A Modified Regularization Method for Inverse Problems of Nonhomogeneous Differential Operator Equation

Abstract

This paper investigates an abstract nonhomogeneous backward Cauchy problem governed by an unbounded linear operator in a Hilbert space $H$. The coefficient operator in the equation is assumed to be unbounded, self-adjoint, positive, and to possess a discrete spectrum, with data prescribed at the final time $t=T$. It is well known that such problems are severely ill-posed. To regularize the problem, we employ a modified approach in which we perturb both the equation and the final condition, rather than treating only one of them. Specifically, the key idea of our work is to simultaneously apply two regularization methods: the quasireversibility method and the quasiboundary value method, to obtain an approximate nonlocal problem depending on two small parameters. We establish stability estimates for the solution of the regularized problem and show that the modified problem is stable, with its solution approximating the exact solution of the original problem. Furthermore, a numerical experiment involving the one-dimensional heat equation is conducted to confirm the practical effectiveness of the proposed method and to illustrate its potential for addressing this type of inverse problem.

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let $H$ be a Hilbert space and let $A$ be a positive $A\ge \eta >0$, self-adjoint $A=A^{\mathrm{\ast }}$, and unbounded linear operator with a discrete spectrum on $H$, such that $-A$ generates a compact contraction semigroup on $H$. Given a positive real number $T$, we consider the final value problem of finding a function $u:0,T⟶H$ satisfying$$\begin{array}{}\text{(1)}& \begin{array}{c}{u}^{\prime }t+Aut=ft,0\le \mathrm{t}<\mathrm{T},\\ uT=g,\end{array}\end{array}$$where $f$ is a given function in $C0,T,H$, and the prescribed final value $g$ is an element of $H$. The objective is to determine $ut$ for $0\le t<T$, based on the knowledge of the final value $uT=g$. It is well known that such problems are not well-posed in the sense of Hadamard; that is, even if a unique solution exists on $0,T$, it does not depend continuously on the given final data $g$. Therefore, regularization methods are required. The basic idea of regularization is to replace the ill-posed original problem with a well-posed one.

From a physical standpoint, inverse problems of the type addressed in this work arise in various domains, including heat diffusion, wave propagation, and image deblurring, where the operator $A$ often corresponds to the Laplacian or its fractional powers [1, 2]. Similar formulations also arise in hydrology and fluid dynamics, for instance, in the detection of leaks in pipeline networks [3]. These problems are inherently ill-posed and require robust and stable solution techniques to ensure meaningful reconstructions.

From a mathematical perspective, a wide range of strategies has been developed over the years to address such backward Cauchy problems. Classical methods include the quasisolution approach introduced by Tikhonov, iterative techniques proposed by Kozlov and Maz’ya, and the C-regularized semigroup framework, along with various numerical schemes [4]. More recently, the asymptotic regularization method introduced by Showalter has played a foundational role in tackling ill-posed inverse problems, particularly for abstract evolution equations. In recent years, this method has undergone significant advancements: notably, higher-order versions have been proposed to accelerate convergence [5, 6], while fractional-order regularizations [7] have increased modeling flexibility, especially in the context of anomalous diffusion processes. Furthermore, stochastic extensions [8] have been introduced to account for uncertainties in data and models, thereby improving robustness and solution accuracy. These developments reflect the evolving nature of regularization strategies and underscore the continued relevance of Showalter’s original framework in modern inverse problem theory.

The homogeneous case has been extensively studied using a variety of methods, whereas the nonhomogeneous case has received significantly less attention, and the related literature remains limited. Although some recent studies have addressed inverse problems for semilinear parabolic systems [9], research on the linear nonhomogeneous backward parabolic problem is still scarce, and only a few works have focused on its regularization. One of the most extensively studied methods in this context is the quasireversibility method (QRM), initially proposed by Lattès and Lions [10], and later refined by Clark and Oppenheimer [11], Miller [12], Payne [13], Showalter [14], Barnes et al. [15], and Huang and Zheng [16]. This method is based on perturbing the ill-posed operator in order to construct stable approximations. QRM has also inspired applications in more complex settings, such as mean field games [17] and mathematical biology, demonstrating its flexibility and continued relevance.

Complementing QRM, the quasiboundary value method (QBVM) focuses on perturbing the final condition to enhance stability. As shown by Showalter [18] and Denche and Bessila [19], this approach provides improved stability estimates and serves as a practical counterpart to QRM.

The regularization method employed in the present study was introduced by Djezzar and Teniou [20, 21] in the context of the homogeneous case, where the operator $A$ has a continuous spectrum. This method yields strongly stable solutions and exhibits faster convergence compared to traditional QRM and QBVM techniques.

The central idea of the modified method consists in perturbing both the equation and the final condition, thereby formulating an approximate nonlocal problem that depends on two small regularization parameters. The resulting problem is subject to a boundary condition involving a derivative of the same order as that appearing in the original equation, as detailed in the following:$$\begin{array}{}\text{(2)}& \begin{array}{c}{v}_{\sigma }^{\prime }t+A_{\alpha }{v}_{\sigma }t=ft,0\le t<T,\\ v_{\sigma }T-\beta v_{\sigma }^{\prime }0=g.\end{array}\end{array}$$

Here, the operator $A$ is replaced by its Yosida approximation $A_{\alpha }=A{I+\alpha A}^{-1}$, and the final condition $uT=g$ is replaced by $v_{\sigma }T-\beta v_{\sigma }^{\prime }0=g$, where $\sigma =\alpha ,\beta ,$ with $\alpha >0$ and $\beta >0$.

The proposed regularization method is structured as follows.

1.1. Description of the Method

• Step 1: Find $v_{\sigma }$, the solution of the perturbed problem,$$\begin{array}{}\text{(3)}& \begin{array}{c}{v}_{\sigma }^{\prime }t+A_{\alpha }{v}_{\sigma }t=ft,0\le t<T,\\ v_{\sigma }T-\beta v_{\sigma }^{\prime }0=g.\end{array}\end{array}$$

• Step 2: Use the initial value$$\begin{array}{}\text{(4)}& v_{\sigma }0=g_{\sigma }={\beta A_{\alpha }+S_{\alpha }T}^{-1}g+\beta f0-{\int }_0^{T}fs{S}_{\alpha }T-sds.\end{array}$$

  where $S_{\alpha }t=e^{-t{A}_{\alpha }}$ denotes the $C_0-$ semigroup generated by $-A_{\alpha }$.

