A modified quasi-reversibility method for inverse source problem of Poisson equation

ABSTRACT

In this article, we consider an inverse source problem for Poisson equation in a strip domain. That is to determine source term in the Poisson equation from a noisy boundary data. This is an ill-posed problem in the sense of Hadamard, i.e., small changes in the data can cause arbitrarily large changes in the results. Before we give the main results about our proposed problem, we display some useful lemmas at first. Then we propose a modified quasi-reversibility regularization method to deal with the inverse source problem and obtain a convergence rate by using an a priori regularization parameter choice rule. Numerical examples are provided to show the effectiveness of the proposed method.

Keywords:

• Inverse source problem
• Poisson equation
• a modified quasi-reversibility regularization method
• a priori choice
• convergence analysis

2010 Mathematics Subject Classifications:

• 35R30
• 65N20

1. Introduction

Inverse source problem is of great importance in many branches of engineering and science; such as heat source determination [1,2], heat conduction problem [3–5], Stephan design problem [6] and pollutant detection. To our best knowledge, there are also a variety of researches on inverse source problems in the Poisson equation adopted numerical methods; for examples, logarithmic potential method [7], the projective method [8], the Green's function method [9], the dual reciprocity boundary element method [10,11] and the method of fundamental solution (MFS) [12–14].

Quasi-reversibility method is originally introduced by Lattes and Lions [15], and later studied by Mel${}^{\mathrm{\prime }}$nikova and Filinkov [16]. The idea consists in replacing the final boundary value problem with an approximate solution of the final boundary value problem. In the initial method of the quasi-reversibility, the author [17] replaced the heat operator $\mathrm{\partial }/\mathrm{\partial }t-{\mathrm{\partial }}^2/\mathrm{\partial }{x}^2$ by a perturbed operator $P\epsilon =(\mathrm{\partial }/\mathrm{\partial }t)+A-\epsilon A{A}^{\ast }$, perturbing the final condition we get an approximate solution from the final boundary value problem with a small parameter ε. The authors [18] take $f(A)=A-ϵA^2$ using logarithmic convexity to obtain well-posed solution as above Lattes and Lions [15]. The final value problem in [19] is considered about perturbing the final conditions to obtain an approximate non-local problem after operator perturbation. In [20], the quasi-reversibility method is to approach the ill-posed second order Cauchy problem depending on a (small) regularization parameter, based on the fundamental solution for a second order elliptic operator. Furthermore they propose the mixed quasi-reversibility method, and give some nice results. The ill-posed problem of the wave equation in [21] is replaced with a boundary value problem for a fourth order equation by using the method of quasi-reversibility. They consider the wave equation $Lu:=\frac{1}{c(x{)}^2}{\mathrm{\partial }}_{tt}u-\mathrm{△}u=f$, constructing Tikhonov functional firstly $J_{\epsilon }(u)=\frac{1}{2}\Vert Lu-f{\Vert }^2+\frac{\epsilon }{2}\Vert u{\Vert }^2$, it is equivalent with the abstract Euler equation $J_{\epsilon }^{\mathrm{\prime }}(u_{\epsilon })(v)=0$, for all $v\in H_0^2(Q_T)$, then through a minimizer $u_{\epsilon }$ of calculation for above equation, they obtain perturb term to approximate the solution of ill-posed problem with a small parameter ε. In [22], from the original quasi-reversibility method, the mixed quasi-reversibility method with variable parameter λ is extended in a system of two second-order equations involving two functions u and λ, the aim is to find an approximation $(u_{\epsilon },\lambda )$ of $(u,\lambda )$ as a solution of the weak formulation and $(\delta ,\epsilon )$ denotes α for small $\epsilon >0$ and $\delta >0$. The method of quasi-reversibility proposed by [23] is a particular case of Tikhonov regularization and $A=\mathrm{△}$ or $\mathrm{△}+k^2$, which provides corresponding error estimate with a priori choice for ε as a function of δ. In [24], the article adds ${\mu }^2{f}_{xx}(x)$ to the left-hand side of the equation $\mathrm{△}u(x,y)-k^{2}u(x,y)-f(x)$, the quasi-reversibility regularization solution and a priori convergence estimate are obtained. There are some important references about inverse source problem by using the quasi-reversibility method recently, such as the inverse source problems for parabolic equations [25,26], and hyperbolic equations [27–29].

