chosen as in Table \ref{tab:1}. 
Here $\xi_{\max}$, $\mu$ and $J$ are the regularization parameters 
of Fourier method, Modified method and Wavelet method, respectively. 
Figure \ref{fig:2} shows that the reconstruction error obtained by 
the different methods for different numbers of discrete points with 
$T=0.01$ and $\epsilon=10^{-3}$. It shows that the number $N$ of 
discrete points (i.e., the step length) also plays the role of 
regularization parameter \cite{Larsdifference}. 
According to the general regularization theory, it should be neither 
too small nor too large. But usually, in some regularization methods, 
the influence of number $N$ is less than the regularization parameter.


\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig3a}\\
\includegraphics[width=0.6\textwidth]{fig3b}\\
\includegraphics[width=0.6\textwidth]{fig3c}
\end{center}
  \caption{Example \ref{examp2}. Exact solution and approximation at $t=0$ 
from  $T=0.01$ (top), 
$T=0.04$ (middle), $T=0.09$ (bottom) with $\epsilon=10^{-2}$.} \label{fig:3}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=0.7\textwidth]{fig4}
\end{center}
  \caption{Example \ref{examp3}. Reconstruction error obtained by the different methods for
 different noisy levels with $T=0.01$, $N=128$.} \label{fig:4}
\end{figure}


\begin{example}[\cite{Qian+Fu+Shi+2007}] \label{examp2} \rm
 The function
\begin{equation}
u(x,t)=e^{-t}\sin(x)
\end{equation}
is the unique solution of the  problem
\begin{equation} \label{eq:2}
\begin{gathered}
u_t= u_{xx}, \quad x\in \mathbb{R}, \quad t>0, \\
u(x,T)=\varphi_T(x):=e^{-T}\sin(x), \quad x\in \mathbb{R}.
\end{gathered}
\end{equation}

 Figure \ref{fig:3} plots the exact solution and the approximation by the 
three methods at $t=0$ from different times $T=0.01, T=0.04, T=0.09$ with 
$\epsilon=10^{-2}$ for Example \ref{examp2} in the interval
 $x\in[-3\pi,3\pi]$. The regularization parameters are also chosen as in 
Table \ref{tab:1}. Figure \ref{fig:4} illustrates that the reconstruction
 error obtained by the different methods for different noisy levels with 
$T=0.01$, $N=128$. It shows that, for the different methods, the error 
between the exact solution and the approximate solution gets smaller as 
the noise in the data decreases.
\end{example}

From the Figures \ref{fig:1} and \ref{fig:3}, we can also see that, for all 
methods, the smaller the $T$, the better the approximation. 
This phenomenon conforms to the theory that, the larger the $T$, 
the more ill-posed the problem.

Unfortunately, for general data $\varphi_T$, it is not easy to find an explicit
 analytical solution to problem \eqref{e11}, so we will construct new examples 
as follows: take a function $\varphi_0(x)\in L^2(\mathbb{R})$ and solve the 
well-posed problem
\begin{equation} \label{eq:1b}
\begin{gathered}
u_t= u_{xx}, \quad  x\in \mathbb{R}, \quad t>0, \\
u(x,0)=\varphi_0(x), \quad x\in \mathbb{R},
\end{gathered}
\end{equation}
to get an approximation to $\varphi_T(x)$. To avoid the inverse crime,
 we use the finite difference to compute this well-posed problem.
Here we discretize problem \eqref{eq:1b} only with respect to the spatial
variable $x$ and leave the time variable $t$ continuous, and then we obtain
a system of ordinary differential equations and we can solve it using an
explicit Runge-Kutta method. See the details in \cite{Qian+Fu+Shi+2007}.
Then we add a random noise to $\varphi_T(x)$ to get the noisy data
$\varphi_T^{\delta}(x)$. At last, we use the proposed regularized technique
to obtain the regularized solution at $t=0$.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig5a}\\
\includegraphics[width=0.6\textwidth]{fig5b}
\end{center}
  \caption{Example \ref{examp3}. Computed input data $\varphi_T(x)$ (top); 
exact and regularized solutions with different $\epsilon$  from 
$T=0.09$ (bottom).} \label{fig:5}
\end{figure}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig6a}\\
\includegraphics[width=0.6\textwidth]{fig6b}
\end{center}
  \caption{Example \ref{examp4}: Computed input data $\varphi_T(x)$ (top); 
Exact and regularized solutions with different $\epsilon$ 
from $T=0.09$ (bottom).} \label{fig:6}
\end{figure}

\begin{example} \label{examp3} \rm
We choose a non-smooth function
\[
\varphi_0(x)=\begin{cases}
1+\frac{x}{3}, & -3 \leq x \leq 0,\\
1-\frac{x}{3}, & 0 < x \leq 3, \\
0, & |x| > 3.
\end{cases}
\]
\end{example}

\begin{example} \label{examp4} \rm
 We consider a discontinuous function
\[
\varphi_0(x)=\begin{cases}
1, & -3 \leq x \leq 0,\\
-1, & 0 < x \leq 3,\\
0, & |x| > 3.
\end{cases}
\]
\end{example}


Figures \ref{fig:5} and \ref{fig:6} illustrate the exact and regularized 
solutions corresponding to different $\epsilon$ for Examples \ref{examp3}
 and \ref{examp4}, 
respectively. The results show that the approximation gets better as 
the noise level $\epsilon$ decreases.
Although $\varphi_0(x)$ in Example \ref{examp3} is not smooth and
$\varphi_0(x)$ in Example \ref{examp4} is even non-continuous, our method is 
also effective for them.

In summary, from the above different kinds of numerical examples, 
we can conclude that the numerical solution is stable, and the Meyer 
wavelet method is an applicable method. This accords with our 
theoretical results.

\subsection*{Acknowledgments}
The authors are very grateful to the anonymous referee for their helpful
comments and suggestions and to Prof. Chu-Li Fu for his constructive 
suggestions. The authors also thank Dr. Yuan-Xiang Zhang and Dr. Yun-Jie Ma 
for helpful discussion. The work is supported by the National Natural Science 
Foundation of China (Nos. 11401456, 41475068), and by 
the Natural Science Basic Research Plan in Shaanxi Province of China 
(No. 2015JQ1016).

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