  This serves as the initial condition for the following problem:$$\begin{array}{}\text{(5)}& \begin{array}{c}{u}_{\sigma }^{\prime }t+A{u}_{\sigma }t=ft,\\ u_{\sigma }0=g_{\sigma }.\end{array}\end{array}$$

• Step 3: Show that$$\begin{array}{c}\text{(6)}& u_{\sigma }T-g⟶0,\text{\hspace{1em}as\hspace{0.17em}}\sigma ⟶0,u_{\sigma }0-u0⟶0,\text{\hspace{1em}as\hspace{0.17em}}\sigma ⟶0,\\ \underset{0\le t\le T}{\sup }{u}_{\sigma }t-ut⟶0,\text{\hspace{1em}as\hspace{0.17em}}\sigma ⟶0.\end{array}$$

Using this modified method, which combines key ideas from both the QRM and the QBVM, we demonstrate that the approximate problem is well-posed in the sense of Hadamard, thereby restoring the stability of the solution with respect to both the equation and the final condition in (1). In addition to proving well-posedness, we derive explicit convergence rates for the proposed regularization scheme.

The remainder of the paper is organized as follows. In Section 2, we present the perturbed problem and provide a detailed description of the regularization strategy. Section 3 is devoted to deriving error estimates and analyzing the convergence behavior of the approximate solution. In Section 4, we present a numerical application to the one-dimensional heat equation, accompanied by a thorough evaluation of the results at different time instances. The paper concludes with an overall assessment of the numerical findings and final remarks in Section 5.

2. Regularization of the Problem

In this paper, we consider $A$ as the standard operator previouslfiy defined and $H$ as a separable Hilbert space such that $-A$ generates contraction semigroup on $H$. Let $v_n$ be an orthonormal eigenbasis of $H$ corresponding to the eigenvalues ${\lambda }_n$ of $A$, where we assume the spectrum satisfies $0<\eta \le {\lambda }_1<{\lambda }_2<\dots <{\text{li}\mathrm{m}}_{n⟶+\infty }{\lambda }_n=\infty$.

We approximate the problem (1) by the following system:$$\begin{array}{}\text{(7)}& \begin{array}{c}{v}_{\sigma }^{\prime }t+A_{\alpha }{v}_{\sigma }t=ft,0\le t<T,\\ v_{\sigma }T-\beta v_{\sigma }^{\prime }0=g,\end{array}\end{array}$$where $A_{\alpha }=A{I+\alpha A}^{-1}$ denotes the Yosida approximation of the operator $A$. We assume that $A$ is injective, which guarantees the completeness of the orthonormal eigenbasis $v_n$ in $H$, thus justifying the use of the following spectral decomposition.

The function $ft\in C0,T,H$ is represented by$$\begin{array}{}\text{(8)}& ft={\displaystyle \sum }_{n=1}^{\infty }{f}_{n}t{v}_n,\end{array}$$where $f_{n}t=ft,v_n$, and$$\begin{array}{}\text{(9)}& ft={\displaystyle \sum }_{n=1}^{\infty }{f}_{n}t={\displaystyle \sum }_{n=1}^{\infty }\underset{0\le t\le T}{\sup }{f}_{n}t,\end{array}$$where the vector $g\in H$ can be expressed as$$\begin{array}{}\text{(10)}& g={\displaystyle \sum }_{n=1}^{\infty }{g}_n{v}_n,\end{array}$$where $g_n=g,v_n$, and$$\begin{array}{}\text{(11)}& g={\displaystyle \sum }_{n=1}^{\infty }{g}_n.\end{array}$$

We assume that $f_{n}t<+\infty$ and $g_n<+\infty$, with $\sigma =\alpha ,\beta$, where $\alpha >0$ and $\beta >0$.

Definition 1.

We define the classical solution of the problem (7) as follows:$$\begin{array}{}\text{(12)}& v_{\sigma }t=S_{\alpha }t{\beta A_{\alpha }+S_{\alpha }T}^{-1}g+\beta f0-{\int }_0^{T}fs{S}_{\alpha }T-sds+{\int }_0^{t}fs{S}_{\alpha }t-sds,\end{array}$$where $S_{\alpha }t$ denotes the semigroup generated by $-A_{\alpha }$, as defined earlier.

This solution can be written as$$\begin{array}{}\text{(13)}& v_{\sigma }t={\displaystyle \sum }_{n=1}^{\infty }\frac{e^{-{\lambda }_n/1+\alpha {\lambda }_{n}t}}{\beta {\lambda }_n/1+\alpha {\lambda }_n+e^{-{\lambda }_n/1+\alpha {\lambda }_{n}T}}{g}_n+\beta f_{n}0-{\int }_0^T{f}_{n}s{e}^{-{\lambda }_n/1+\alpha {\lambda }_{n}T-s}ds+{\int }_0^t{f}_{n}s{e}^{-{\lambda }_n/1+\alpha {\lambda }_{n}t-s}ds.\end{array}$$

If we denote$$\begin{array}{}\text{(14)}& I_{t,\alpha }{f}_n={\int }_0^t{f}_{n}s{e}^{-{\lambda }_n/1+\alpha {\lambda }_{n}t-s}ds,\end{array}$$then, the function $v_{\sigma }t$ can be written as follows:$$\begin{array}{}\text{(15)}& v_{\sigma }t={\displaystyle \sum }_{n=1}^{\infty }\frac{e^{-{\lambda }_n/1+\alpha {\lambda }_{n}t}}{\beta {\lambda }_n/1+\alpha {\lambda }_n+e^{-{\lambda }_n/1+\alpha {\lambda }_{n}T}}{g}_n+\beta f_{n}0-I_{T,\alpha }{f}_n+I_{t,\alpha }{f}_n{v}_n.\end{array}$$

Theorem 1.

For all $f\in C0,T,H$ and $g\in H$, the function $v_{\sigma }t$ given by equation (12) is the unique solution of the perturbed problem (7), and the series (15) is convergent. Furthermore, we have$$\begin{array}{}\text{(16)}& \underset{0\le t\le T}{\text{su}\mathrm{p}}{v_{\sigma }t}^2\le C_{\alpha ,\beta ,T}{g}^2+f^2,\end{array}$$where $C_{\alpha ,\beta ,T}=4{\beta +T}^2/{\beta \eta /1+\alpha \eta +e^{-T/\alpha }}^2+2T^2$.

Proof 1.

We consider the following classical Cauchy problem:$$\begin{array}{}\text{(17)}& \begin{array}{c}{v}_{\sigma }^{\prime }t+A_{\alpha }{v}_{\sigma }t=ft,0\le t<T,\\ v_{\sigma }0={\beta A_{\alpha }+S_{\alpha }T}^{-1}g+\beta f0-{\int }_0^{T}fs{S}_{\alpha }T-sds.\end{array}\end{array}$$

This problem is well-posed, and it is clearly evident that $v_{\sigma }t$ is its unique solution. Moreover, we compute that$$\begin{array}{c}\text{(18)}& v_{\sigma }T-\beta v_{\sigma }^{\prime }0=S_{\alpha }T{\beta A_{\alpha }+S_{\alpha }T}^{-1}g+\beta f0-{\int }_0^{T}fs{S}_{\alpha }T-sds+{\int }_0^{T}fs{S}_{\alpha }T-sds-\beta f0-A_{\alpha }{v}_{\sigma }0=S_{\alpha }T{\beta A_{\alpha }+S_{\alpha }T}^{-1}g+\beta f0-{\int }_0^{T}fs{S}_{\alpha }T-sds+{\int }_0^{T}fs{S}_{\alpha }T-sds\\ -\beta f0+\beta A_{\alpha }{\beta A_{\alpha }+S_{\alpha }T}^{-1}g+\beta f0-{\int }_0^{T}fs{S}_{\alpha }T-sds\\ =g.\end{array}$$

Thus, the perturbed boundary condition is satisfied.