In this article, we consider the following inverse problem: (1) $\{\begin{array}{lc}-u_{xx}-u_{yy}=f(x),& 0<x<\pi ,0<y<\mathrm{\infty },\\ u(0,y)=u(\pi ,y)=0,& 0\le y<\mathrm{\infty },\\ u(x,0)=0,u(x,y){|}_{y\to \mathrm{\infty }}bounded,& 0\le x\le \pi ,\\ u(x,1)=g(x),& 0\le x\le \pi ,\end{array}$(1) to find a pair of function $(u(x,y),f(x))$ which satisfies the Poisson equation on above conditions. Subsequently we will study the above problem, where we perturb the equation to form an approximate problem depending on a small parameter, before that we need to give the following preparations.

Generally, the input data $g(x)$ with a noise level δ is merely measured in $L^2(0,\pi )$, and we give that (2) $\Vert g-g_{\delta }\Vert \le \delta .$(2) We obtain that the solution of problem (1(1) $\{\begin{array}{lc}-u_{xx}-u_{yy}=f(x),& 0<x<\pi ,0<y<\mathrm{\infty },\\ u(0,y)=u(\pi ,y)=0,& 0\le y<\mathrm{\infty },\\ u(x,0)=0,u(x,y){|}_{y\to \mathrm{\infty }}bounded,& 0\le x\le \pi ,\\ u(x,1)=g(x),& 0\le x\le \pi ,\end{array}$(1) ) using separation of variables has the following form: (3) $u(x,y)=\sum _{n=1}^{\mathrm{\infty }}\frac{1-e^{-ny}}{n^2}(f,X_n)X_n,$(3) where $$\begin{gathered} \{X_n=\sqrt{\frac{2}{\pi }}\sin nx,(n=1,2,\dots )\}, \\ (f,X_n)=\sqrt{\frac{2}{\pi }}{\int }_0^{\pi }f(x)\sin nx\mathrm{d}x. \end{gathered}$$We define the operator $K:f\to g$, then we have (4) $g(x)=Kf(x)=\sum _{n=1}^{+\mathrm{\infty }}(g,X_n)X_n=\sum _{n=1}^{\mathrm{\infty }}\frac{1-e^{-n}}{n^2}(f,X_n)X_n.$(4) The singular values $\{{\sigma }_n{\}}_{n=1}^{\mathrm{\infty }}$ of K satisfy ${\sigma }_n=\frac{1-e^{-n}}{n^2},$correspondingly $(g,X_n)=\frac{1-e^{-n}}{n^2}(f,X_n)(X_n,X_n),$i.e. $(f,X_n)={\sigma }_n^{-1}(g,X_n),$then, $f(x)=K^{-1}g(x)=\sum _{n=1}^{+\mathrm{\infty }}\frac{1}{{\sigma }_n}(g,X_n)X_n=\sum _{n=1}^{+\mathrm{\infty }}\frac{n^2}{1-e^{-n}}(g,X_n)X_n.$From [30], the solution does not depend on the data continuously, the problem (1(1) $\{\begin{array}{lc}-u_{xx}-u_{yy}=f(x),& 0<x<\pi ,0<y<\mathrm{\infty },\\ u(0,y)=u(\pi ,y)=0,& 0\le y<\mathrm{\infty },\\ u(x,0)=0,u(x,y){|}_{y\to \mathrm{\infty }}bounded,& 0\le x\le \pi ,\\ u(x,1)=g(x),& 0\le x\le \pi ,\end{array}$(1) ) is ill-posed. Several articles impose regularization method to deal with ill-posed problem (1(1) $\{\begin{array}{lc}-u_{xx}-u_{yy}=f(x),& 0<x<\pi ,0<y<\mathrm{\infty },\\ u(0,y)=u(\pi ,y)=0,& 0\le y<\mathrm{\infty },\\ u(x,0)=0,u(x,y){|}_{y\to \mathrm{\infty }}bounded,& 0\le x\le \pi ,\\ u(x,1)=g(x),& 0\le x\le \pi ,\end{array}$(1) ): for examples, the Tikhonov regularization method [31,32], the super order regularization method [33], the quasi-boundary value regularization method [34,35], the quasi-reversibility method [36], the modified regularization method [30,37], the truncation method [38]. Recently, Boussetila and Rebbani [39] propose a modified quasi-reversibility method, and it is employed by Huang [40] and Fury [41] and Trong and Tuan [42] in the case of the autonomous Cauchy problem.