Next, for each $0\le t\le T$, we observe that$$\begin{array}{}\text{(19)}& I_{t,\alpha }{f}_n\le T{f}_n,\end{array}$$since the exponential function is bounded by 1.

Moreover, we have$$\begin{array}{}\text{(20)}& \frac{e^{-{\lambda }_n/1+\alpha {\lambda }_{n}t}}{\beta {\lambda }_n/1+\alpha {\lambda }_n+e^{-{\lambda }_n/1+\alpha {\lambda }_{n}T}}\le \frac{1}{\beta {\lambda }_n/1+\alpha {\lambda }_n+e^{-{\lambda }_n/1+\alpha {\lambda }_{n}T}}.\end{array}$$

Now, since$$\begin{array}{}\text{(21)}& -\frac{{\lambda }_{n}T}{1+\alpha {\lambda }_n}\ge -\frac{T}{\alpha },\end{array}$$it follows that$$\begin{array}{}\text{(22)}& e^{-{\lambda }_n/1+\alpha {\lambda }_{n}T}\ge e^{-T/\alpha }.\end{array}$$

Also, the function$$\begin{array}{}\text{(23)}& \lambda ⟼\frac{\beta {\lambda }_n}{1+\alpha {\lambda }_n},\end{array}$$is increasing with respect to ${\lambda }_n$, and reaches its minimum at ${\lambda }_n=\eta$. Therefore, we obtain the bound$$\begin{array}{}\text{(24)}& \frac{e^{-{\lambda }_n/1+\alpha {\lambda }_{n}t}}{\beta {\lambda }_n/1+\alpha {\lambda }_n+e^{-{\lambda }_n/1+\alpha {\lambda }_{n}T}}\le \frac{1}{\beta \eta /1+\alpha \eta +e^{-T/\alpha }}.\end{array}$$

By applying the Cauchy criterion of convergence in norm, we evaluate that$$\begin{array}{}\text{(25)}& \underset{0\le t\le T}{\text{su}\mathrm{p}}{\underset{n=m}{{\displaystyle \sum }^{m+p}}{v}_{\sigma ,n}t}^2=\underset{0\le t\le T}{\text{su}\mathrm{p}}{\underset{n=m}{{\displaystyle \sum }^{m+p}}\begin{array}{c}\frac{e^{-{\lambda }_n/1+\alpha {\lambda }_{n}t}}{\beta {\lambda }_n/1+\alpha {\lambda }_n+e^{-{\lambda }_n/1+\alpha {\lambda }_{n}T}}{g}_n+\beta f_{n}0-I_{T,\alpha }{f}_n\\ +I_{t,\alpha }{f}_n\end{array}{v}_n}^2\le 2\underset{0\le t\le T}{\text{su}\mathrm{p}}{{\displaystyle \sum }_{n=m}^{m+p}\frac{e^{-{\lambda }_n/1+\alpha {\lambda }_{n}t}}{\beta {\lambda }_n/1+\alpha {\lambda }_n+e^{-{\lambda }_n/1+\alpha {\lambda }_{n}T}}{g}_n+\beta f_{n}0-I_{T,\alpha }{f}_n{v}_n}^2+2\underset{0\le t\le T}{\text{su}\mathrm{p}}{{\displaystyle \sum }_{n=m}^{m+p}{I}_{t,\alpha }{f}_n{v}_n}^2.\end{array}$$

It then follows from (19) and (24) that$$\begin{array}{}\text{(26)}& \underset{0\le t\le T}{\text{su}\mathrm{p}}{v_{\sigma }t}^2\le 4\frac{1}{{\beta \eta /1+\alpha \eta +e^{-T/\alpha }}^2}{g}^2+4\frac{{\beta +T}^2}{{\beta \eta /1+\alpha \eta +e^{-T/\alpha }}^2}+2T^2{f}^2.\end{array}$$

We may upper-bound this expression by$$\begin{array}{}\text{(27)}& \underset{0\le t\le T}{\text{su}\mathrm{p}}{v_{\sigma }t}^2\le \text{ma}\mathrm{x}\frac{4}{{\beta \eta /1+\alpha \eta +e^{-T/\alpha }}^2},\frac{4{\beta +T}^2}{{\beta \eta /1+\alpha \eta +e^{-T/\alpha }}^2}+2T^2{g}^2+f^2.\end{array}$$

Hence, we obtain the following bound$$\begin{array}{}\text{(28)}& \underset{0\le t\le T}{\text{su}\mathrm{p}}{v_{\sigma }t}^2\le \frac{4{\beta +T}^2}{{\beta \eta /1+\alpha \eta +e^{-T/\alpha }}^2}+2T^2{g}^2+f^2,\end{array}$$which yields the desired estimate as$$\begin{array}{}\text{(29)}& \underset{0\le t\le T}{\text{su}\mathrm{p}}{v_{\sigma }t}^2\le C_{\alpha ,\beta ,T}{g}^2+f^2,\end{array}$$where$$\begin{array}{}\text{(30)}& C_{\alpha ,\beta ,T}=\frac{4{\beta +T}^2}{{\beta \eta /1+\alpha \eta +e^{-T/\alpha }}^2}+2T^2.\end{array}$$

Therefore, for all $g\in H$ and $f\in C0,T,H$, the series defining $v_{\sigma }t$ converges and defines the strong solution of problem (7).

This concludes the proof.

Using the initial value$$\begin{array}{}\text{(31)}& v_{\sigma }0=g_{\sigma }={\beta A_{\alpha }+S_{\alpha }T}^{-1}g+\beta f0-I_{T,\alpha }f,\end{array}$$we define the following problem:$$\begin{array}{}\text{(32)}& \begin{array}{c}{u}_{\sigma }^{\prime }t+A{u}_{\sigma }t=ft,\\ u_{\sigma }0=g_{\sigma },\end{array}\end{array}$$where $St=e^{-tA}$ is the $C_0$-semigroup generated by $-A$.

If we denote$$\begin{array}{}\text{(33)}& I_{t}f={\int }_0^{t}fs{e}^{-At-s}ds={\int }_0^{t}fsSt-sds,\end{array}$$we can now state the following theorem.

Theorem 2.

The problem (32) is well-posed, and its unique solution is given by$$\begin{array}{}\text{(34)}& u_{\sigma }t=St{g}_{\sigma }+I_{t}f,\end{array}$$that is,$$\begin{array}{}\text{(35)}& u_{\sigma }t={\displaystyle \sum }_{n=1}^{\infty }\frac{e^{-{\lambda }_n}}{\beta {\lambda }_n/1+\alpha {\lambda }_n+e^{-{\lambda }_n/1+\alpha {\lambda }_{n}T}}{g}_n+\beta f_{n}0-{\int }_0^{T}fs{e}^{-{\lambda }_n/1+\alpha {\lambda }_{n}T-s}ds+{\int }_0^t{f}_{n}s{e}^{-{\lambda }_{n}t-s}ds{v}_n.\end{array}$$

Furthermore, we have the following estimate:$$\begin{array}{}\text{(36)}& {u_{\sigma }t}^2\le C_{\alpha ,\beta ,T}{g}^2+f^2,\end{array}$$where $C_{\alpha ,\beta ,T}=4{\beta +T}^2/{\beta \eta /1+\alpha \eta +e^{-T/\alpha }}^2+2T^2$.

Proof 2.

First, the Cauchy problem (32) is a classical Cauchy problem in the Hilbert space $H$ associated with a $C_0$-semigroup generated by $-A$. Since $f\in C0,T,H$ and $g_{\sigma }\in H$, it follows by a straightforward computation that (34) is a classical solution of problem (32).

Now, we can write that$$\begin{array}{c}\text{(37)}& \underset{0\le t\le T}{\text{su}\mathrm{p}}{\underset{n=m}{{\displaystyle \sum }^{m+p}}{u}_{\sigma ,n}t}^2=\underset{0\le t\le T}{\text{su}\mathrm{p}}{\underset{n=m}{{\displaystyle \sum }^{m+p}}\frac{e^{-{\lambda }_n}}{\beta {\lambda }_n/1+\alpha {\lambda }_n+e^{-{\lambda }_n/1+\alpha {\lambda }_{n}T}}{g}_n+\beta f_{n}0-{\int }_0^{T}fs{e}^{-{\lambda }_n/1+\alpha {\lambda }_{n}T-s}ds+{\int }_0^t{f}_{n}s{e}^{-{\lambda }_{n}t-s}ds{v}_n}^2\le 2\underset{0\le t\le T}{\text{su}\mathrm{p}}{{\displaystyle \sum }_{n=m}^{m+p}\frac{e^{-{\lambda }_n}}{\beta {\lambda }_n/1+\alpha {\lambda }_n+e^{-{\lambda }_n/1+\alpha {\lambda }_{n}T}}{g}_n+\beta f_{n}0-{\int }_0^{T}fs{e}^{-{\lambda }_n/1+\alpha {\lambda }_{n}T-s}ds{v}_n}^2\\ +2\underset{0\le t\le T}{\text{su}\mathrm{p}}{{\displaystyle \sum }_{n=m}^{m+p}{\int }_0^t{f}_{n}s{e}^{-{\lambda }_{n}t-s}ds{v}_n}^2.\end{array}$$

We have$$\begin{array}{}\text{(38)}& \frac{e^{-{\lambda }_n}}{\beta {\lambda }_n/1+\alpha {\lambda }_n+e^{-{\lambda }_n/1+\alpha {\lambda }_{n}T}}\le \frac{1}{\beta \eta /1+\alpha \eta +e^{-T/\alpha }},\end{array}$$and we use the inequality$$\begin{array}{}\text{(39)}& I_t{f}_n\le T{f}_n.\end{array}$$

Hence, we obtain the estimate$$\begin{array}{}\text{(40)}& \underset{0\le t\le T}{\text{su}\mathrm{p}}{u_{\sigma }t}^2\le 4\frac{1}{{\beta \eta /1+\alpha \eta +e^{-T/\alpha }}^2}{g}^2+4\frac{{\beta +T}^2}{{\beta \eta /1+\alpha \eta +e^{-T/\alpha }}^2}+2T^2{f}^2.\end{array}$$

An upper bound for this expression is given by$$\begin{array}{}\text{(41)}& \underset{0\le t\le T}{\text{su}\mathrm{p}}{u_{\sigma }t}^2\le \text{ma}\mathrm{x}\frac{4}{{\beta \eta /1+\alpha \eta +e^{-T/\alpha }}^2},\frac{4{\beta +T}^2}{{\beta \eta /1+\alpha \eta +e^{-T/\alpha }}^2}+2T^2{g}^2+f^2,\end{array}$$as a result, we obtain$$\begin{array}{c}\text{(42)}& \underset{0\le t\le T}{\text{su}\mathrm{p}}{u_{\sigma }t}^2\le \frac{4{\beta +T}^2}{{\beta \eta /1+\alpha \eta +e^{-T/\alpha }}^2}+2T^2{g}^2+f^2,\underset{0\le t\le T}{\text{su}\mathrm{p}}{u_{\sigma }t}^2\le C_{\alpha ,\beta ,T}{g}^2+f^2,\end{array}$$where$$\begin{array}{}\text{(43)}& C_{\alpha ,\beta ,T}=\frac{4{\beta +T}^2}{{\beta \eta /1+\alpha \eta +e^{-T/\alpha }}^2}+2T^2.\end{array}$$

Thus, the proof is complete.

3. Error Estimates

Theorem 3.

For any $g\in H$, $u_{\sigma }T$ converges to $g$ as $\sigma ⟶0$, that is,$$\begin{array}{}\text{(44)}& \underset{\sigma ⟶0}{\text{li}\mathrm{m}}{u}_{\sigma }T-g=0.\end{array}$$

Proof 3.

From previous results, the approximate solution at final time is given by$$\begin{array}{}\text{(45)}& u_{\sigma }T=ST{\beta A_{\alpha }+S_{\alpha }T}^{-1}g+\beta f0-I_{T,\alpha }f+I_{T}f,\end{array}$$then,$$\begin{array}{}\text{(46)}& u_{\sigma }T-g={\displaystyle \sum }_{n\ge 1}\frac{e^{-{\lambda }_{n}T}}{\beta {\lambda }_n/1+\alpha {\lambda }_n+e^{-{\lambda }_{n}T/1+\alpha {\lambda }_n}}-1g_n+\frac{e^{-{\lambda }_{n}T}}{\beta {\lambda }_n/1+\alpha {\lambda }_n+e^{-{\lambda }_{n}T/1+\alpha {\lambda }_n}}\beta f_{n}0-I_{T,\alpha }f+I_{T}f{v}_n.\end{array}$$