Figure 1. The exact and approximate solutions with M = 50, N = 5 and various noise level for Example 1 where $x\in [0,\pi ]$. (a): heat source for p = 1; (b): heat source for p = 3.

Figure 2. The exact and approximate solutions with M = 50, N = 5 and various noise level for Example 2 where $x\in [0,\pi ]$. (a): heat source for p = 1; (b): heat source for p = 3.

Figure 3. The exact and approximate solutions with M = 50, N = 5 and various noise level for Example 2 where $x\in [0,\pi ]$. (a): heat source for p = 1; (b): heat source for p = 3.

In this article, we will use a modified quasi-reversibility method to deal with identifying the unknown source of the problem (1(1) $\{\begin{array}{lc}-u_{xx}-u_{yy}=f(x),& 0<x<\pi ,0<y<\mathrm{\infty },\\ u(0,y)=u(\pi ,y)=0,& 0\le y<\mathrm{\infty },\\ u(x,0)=0,u(x,y){|}_{y\to \mathrm{\infty }}bounded,& 0\le x\le \pi ,\\ u(x,1)=g(x),& 0\le x\le \pi ,\end{array}$(1) ). Before doing that, we need to define an a priori bound on unknown source, (5) $\Vert f(x){\Vert }_{H^p(0,\pi )}\le E,p>0,$(5) where E>0 is a constant and $\Vert \cdot {\Vert }_{H^p(0,\pi )}$ denotes the norm in Sobolev space which is defined as follows [43]: $\Vert f(x){\Vert }_{H^p(0,\pi )}={(\sum _{n=1}^{+\mathrm{\infty }}(1+n^2{)}^p|⟨f,X_n⟩{|}^2)}^{\frac{1}{2}}.$This article is organized as follows. Section 2 gives some preliminary results. In Section 3, a regularization solution and error estimation of the inverse problem are provided by a modified quasi-reversibility method. Section 4 gives some examples to illustrate the accuracy and efficiency of the proposed method in problem (1(1) $\{\begin{array}{lc}-u_{xx}-u_{yy}=f(x),& 0<x<\pi ,0<y<\mathrm{\infty },\\ u(0,y)=u(\pi ,y)=0,& 0\le y<\mathrm{\infty },\\ u(x,0)=0,u(x,y){|}_{y\to \mathrm{\infty }}bounded,& 0\le x\le \pi ,\\ u(x,1)=g(x),& 0\le x\le \pi ,\end{array}$(1) ). Section 5 puts an end to this paper with a brief conclusion.

2. Some auxiliary results

In this section, we give four important lemmas as follows.

Lemma 2.1

For $n\ge 1$, $\frac{1}{1-e^{-n}}<2.$

Lemma 2.2

If $\mu >0$, $n\ge 1$, $\frac{1}{1-e^{\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}}}\le (1+\mu )\cdot e^{\frac{1}{\mu }}.$

Proof.