By performing the necessary computations, we can write that$$\begin{array}{}\text{(47)}& u_{\sigma }T-g={\displaystyle \sum }_{n\ge 1}-\frac{\beta {\lambda }_n/1+\alpha {\lambda }_n}{\beta {\lambda }_n/1+\alpha {\lambda }_n+e^{-{\lambda }_{n}T/1+\alpha {\lambda }_n}}+\frac{e^{-{\lambda }_{n}T/1+\alpha {\lambda }_n}-e^{-{\lambda }_{n}T}}{\beta {\lambda }_n/1+\alpha {\lambda }_n+e^{-{\lambda }_{n}T/1+\alpha {\lambda }_n}}{g}_n+\frac{e^{-{\lambda }_{n}T}}{\beta {\lambda }_n/1+\alpha {\lambda }_n+e^{-{\lambda }_{n}T/1+\alpha {\lambda }_n}}\beta f_{n}0-I_{T,\alpha }{f}_n+I_T{f}_n{v}_n.\end{array}$$

If we denote$$\begin{array}{c}\text{(48)}& F_{\sigma }{\lambda }_n=\frac{\beta {\lambda }_n/1+\alpha {\lambda }_n}{\beta {\lambda }_n/1+\alpha {\lambda }_n+e^{-{\lambda }_{n}T/1+\alpha {\lambda }_n}},G_{\sigma }{\lambda }_n=\frac{e^{-{\lambda }_{n}T/1+\alpha {\lambda }_n}-e^{-{\lambda }_{n}T}}{\beta {\lambda }_n/1+\alpha {\lambda }_n+e^{-{\lambda }_{n}T/1+\alpha {\lambda }_n}},\\ H_{\sigma }{\lambda }_n=\frac{e^{-{\lambda }_{n}T}}{\beta {\lambda }_n/1+\alpha {\lambda }_n+e^{-{\lambda }_{n}T/1+\alpha {\lambda }_n}},\end{array}$$then,$$\begin{array}{}\text{(49)}& {u_{\sigma }T-g}^2\le 2{\displaystyle \sum }_{n\ge 1}{F_{\sigma }{\lambda }_n+G_{\sigma }{\lambda }_n{g}_n}^2+2{\displaystyle \sum }_{n\ge 1}{H_{\sigma }{\lambda }_n\beta f_{n}0-I_{T,\alpha }{f}_n+I_T{f}_n}^2.\end{array}$$

It is clear that$$\begin{array}{c}\text{(50)}& F_{\sigma }{\lambda }_n\le \beta e^{{\lambda }_{n}T},G_{\sigma }{\lambda }_n\le \alpha T{\lambda }_n^2,\\ H_{\sigma }{\lambda }_n\le \frac{1}{\beta {\lambda }_n/1+\alpha {\lambda }_n}\le 1,\end{array}$$then,$$\begin{array}{}\text{(51)}& {u_{\sigma }T-g}^2\le 4{\beta }^2{e}^{2{\lambda }_{N}T}+{\alpha }^2{T}^2{\lambda }_N^4{g}^2+4{\beta +T}^2+T^2{f}^2,\end{array}$$where ${\lambda }_N=\text{ma}\mathrm{x}{\lambda }_n$.

By choosing$$\begin{array}{}\text{(52)}& {\beta }^2\le \frac{{\epsilon }^2}{8}\frac{1}{f^2},\end{array}$$we immediately obtain$$\begin{array}{}\text{(53)}& {u_{\sigma }T-g}^2⟶0,\text{\hspace{1em}as\hspace{0.17em}}\sigma ⟶0.\end{array}$$

Now, we define the function$$\begin{array}{c}\text{(54)}& {F:\mathbb{R}}_{+}\times {\mathbb{R}}_{+}⟶H\sigma =\alpha ,\beta ⟼F\sigma =\begin{array}{cc}{u}_{\sigma }0=g_{\sigma },& \text{if\hspace{0.17em}}\sigma \ne 0,0,\\ u0=g_0,& \text{if\hspace{0.17em}}\sigma =0,0.\end{array}.\end{array}$$

We then state the following theorem.

Theorem 4.

For all $g\in H$, the problem (1) has a solution $u$ if and only if the function $F$ is continuous at $0,0$ (i.e., $u_{\sigma }0⟶u0$ as $\sigma ⟶0$).

Furthermore, $u_{\sigma }t⟶ut$ as $\sigma ⟶0$ uniformly in $t$.

Proof 4.

We assume that the function $F$ is continuous at $0,0$.$$\begin{array}{c}\text{(55)}& \underset{\sigma ⟶0}{\text{li}\mathrm{m}}{g}_{\sigma }=g_0,g_0<\infty .\end{array}$$

We define that$$\begin{array}{}\text{(56)}& wt=St{g}_0+I_{t}f.\end{array}$$

Thus, we have$$\begin{array}{c}\text{(57)}& wt-u_{\sigma }t=St{g}_0-g_{\sigma },wt-u_{\sigma }t\le g_0-g_{\sigma },\end{array}$$then,$$\begin{array}{}\text{(58)}& \underset{0\le t\le T}{\text{su}\mathrm{p}}wt-u_{\sigma }t\le g_0-g_{\sigma }\underset{\sigma ⟶0}{⟶}0.\end{array}$$

Therefore,$$\begin{array}{}\text{(59)}& \underset{\sigma ⟶0}{\text{li}\mathrm{m}}{u}_{\sigma }T=g,\end{array}$$from Theorem 3, and$$\begin{array}{}\text{(60)}& \underset{\sigma ⟶0}{\text{li}\mathrm{m}}{u}_{\sigma }T=wT.\end{array}$$

Thus, we conclude that$$\begin{array}{c}\text{(61)}& wT=g,wt=St{g}_0+I_{t}f,\end{array}$$is a solution to the problem (1).

Now, for the reverse implication, we assume that$$\begin{array}{}\text{(62)}& ut=e^{T-tA}g+{\int }_0^{t}fs{e}^{-At-s}ds,\end{array}$$is a solution to problem (1), and we demonstrate the continuity of the function $F$.

We have$$\begin{array}{}\text{(63)}& u_{\sigma }t=St{g}_{\sigma }+I_{t}f,\end{array}$$and$$\begin{array}{c}\text{(64)}& {\sigma }_1={\alpha }_1,{\beta }_1,{\sigma }_2={\alpha }_2,{\beta }_2,\end{array}$$then,$$\begin{array}{c}\text{(65)}& {u_{{\sigma }_1}0-u_{{\sigma }_2}0}^2={{\displaystyle \sum }_{n\ge 1}\frac{1}{{\beta }_1{\lambda }_n/1+{\alpha }_1{\lambda }_n+e^{-{\lambda }_{n}T/1+{\alpha }_1{\lambda }_n}}{g}_n+{\beta }_1{f}_{n}0-{\int }_0^T{f}_{n}s{e}^{-{\lambda }_{n}T-s/1+{\alpha }_1{\lambda }_n}ds{v}_n-{\displaystyle \sum }_{n\ge 1}\frac{1}{{\beta }_2{\lambda }_n/1+{\alpha }_2{\lambda }_n+e^{-{\lambda }_{n}T/1+{\alpha }_2{\lambda }_n}}{g}_n+{\beta }_2{f}_{n}0-{\int }_0^T{f}_{n}s{e}^{-{\lambda }_{n}T-s/1+{\alpha }_2{\lambda }_n}ds{v}_n}^2\le 2{{\displaystyle \sum }_{n\ge 1}\frac{g_n+{\beta }_1{f}_{n}0-{\int }_0^T{f}_{n}s{e}^{-{\lambda }_{n}T-s/1+{\alpha }_1{\lambda }_n}ds{\beta }_2{\lambda }_n/1+{\alpha }_2{\lambda }_n+e^{-{\lambda }_{n}T/1+{\alpha }_2{\lambda }_n}}{{\beta }_1{\lambda }_n/1+{\alpha }_1{\lambda }_n+e^{-{\lambda }_{n}T/1+{\alpha }_1{\lambda }_n}{\beta }_2{\lambda }_n/1+{\alpha }_2{\lambda }_n+e^{-{\lambda }_{n}T/1+{\alpha }_2{\lambda }_n}}{v}_n}^2\\ +2{{\displaystyle \sum }_{n\ge 1}\frac{g_n+{\beta }_2{f}_{n}0-{\int }_0^T{f}_{n}s{e}^{-{\lambda }_{n}T-s/1+{\alpha }_2{\lambda }_n}ds{\beta }_1{\lambda }_n/1+{\alpha }_1{\lambda }_n+e^{-{\lambda }_{n}T/1+{\alpha }_1{\lambda }_n}}{{\beta }_1{\lambda }_n/1+{\alpha }_1{\lambda }_n+e^{-{\lambda }_{n}T/1+{\alpha }_1{\lambda }_n}{\beta }_2{\lambda }_n/1+{\alpha }_2{\lambda }_n+e^{-{\lambda }_{n}T/1+{\alpha }_2{\lambda }_n}}{v}_n}^2.\end{array}$$

If we set$$\begin{array}{}\text{(66)}& F_{{\sigma }_1,{\sigma }_2}{\lambda }_n=\frac{g_n+{\beta }_1{f}_{n}0-{\int }_0^T{f}_{n}s{e}^{-{\lambda }_{n}T-s/1+{\alpha }_1{\lambda }_n}ds{\beta }_2{\lambda }_n/1+{\alpha }_2{\lambda }_n+e^{-{\lambda }_{n}T/1+{\alpha }_2{\lambda }_n}}{{\beta }_1{\lambda }_n/1+{\alpha }_1{\lambda }_n+e^{-{\lambda }_{n}T/1+{\alpha }_1{\lambda }_n}{\beta }_2{\lambda }_n/1+{\alpha }_2{\lambda }_n+e^{-{\lambda }_{n}T/1+{\alpha }_2{\lambda }_n}},\end{array}$$and$$\begin{array}{}\text{(67)}& G_{{\sigma }_1,{\sigma }_2}{\lambda }_n=\frac{g_n+{\beta }_2{f}_{n}0-{\int }_0^T{f}_{n}s{e}^{-{\lambda }_{n}T-s/1+{\alpha }_2{\lambda }_n}ds{\beta }_1{\lambda }_n/1+{\alpha }_1{\lambda }_n+e^{-{\lambda }_{n}T/1+{\alpha }_1{\lambda }_n}}{{\beta }_1{\lambda }_n/1+{\alpha }_1{\lambda }_n+e^{-1+{\alpha }_1{\lambda }_n/1+{\alpha }_1{\lambda }_n}{\beta }_2{\lambda }_n/1+{\alpha }_2{\lambda }_n+e^{-{\lambda }_{n}T/1+{\alpha }_2{\lambda }_n}},\end{array}$$then,$$\begin{array}{}\text{(68)}& {u_{{\sigma }_1}0-u_{{\sigma }_2}0}^2\le 2{{\displaystyle \sum }_{n\ge 1}{F}_{{\sigma }_1,{\sigma }_2}{\lambda }_n{v}_n}^2+2{{\displaystyle \sum }_{n\ge 1}{G}_{{\sigma }_1,{\sigma }_2}{\lambda }_n{v}_n}^2.\end{array}$$

Using straightforward calculations, we obtain$$\begin{array}{}\text{(69)}& F_{{\sigma }_1,{\sigma }_2}{\lambda }_n\le e^{2{\lambda }_{N}T}\frac{{\beta }_2}{{\alpha }_2}+1g_n+{\beta }_1+T{f}_n,\end{array}$$and$$\begin{array}{}\text{(70)}& G_{{\sigma }_1,{\sigma }_2}{\lambda }_n\le e^{2{\lambda }_{N}T}\frac{{\beta }_1}{{\alpha }_1}+1g_n+{\beta }_2+T{f}_n,\end{array}$$thus,$$\begin{array}{}\text{(71)}& {u_{{\sigma }_1}0-u_{{\sigma }_2}0}^2\le 4e^{4{\lambda }_N^2{T}^2}{\frac{{\beta }_2}{{\alpha }_2}+1}^2{g}^2+{{\beta }_1+T}^2{f}^2+4e^{4{\lambda }_N^2{T}^2}{\frac{{\beta }_1}{{\alpha }_1}+1}^2{g}^2+{{\beta }_2+T}^2{f}^2.\end{array}$$

By choosing ${\alpha }_1,{\beta }_1,{\alpha }_2,{\beta }_2$ according to the following relations:$$\begin{array}{}\text{(72)}& \frac{{\beta }_i}{{\alpha }_i}\le \frac{\epsilon }{\sqrt{8}}\frac{1}{e^{2{\lambda }_{N}T}}\frac{1}{{g^2+T^2{f}^2}^{1/2}},i=1,2,\end{array}$$and ensuring ${\beta }_i\ll {\alpha }_i$, we obtain$$\begin{array}{}\text{(73)}& {u_{{\sigma }_1}0-u_{{\sigma }_2}0}^2\le {\epsilon }^2,\end{array}$$which means that $u_{\sigma }0$ is a Cauchy sequence in $H$ and therefore it is convergent.

In particular, taking ${\sigma }_1=\sigma$ and ${\sigma }_2={\sigma }_0=0,0$, we have$$\begin{array}{}\text{(74)}& {u_{\sigma }0-u_{{\sigma }_0}0}^2={g_{\sigma }-g_0}^2\le {\epsilon }^2,\end{array}$$thus,$$\begin{array}{}\text{(75)}& \underset{\sigma ⟶0}{\text{li}\mathrm{m}}{u}_{\sigma }0=u0,\end{array}$$which means that the function $F$ is continuous at $0,0$.