$\begin{array}{rl}\frac{1}{1-e^{\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}}}& \le \frac{\sqrt{1+{\mu }^2{n}^2}\cdot e^{\frac{n}{\sqrt{1+{\mu }^2{n}^2}}}}{n}\\ & \le \sqrt{\frac{1}{n^2}+{\mu }^2}\cdot e^{\sqrt{\frac{1}{\frac{1}{n^2}+{\mu }^2}}}\\ & \le (1+\mu )\cdot e^{1/\mu }.\end{array}$

Lemma 2.3

[30]

If $\mu >0$, $n\ge 1$, p>0, $\frac{{\mu }^2{n}^2}{(1+{\mu }^2{n}^2)(1+n^2{)}^{\frac{p}{2}}}<max\{{\mu }^2,{\mu }^p\}.$

Lemma 2.4

If $\mu >0$, $n\ge 1$, $e^{\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}}-e^{-n}<n\cdot \frac{{\mu }^2{n}^2}{1+{\mu }^2{n}^2}.$

Proof.

$\begin{array}{rl}{e}^{\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}}-e^{-n}& =(n-\frac{n}{\sqrt{1+{\mu }^2{n}^2}})\cdot e^{\zeta }\\ & \le n\cdot \frac{(\sqrt{1+{\mu }^2{n}^2}-1)(\sqrt{1+{\mu }^2{n}^2}+1)}{\sqrt{1+{\mu }^2{n}^2}(\sqrt{1+{\mu }^2{n}^2}+1)}\\ & <n\cdot \frac{{\mu }^2{n}^2}{1+{\mu }^2{n}^2},\end{array}$where $\zeta \in [-n,\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}].$

3. A modified regularization method and convergence estimates

We will investigate the following problem: (6) $\{\begin{array}{lc}-u_{xx}-u_{yy}+{\mu }^2{u}_{xxxx}=f(x)-{\mu }^2/(1+{\mu }^2)(f_{xx}(x)+f(x)),& 0<x<\pi ,0<y<\mathrm{\infty },\\ u(0,y)=u(\pi ,y)=0,& 0\le y<\mathrm{\infty },\\ u(x,0)=0,u(x,y){|}_{y\to \mathrm{\infty }}bounded,& 0\le x\le \pi ,\\ u(x,1)=g^{\delta }(x),& 0\le x\le \pi .\end{array}$(6) By separation of variables, we obtain that (7) $f_{\mu }^{\delta }(x)=\sum _{n=1}^{+\mathrm{\infty }}\frac{(1+{\mu }^2)n^2}{(1+{\mu }^2{n}^2)(1-e^{\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}})}(g_{\delta },X_n)X_n,$(7) which is called the modified regularized solution of problem (1(1) $\{\begin{array}{lc}-u_{xx}-u_{yy}=f(x),& 0<x<\pi ,0<y<\mathrm{\infty },\\ u(0,y)=u(\pi ,y)=0,& 0\le y<\mathrm{\infty },\\ u(x,0)=0,u(x,y){|}_{y\to \mathrm{\infty }}bounded,& 0\le x\le \pi ,\\ u(x,1)=g(x),& 0\le x\le \pi ,\end{array}$(1) ), correspondingly (8) $f_{\mu }(x)=\sum _{n=1}^{+\mathrm{\infty }}\frac{(1+{\mu }^2)n^2}{(1+{\mu }^2{n}^2)(1-e^{\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}})}(g,X_n)X_n.$(8)