Based on the previous relations, we can conclude that$$\begin{array}{}\text{(76)}& \underset{0\le t\le T}{\text{su}\mathrm{p}}{u}_{\sigma }t-ut\le g_{\sigma }-g_0\underset{\sigma ⟶0}{⟶}0.\end{array}$$

Therefore, $u_{\sigma }t⟶ut$ as $\sigma ⟶0$ uniformly in $t$.

This completes the proof.

Remark 1.

Choice of the regularization parameters.

The parameters $\alpha$ and $\beta$ are not chosen arbitraril, but are selected to satisfy the stability and convergence conditions derived in our analysis. More precisely, to ensure the final-time estimate holds, we require$$\begin{array}{}\text{(77)}& \beta \le \frac{\epsilon }{\sqrt{8}}\frac{1}{f}.\end{array}$$

While for the initial time $t=0$ and for any intermediate time $t\in 0,T$, the estimate is satisfied under the condition$$\begin{array}{}\text{(78)}& \frac{\beta }{\alpha }\le \frac{\epsilon }{\sqrt{8}}\frac{1}{e^{2{\lambda }_{N}T}}\frac{1}{{g^2+T^2{f}^2}^{1/2}}.\end{array}$$

These conditions guide the practical selection of $\alpha$ and $\beta$ in numerical simulations, ensuring consistency with the theoretical convergence guarantees. In particular, we take $\beta =c\alpha$, with $c>0$ chosen small enough to satisfy both inequalities for a given noise level $\epsilon$.

4. Numerical Application

4.1. Application to the Heat Equation

In this section, we apply the theoretical results established earlier to an inverse problem associated with the heat equation.

Let $T>0$ be a fixed time. We consider the following problem: find a function $ux,t$ defined on $t,x\in 0,T\times 0,1$, satisfying$$\begin{array}{}\text{(79)}& \begin{array}{cc}{u}_{t}t,x-u_{xx}t,x=e^{-\mu t}\text{\hspace{0.17em}si}\mathrm{n}\omega t\text{si}\mathrm{n}\pi x,& t,x\in 0,T\times 0,1,\\ ut;0=ut,1=0,& t\in 0,T,\\ ux,T=\sqrt{2}\text{si}\mathrm{n}\pi x,& x\in 0,1,\end{array}\end{array}$$where $\mu =0.5$, $\omega =2\pi$, and $T=1$. This is an ill-posed inverse problem for the heat equation, where the source term is given by $e^{-\mu t}\text{\hspace{0.17em}si}\mathrm{n}\omega t\text{si}\mathrm{n}\pi x$, and the final data by $gx=\sqrt{2}\text{si}\mathrm{n}\pi x$.

Our aim is to reformulate problem (79) as an abstract final value problem of the form$$\begin{array}{}\text{(80)}& \begin{array}{c}{u}^{\prime }t+Aut=ft,0\le \mathrm{t}<\mathrm{T},\\ uT=g,\end{array}\end{array}$$where $A={\mathrm{\partial }}^2/\mathrm{\partial }{x}^2$ is a positive, self-adjoint operator defined on $L^{2}0,1$ with Dirichlet boundary conditions. The eigenvalues of $A$ are ${\lambda }_n={n\pi }^2$, and the corresponding eigenfunctions $v_{n}x=\sqrt{2}\text{si}\mathrm{n}n\pi x$, for $n\in {\mathbb{N}}^{\mathrm{\ast }}$, form an orthonormal basis of $L^{2}0,1$.

It is well known that this type of final value problem is ill-posed in the sense of Hadamard.

Let $f_{n}t$ and $g_n$ denote the Fourier coefficients of $ft,x$ and $gx$, respectively, with respect to this basis,$$\begin{array}{c}\text{(81)}& f_{n}t,x=\begin{array}{cc}\frac{\sqrt{2}}{2}{e}^{-\mu t}\text{\hspace{0.17em}si}\mathrm{n}2\pi t,& \text{for\hspace{0.17em}}n=1,\\ 0,& \text{fo}\mathrm{r}n\ge 2,\end{array}{g}_{n}x=\begin{array}{cc}1,& \text{for\hspace{0.17em}}n=1,\\ 0,& \text{fo}\mathrm{r}n\ge 2.\end{array}.\end{array}$$

The exact solution of this problem is given by$$\begin{array}{}\text{(82)}& ut,x=e^{-{\pi }^{2}T-t}-{\int }_t^1{e}^{-{\pi }^{2}s-t}\frac{\sqrt{2}}{2}\text{si}\mathrm{n}2\pi sds\sqrt{2}\text{si}\mathrm{n}\pi x.\end{array}$$

The regularized solution based on spectral data can be written as$$\begin{array}{}\text{(83)}& u_{\sigma }t,x=\frac{e^{-{\pi }^2}}{\beta \mu +e^{-{\pi }^2/1+\alpha {\pi }^{2}T}}1-\frac{\sqrt{2}}{2}{\int }_0^1{e}^{-\mu }\text{si}\mathrm{n}2\pi s{e}^{-{\pi }^2/1+\alpha {\pi }^{2}1-s}ds++\frac{\sqrt{2}}{2}{\int }_0^t{e}^{-\mu s}\text{si}\mathrm{n}2\pi s{e}^{-{\pi }^{2}t-s}ds\sqrt{2}\text{si}\mathrm{n}\pi x.\end{array}$$

4.2. Numerical Assessment at Final, Intermediate, and Initial Time Levels

4.2.1. Final Time $t=1$

We first compute the regularized numerical solution for the final time $T=1$, using selected values of $\alpha$ and $\beta$. The corresponding results are shown in Table 1.

Table 1

Results at final time.

$\alpha$	$\beta$	$u1,x$	$u_{\sigma }1,x$	${u1,x-u_{\sigma }1,x}_{L^2}$
$2.7\times {10}^{-3}$	${10}^{-3}$	$-0.0298\text{\hspace{0.17em}si}\mathrm{n}\pi x$	$-0.0221\text{\hspace{0.17em}si}\mathrm{n}\pi x$	0.00545
$2.7\times {10}^{-2}$	${10}^{-2}$	$-0.0298\text{\hspace{0.17em}si}\mathrm{n}\pi x$	$-0.0288\text{\hspace{0.17em}si}\mathrm{n}\pi x$	0.00065
$2.7\times {10}^{-1}$	${10}^{-1}$	$-0.0298\text{\hspace{0.17em}si}\mathrm{n}\pi x$	$-0.0295\text{\hspace{0.17em}si}\mathrm{n}\pi x$	0.00015

This table illustrates the performance of the proposed regularization method at the final time $T=1$, where the exact solution is given by $u1,x=-0.0298\text{\hspace{0.17em}si}\mathrm{n}\pi x$. The regularized solution $u_{\sigma }1,x$ is computed for three values of $\alpha$, and three values of β. The results show a clear improvement in accuracy as $\alpha$ increases. For $\alpha =2.7\times {10}^{-3}$, the error in $L^2$-norm is relatively high (Figure 1). This indicates that for small regularization parameters, the solution remains sensitive to the ill-posed nature of the problem. However, when $\alpha$ is increased to $2.7\times {10}^{-1}$, the error significantly decreases to $1.5\times {10}^{-4}$ (Figure 2), demonstrating the stabilizing effect of the regularization and the ability of the method to approximate the final state accurately, as similarly illustrated in Figure 3.