Theorem 3.1

Let $g_{\delta }$ be measured data at y = 1 satisfying (2(2) $\Vert g-g_{\delta }\Vert \le \delta .$(2) ) and the a priori condition (5(5) $\Vert f(x){\Vert }_{H^p(0,\pi )}\le E,p>0,$(5) ) hold for p>0, if selecting $\mu =\{\begin{array}{lc}{\displaystyle \frac{1}{\ln (\frac{E}{\delta }{)}^{\frac{1}{p+2}}},}& 0<p\le 2,\\ {\displaystyle \frac{1}{\ln (\frac{E}{\delta }{)}^{\frac{1}{4}}},}& p>2,\end{array}$then we obtain the following error estimate: $\begin{array}{rl}& \Vert f_{\mu }^{\delta }-f\Vert \\ & \le \{\begin{array}{l}{\displaystyle (1+\frac{1}{\ln (\frac{E}{\delta }{)}^{\frac{1}{p+2}}}){\left(\frac{E}{\delta }\right)}^{\frac{1}{p+2}}(\delta +\frac{E}{(\ln (\frac{E}{\delta }{)}^{\frac{1}{p+2}}{)}^p}(1+\frac{1}{(\ln (\frac{E}{\delta }{)}^{\frac{1}{p+2}}{)}^2})),}\\ 0<p\le 2,\\ {\displaystyle (1+\frac{1}{\ln (\frac{E}{\delta }{)}^{\frac{1}{4}}}){\left(\frac{E}{\delta }\right)}^{\frac{1}{4}}(\delta +E(\frac{1}{\ln (\frac{E}{\delta }{)}^{\frac{1}{4}}}{)}^2(1+\frac{1}{(\ln (\frac{E}{\delta }{)}^{\frac{1}{4}}{)}^2})),}\\ p>2.\end{array}\end{array}$

Proof.