[figure(s) omitted; refer to PDF]

4.2.2. Intermediate Time $t=0.5$

Table 2 presents the comparison between the exact and regularized solutions at the intermediate time $t=0.5$, with $u0.5,x=0.0486\text{\hspace{0.17em}si}\mathrm{n}\pi x$.

Table 2

Results at intermediate time.

$\alpha$	$\beta$	$u0.5,x$	$u_{\sigma }0.5,x$	${u0.5,x-u_{\sigma }0.5,x}_{L^2}$
$2.7\times {10}^{-3}$	${10}^{-3}$	$0.0486\text{\hspace{0.17em}si}\mathrm{n}\pi x$	$0.0463\text{\hspace{0.17em}si}\mathrm{n}\pi x$	0.0017
$2.7\times {10}^{-2}$	${10}^{-2}$	$0.0486\text{\hspace{0.17em}si}\mathrm{n}\pi x$	$0.0397\text{\hspace{0.17em}si}\mathrm{n}\pi x$	0.0065
$2.7\times {10}^{-1}$	${10}^{-2}$	$0.0486\text{\hspace{0.17em}si}\mathrm{n}\pi x$	$0.0458\text{\hspace{0.17em}si}\mathrm{n}\pi x$	0.0033

Interestingly, the lowest error is not achieved for the smallest $\alpha$, which yields an error of $1.7\times {10}^{-3}$, but for the moderate value $\alpha =2.7\times {10}^{-1}$, where the error is reduced to $3.3\times {10}^{-3}$ (Figure 4). The case $\alpha =2.7\times {10}^{-2}$, however, shows a slightly larger error of $6.5\times {10}^{-3}$ (Figure 5). These results suggest that the accuracy at intermediate times may depend more subtly on the balance between approximation quality and regularization strength. Notably, even at half-time, the regularized solution retains a satisfactory fidelity to the true solution, as shown in Figure 6.

[figure(s) omitted; refer to PDF]

4.2.3. Initial Time $t=0$

Table 3 contains the most significant results, as the evaluation at $t=0$ tests the method’s ability to reconstruct the initial state of the system from final time data. The exact solution at this point is $u0,x=\text{si}\mathrm{n}\pi x$.

Table 3

Results at initial time.

$\alpha$	$\beta$	$u0,x$	$u_{\sigma }0,x$	${u0,x-u_{\sigma }0,x}_{L^2}$
$0.76\times {10}^{-2}$	$2.67\times {10}^{-9}$	$\text{si}\mathrm{n}\pi x$	$1.03\text{\hspace{0.17em}si}\mathrm{n}\pi x$	0.021
$0.78\times {10}^{-2}$	$2.67\times {10}^{-9}$	$\text{si}\mathrm{n}\pi x$	$1.01\text{\hspace{0.17em}si}\mathrm{n}\pi x$	0.0071
$0.80\times {10}^{-2}$	$2.67\times {10}^{-9}$	$\text{si}\mathrm{n}\pi x$	$0.998\text{\hspace{0.17em}si}\mathrm{n}\pi x$	0.0014

The regularized approximations are remarkably close, as shown in Figures 7 and 8, particularly for larger values of $\alpha$. For instance, when $\alpha =0.80\times {10}^{-2}$, the error in $L^2$-norm is only $1.4\times {10}^{-3}$ (Figure 9), and the amplitude difference is minimal ($u_{\sigma }0,x=0.998\text{\hspace{0.17em}si}\mathrm{n}\pi x$). These results provide strong evidence that the proposed regularization method is capable of reconstructing the full temporal profile of the solution, including its initial state, with high accuracy. Given the severe ill-posedness of backward heat problems, this level of performance is particularly noteworthy.

[figure(s) omitted; refer to PDF]

4.3. Overall Evaluation

The numerical experiments conducted at three distinct time levels “final, intermediate, and initial” confirm the robustness and reliability of the proposed regularization method. The solution remains stable and accurate across time, with a clear trend of error reduction as the regularization parameters are appropriately chosen. Notably, the selection of the parameters $\alpha$ and $\beta$ was guided by theoretical estimates derived in the analytical study, rather than being arbitrary. This coherence between theoretical framework and computational implementation reinforces the consistency and validity of the method. Most importantly, the method demonstrates a strong capacity to recover the initial condition from terminal data, which is a key benchmark for assessing the effectiveness of any regularization strategy applied to ill-posed inverse problems. The graphical results further corroborate these findings, showing good agreement between the exact and regularized solutions across the spatial domain. Altogether, these results validate the theoretical expectations and highlight the practical potential of the proposed approach for severely ill-posed Cauchy problems associated with parabolic equations.

5. Conclusion

In this paper, we investigated the regularization of a class of nonhomogeneous backward parabolic equations posed in abstract Hilbert spaces. We proposed a modified regularization strategy that combines essential features of the QRM and the QBVM by simultaneously perturbing both the differential operator and the final condition. This dual-perturbation framework ensures the well-posedness of the regularized problem and guarantees convergence toward the exact solution under classical solvability assumptions. Moreover, we rigorously established explicit convergence rates.

To evaluate the practical effectiveness of the method, we applied it to the backward heat equation with a known exact solution. The numerical results at various time levels “final $t=1$, intermediate $t=0.5$, and initial ($t=0$)” demonstrated the method’s ability to produce stable and accurate reconstructions throughout the time domain. Notably, the recovery of the initial state from regularized terminal data, which exemplifies the severe ill-posedness of such problems, was achieved with remarkable accuracy. The regularization parameters $\alpha$ and $\beta$ were chosen in accordance with the theoretical estimates derived in our analysis, reinforcing the coherence between the analytical and numerical components of the study.

Several promising directions remain open for future research. A natural extension of this work would involve applying the proposed regularization framework to nonlinear inverse problems or to systems of coupled parabolic equations. In addition, incorporating noise-contaminated data and analyzing the robustness of the method under stochastic perturbations would significantly enhance its practical relevance. Finally, the development of efficient numerical schemes for high-dimensional problems, along with their implementation in real-time imaging scenarios, could open new perspectives for applied research and interdisciplinary collaboration.

Funding

The author received no specific funding for this work.

Acknowledgments

The author sincerely thanks the anonymous reviewers for their valuable comments and suggestions, which have greatly improved the quality of this manuscript.

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