By the triangle inequality, we know $\Vert f_{\mu }^{\delta }-f\Vert \le \Vert f_{\mu }^{\delta }-f_{\mu }\Vert +\Vert f_{\mu }-f\Vert .$Firstly, we give an estimate for the first term as follows: $\begin{array}{rl}& \Vert f_{\mu }^{\delta }-f_{\mu }\Vert \\ & =\Vert \sum _{n=1}^{+\mathrm{\infty }}\frac{(1+{\mu }^2)n^2}{(1+{\mu }^2{n}^2)(1-e^{\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}})}(g_{\delta },X_n)X_n\\ & -\sum _{n=1}^{+\mathrm{\infty }}\frac{(1+{\mu }^2)n^2}{(1+{\mu }^2{n}^2)(1-e^{\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}})}(g,X_n)X_n\Vert \\ & \le \underset{n\ge 1}{sup}\frac{(1+{\mu }^2)n^2}{(1+{\mu }^2{n}^2)(1-e^{\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}})}\Vert \sum _{n=1}^{+\mathrm{\infty }}(g,X_n)X_n\Vert \\ & \le \delta (1+{\mu }^2)\underset{n\ge 1}{sup}\left\{\frac{n\cdot \sqrt{1+{\mu }^2{n}^2}{e}^{\frac{n}{\sqrt{1+{\mu }^2{n}^2}}}}{1+{\mu }^2{n}^2}\right\}\\ & \le \delta \frac{(1+{\mu }^2)\cdot e^{\frac{1}{\mu }}}{\sqrt{1+{\mu }^2{n}^2}}\\ & \le \delta \sqrt{1+{\mu }^2}\cdot e^{\frac{1}{\mu }}\\ & \le \delta (1+\mu )\cdot e^{\frac{1}{\mu }}.\end{array}$According to Lemmas 2.1–2.4 and an a priori bound condition of unknown source, we obtain $\begin{array}{rl}& \Vert f_{\mu }-f\Vert \\ & =\Vert \sum _{n=1}^{+\mathrm{\infty }}\frac{(1+{\mu }^2)n^2}{(1+{\mu }^2{n}^2)(1-e^{\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}})}(g,X_n)X_n-\sum _{n=1}^{+\mathrm{\infty }}\frac{n^2}{1-e^{-n}}(g,X_n)X_n\Vert \\ & =\Vert \sum _{n=1}^{+\mathrm{\infty }}\frac{{\mu }^2{n}^2(e^{\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}}-1)+e^{\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}}-e^{-n}+{\mu }^2(1-e^{-n})}{(1+{\mu }^2{n}^2)(1-e^{\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}})(1+n^2{)}^{\frac{p}{2}}}\\ & \cdot \frac{n^2}{1-e^{-n}}(1+n^2{)}^{\frac{p}{2}}(g,X_n)X_n\Vert \\ & =\underset{n\ge 1}{sup}\{\frac{-{\mu }^2{n}^2}{(1+{\mu }^2{n}^2)(1+n^2{)}^{\frac{p}{2}}}+\frac{e^{\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}}-e^{-n}}{(1+{\mu }^2{n}^2)(1-e^{\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}})(1+n^2{)}^{\frac{p}{2}}}\\ & +\frac{{\mu }^2(1-e^{-n})}{(1+{\mu }^2{n}^2)(1-e^{\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}})(1+n^2{)}^{\frac{p}{2}}}\}\Vert \sum _{n=1}^{+\mathrm{\infty }}\frac{n^2}{1-e^{-n}}(1+n^2{)}^{\frac{p}{2}}(g,X_n)X_n\Vert \\ & \le \underset{n\ge 1}{sup}\left\{\frac{{\mu }^2{n}^2}{(1+{\mu }^2{n}^2)(1+n^2{)}^{\frac{p}{2}}}(\frac{e^{\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}}-e^{-n}}{{\mu }^2{n}^2(1-e^{\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}})}+\frac{1-e^{-n}}{n^2(1-e^{\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}})})\right\}\cdot E\\ & \le E\underset{n\ge 1}{sup}\left\{\frac{{\mu }^2{n}^2}{(1+{\mu }^2{n}^2)(1+n^2{)}^{\frac{p}{2}}}(\frac{n\cdot (1+\mu )}{1+{\mu }^2{n}^2}\cdot e^{\frac{1}{\mu }}+\frac{1+\mu }{n}\cdot e^{\frac{1}{\mu }})\right\}\\ & \le Emax\{{\mu }^2,{\mu }^p\}{e}^{\frac{1}{\mu }}(1+\mu )(\frac{1}{{\mu }^2}+1).\end{array}$Combining above two estimates, we have $\begin{array}{rl}& \Vert f_{\mu }^{\delta }-f\Vert \\ & \le \delta (1+\mu )\cdot e^{\frac{1}{\mu }}+E(max\{{\mu }^2,{\mu }^p\}{e}^{\frac{1}{\mu }}(1+\mu )(\frac{1}{{\mu }^2}+1))\\ & \le \{\begin{array}{lc}(1+\frac{1}{\ln (\frac{E}{\delta }{)}^{\frac{1}{p+2}}}){\left(\frac{E}{\delta }\right)}^{\frac{1}{p+2}}(\delta +\frac{E}{(\ln (\frac{E}{\delta }{)}^{\frac{1}{p+2}}{)}^p}(1+\frac{1}{(\ln (\frac{E}{\delta }{)}^{\frac{1}{p+2}}{)}^2})),& 0<p\le 2,\\ (1+\frac{1}{\ln (\frac{E}{\delta }{)}^{\frac{1}{4}}}){\left(\frac{E}{\delta }\right)}^{\frac{1}{4}}(\delta +E(\frac{1}{\ln (\frac{E}{\delta }{)}^{\frac{1}{4}}}{)}^2(1+\frac{1}{(\ln (\frac{E}{\delta }{)}^{\frac{1}{4}}{)}^2})),& p>2.\end{array}\end{array}$

Based on the above discussion, we need to illustrate them with some examples in the next section.

4. Numerical verification

In this section, we give some different examples on the basis of the following preparation process.

From (4(4) $g(x)=Kf(x)=\sum _{n=1}^{+\mathrm{\infty }}(g,X_n)X_n=\sum _{n=1}^{\mathrm{\infty }}\frac{1-e^{-n}}{n^2}(f,X_n)X_n.$(4) ), we know that (9) $\begin{array}{rl}(Kf)(x)& =\sum _{n=1}^{+\mathrm{\infty }}\frac{1-e^{-n}}{n^2}(f,X_n)X_n\end{array}$(9) (10) $\begin{array}{rl}& =\frac{2}{\pi }{\int }_0^{\pi }\sum _{n=1}^{+\mathrm{\infty }}\frac{1-e^{-n}}{n^2}f(s)\sin ns\sin nx\mathrm{d}s=g(x).\end{array}$(10) We use the rectangle formula to approach the integral and do an approximate truncation for the series by choosing the sum of the front N terms. By considering an equidistant grid $0=x_1<\cdots <x_M=\pi$, $(x_i=\frac{i-1}{M-1}\pi ,i=1,\dots ,M)$, we get (11) $\frac{2}{\pi }\sum _{i=1}^M\sum _{n=1}^N\frac{1-e^{-n}}{n^2}f(x_i)\sin n{x}_i\sin n{x}_{j}h=g(x_j),$(11) where $h=\frac{\pi }{M-1}$. Correspondingly, we obtain (12) $f_{\mu }^{\delta }(x_j)=\frac{2}{\pi }\sum _{i=1}^M\sum _{n=1}^N\frac{(1+{\mu }^2)n^2}{(1+{\mu }^2{n}^2)(1-e^{\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}})}{g}^{\delta }(x_i)\sin n{x}_i\sin n{x}_{j}h.$(12) Adding a random distribute perturbation to each data function, we obtain $g^{\delta }$, i.e. $g^{\delta }=g+\epsilon \mathrm{randn}(\mathrm{size}(g)).$The total noise level δ can be measured in the sense of root mean square error(RMSE) according to $\delta =\Vert g^{\delta }-g{\Vert }_2={(\frac{1}{M}\sum _{n=1}^M(g_n-g^{\delta }{)}^2)}^{\frac{1}{2}}.$In order to research the effect of numerical computations, we compute the relative root mean squares error (RRMSE) of $f(x)$ by (13) $RRMSE(f)=\frac{\sqrt{\sum _{i=1}^M(f_{\mu }^{\delta }({\overline{x}}_i)-f({\overline{x}}_i){)}^2}}{\sqrt{\sum _{i=1}^M(f({\overline{x}}_i){)}^2}},$(13) where $\{{\overline{x}}_i{\}}_{i=1}^M$ is a set of discrete points in internal $[0,\pi ]$.

The numerical examples are constructed in the following way: First we select the exact solution $f(x)$ and obtain the exact data function $g(x)$ using (11(11) $\frac{2}{\pi }\sum _{i=1}^M\sum _{n=1}^N\frac{1-e^{-n}}{n^2}f(x_i)\sin n{x}_i\sin n{x}_{j}h=g(x_j),$(11) ). Then we add a normally distributed perturbation to each data function giving vector $g^{\delta }$. Finally we obtain the regularization solutions using (12(12) $f_{\mu }^{\delta }(x_j)=\frac{2}{\pi }\sum _{i=1}^M\sum _{n=1}^N\frac{(1+{\mu }^2)n^2}{(1+{\mu }^2{n}^2)(1-e^{\frac{-n}{\sqrt{1+{\mu }^2{n}^2}}})}{g}^{\delta }(x_i)\sin n{x}_i\sin n{x}_{j}h.$(12) ).

Example 1

We suppose that the solution of equation $u(x,y)=(1-\exp (-y))\sin (x)$ and the source function $f(x)=\sin (x)$, easily know that the data function $g(x)=(1-\exp (-1))\sin (x)$, we choose $x\in [0,\pi ]$ in this example.

Table 1. δ, μ, $RRMSE(f)$ with respect to various values of ε while p = 1, M = 50, N = 5 and for Example 1.

ε	0.01	0.0.001	0.0001	0.00001
μ	0.0279	0.0202	0.0152	0.0127
δ	0.0113	0.0011	0.0001	$8.7\times {10}^{-6}$
$RRMSE(f)$	0.0092	0.0016	0.0001	$1.05\times {10}^{-5}$

Table 2. δ, μ, $RRMSE(f)$ (relative error of the source term) with respect to various values of ε while p = 3, M = 50, N = 5 and for Example 1.

ε	0.01	0.0.001	0.0001	0.00001
μ	0.1065	0.0985	0.0919	0.0878
δ	0.0110	0.0012	0.0001	$1.03\times {10}^{-5}$
$RRMSE(f)$	0.0070	0.0006	0.0004	$3.557\times {10}^{-4}$

Example 2

Consider the reconstruction of a Gaussian normal distribution: $f(x)=\frac{1}{\alpha \sqrt{2\pi }}{e}^{-(x-\beta {)}^2/(2{\alpha }^2)}$where $\alpha =\frac{\pi }{4}$, $\beta =\frac{\pi }{2}.$

Table 3. δ, μ, $RRMSE(f)$ with respect to various values of ε while p = 1, M = 50, N = 5 and the $RRMSE(f)$ approaches 0.0093 when $\epsilon \le 1\times {10}^{-4}$ for Example 2.

ε	0.01	0.001	0.0001	0.00001
μ	0.0952	0.0844	0.0774	0.0722
δ	0.0099	0.0010	0.0001	$9.709\times {10}^{-6}$
$RRMSE(f)$	0.0169	0.0095	0.0093	0.0093

Table 4. δ, μ, $RRMSE(f)$ with respect to various values of ε while p = 3, M = 50, N = 5 and the $RRMSE(f)$ approaches 0.1053 when $\epsilon \le 1\times {10}^{-9}$ for Example 2.

ε	0.01	0.001	0.0001	0.00001
μ	0.1280	0.1100	0.1010	0.0942
δ	0.0099	0.0009	0.0001	$1.036\times {10}^{-5}$
$RRMSE(f)$	0.1663	0.1063	0.1059	0.1056

Example 3

Consider the reconstruction of a piecewise smooth source: $f(x)=\{\begin{array}{lc}0,& 0\le x<\frac{\pi }{4}\\ \frac{4}{\pi }x-1,& \frac{\pi }{4}<x\le \frac{\pi }{2}\\ 3-\frac{4}{\pi }x,& \frac{\pi }{2}<x\le \frac{3\pi }{4}\\ 0,& \frac{3\pi }{4}<x\le \pi .\end{array}$

Table 5. δ, μ, $RRMSE(f)$ with respect to various values of ε while p = 1, M = 50, N = 5 and the $RRMSE(f)$ approaches 0.0134 when $\epsilon \le 1\times {10}^{-11}$ for Example 3.

ε	0.01	0.001	0.0001	0.00001
μ	0.0955	0.0857	0.0779	0.0725
δ	0.0085	0.0011	0.0001	$9.2\times {10}^{-6}$
$RRMSE(f)$	0.0165	0.0148	0.0145	0.0144

Table 6. δ, μ, $RRMSE(f)$ with respect to various values of ε while p = 3, M = 50, N = 5 and the $RRMSE(f)$ approaches 0.0889 when $\epsilon \le 1\times {10}^{-37}$ for Example 3.

ε	0.01	0.001	0.0001	0.00001
μ	0.1250	0.1091	0.0997	0.0932
δ	0.0110	0.0011	0.0001	$9.46\times {10}^{-6}$
$RRMSE(f)$	0.2365	0.1283	0.1167	0.1110

For Examples 1–3, we illustrate the comparisons about exact solutions and regularized solutions by a priori regularization parameter choice rule with different noise levels and different cases of p. We can find that the smaller ε is, the better the computed approximation is. For both continuous and discontinuous cases in Examples 2–3, it can also be seen that the well-known Gibbs phenomenon occurs and the approximate solutions near non-smooth and discontinuous points are less ideal.

5. Conclusion

In this article, we use a modified quasi-reversibility method to identify an unknown source term depending only on one variable in two dimensional Poisson equation. It is shown that with a certain choice of the parameter μ, a stability estimate is obtained. Meanwhile, three examples verify the efficiency and accuracy of our proposed method.

Acknowledgments

The authors gratefully thank the referees for their valuable constructive comments which improve greatly the quality of the paper. The work described in this article was supported by the NNSF of China (11326234), NSF of Gansu Province (145RJZA099), Scientific research project of Higher School in Gansu Province (2014A-012), and Project of NWNU-LKQN2020-08.

Disclosure statement

No potential conflict of interest was reported by the authors.