Identification of Source Term from Part of the Boundary Conditions

Abstract

This paper identifies a source term depending on spatial variable in a heat equation from just part of the boundary conditions. The measurement data are specified at an internal moment of time. The ill-posedness of the problem is higher than most of the previous source identification problems. This is because the problem becomes a noncharacteristic Cauchy problem for the heat equation if the source term is given, which is known as severely ill-posed. The method of fundamental solutions (MFS) in conjunction with the classical Tikhonov regularization method is proposed to reconstruct a stable approximation. The fundamental solutions for the heat equation are spherically symmetric in spatial variable and satisfy the equation automatically, and thus only the boundary conditions need to be satisfied. This characteristic allows the discretization to be performed only on boundary-like geometry and improve the computational efficiency. In this paper, several numerical examples are listed to show the feasibility and effectiveness of the suggested method.

1. Introduction

In the mathematical model, the source term in the heat equation often represents the measurable properties of physical or chemical reactions. It is generally difficult to determine if these kinds of reactions possibly happened during the solute migration processes directly. Thus, the source term is usually unknown and has to be resolved. This problem is referred to as the inverse source problem. It arises in many engineering fields, such as the inverse source calibration problem in weld modeling [1,2], estimation of an integrated modular motor drive in robotic application [3], determination of pollution source in groundwater resources management [4,5] and so on. Given the strong practical background, the study on the inverse problem has always been a focus of significant interest both in the fields of mathematics and engineering.

It is well known that the inverse source problem is ill-posed, i.e., any small perturbations of the measurement data would probably cause large deviation of the solution. It is precisely this instability that poses a challenge to theoretical analysis and numerical computation. Isakov [6] gave a complete description of this problem from the aspect of abstract function spaces. Some earlier results about the uniqueness and conditional stability are [7,8]. Recently, the authors [9] reconstructed a heat source just from local measurement and obtained the uniqueness and conditional stability estimate. For the dynamic boundary conditions, the authors [10,11] established the existence and uniqueness result of a quasi-solution related to the heat equation and the Schrödinger equation, respectively. As for the general regularization analysis, there are the method of truncated integration [12], the Fourier regularization method [13], a simplified Tikhonov regularization method [14], the Landweber iteration regularization method [15] and so on. In the numerical computation aspect, several numerical schemes have been used to solve the inverse heat source problem, for instance, the finite difference method [16,17], variational method [18,19], an iterative method [20,21], the boundary element method [22,23], the MFS [24,25] and so on. The MFS is an inherently meshless numerical scheme. That is, it does not require domain discretization, and thus it has a simple computational implementation. Beyond this benefit, it is easy to apply to resolve high order or high-dimensional partial differential equations including in the spherically symmetric regions. Hon and Wei [26] first used the MFS to solve the inverse heat conduction problem. After this, Yan et al. successfully implemented this approach to determine the time-dependent [24] and space-dependent [25] source term related to heat equations, respectively.

Most of the known results focus on the complete boundary conditions, i.e., those problems are mildly ill-posed. However, in many real applications, the surface of the origin is inaccessible for measurements. In such cases, one is restricted to part of the boundary measurements, which would not fully reflect the interesting applied situation. At this time, the inverse problems are severely ill-posed. In this paper, we apply the MFS to identify an unknown spatially dependent heat source and overdetermine the solution in the whole origin from just part of the boundary measurements.

2. Formulation of the Inverse Source Problem

Let $\mathrm{\Omega }\in {\mathbb{R}}^d$ be a bounded open domain with $C^2$-smooth boundary $\partial \mathrm{\Omega }$, and suppose that $\mathrm{\Gamma }\in \partial \mathrm{\Omega }$ is a subboundary. For simplicity, we denote $Q=\mathrm{\Omega }\times (0,t_f)$. For $x\in \partial \mathrm{\Omega }$, let $\nu =\nu \left(x\right)$ be the unit outward normal vector to $\partial \mathrm{\Omega }$. Consider the following problem related to a general parabolic equation

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{a}_0{u}_t-div(a\nabla u)+b·\nabla u+cu=\beta (x,t)F(x,t),\hfill & (x,t)\in Q,\hfill \\ u(x,0)=0,\hfill & x\in \mathrm{\Omega },\hfill \\ u(x,t)=g(x,t),\hfill & x\in \mathrm{\Gamma },0<t\le t_f,\hfill \\ {\displaystyle \frac{\partial u}{\partial \nu }}(x,t)=h(x,t),\hfill & x\in \mathrm{\Gamma },0<t\le t_f,\hfill \end{array}\right.\end{array}$$

(1)

in which a is a strictly positive and symmetric matrix function with elements in $C^1\left(\overline{Q}\right)$, $a_0,a,b,c\in C^1\left(\overline{Q}\right)$, $\beta ,{\beta }_t\in C\left(\overline{Q}\right)$, and $g,h$, are given functions. Note that if the source term F is given, the problem (1) is a noncharacteristic Cauchy problem for the parabolic equation, which is a highly ill-posed problem. When the information about F is missing, the problem (1) is under-determined mathematically and the degree of ill-posedness becomes more severer. Some supplementary information is required to completely determine the source term and the solution. In this paper, the additional temperature measurement is given on a fixed moment of time $t_0\in (0,t_f)$ by

$$u(x,t_0)=\phi \left(x\right),x\in \mathrm{\Omega }.$$

(2)

Remark 1.

According to Theorem 9.2.6 in [27], under the same regularity properties of $a_0,a,b,c$ and β as the above, if there exists some positive number ${ϵ}_0$, such that $\beta (x,t_0)>{ϵ}_0$ on Ω, and $F_t\equiv 0$ in Q. Then the solution pair $(u,F)\in H_{2,1;2}\left(Q\right)\times L_2\left(\mathrm{\Omega }\right)$ of problems (1) and (2) is uniquely determined by the Cauchy data $g,h$ and the intermediate time data φ.

A proof of this theorem is quite technical and long. The general ideas are to differentiate the equation in (1) with respect to t twice and use the Carleman estimate, which holds in level sets of some weight functions with special properties on a subdomain of $\mathrm{\Omega }$, combining with the properties of weight function; the author obtained that F is uniquely determined on this subdomain. Finally, the result could be received by the principle of analytic continuation.

This theorem illustrates the localized nature of the uniqueness result, as it is not necessary to know the lateral boundary data outside $\mathrm{\Gamma }$. Thus, the considered space region may be a subset of a larger domain.

For simplicity, we consider the equation in (1) with identity matrix a and functions $a_0=\beta =1$ and $c=0$ and vector function $b=0$, and F only depends on the spatial variable. That is the following problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t-\mathrm{\Delta }u=f\left(x\right),\hfill & (x,t)\in Q,\hfill \\ u(x,0)=0,\hfill & x\in \mathrm{\Omega },\hfill \\ u(x,t)=g(x,t),\hfill & x\in \mathrm{\Gamma },0<t\le t_f,\hfill \\ {\displaystyle \frac{\partial u}{\partial \nu }}(x,t)=h(x,t),\hfill & x\in \mathrm{\Gamma },0<t\le t_f.\hfill \end{array}\right.\end{array}$$

(3)

The purpose of this paper is to determine u and f from the measured intermediate time data $\phi \left(x\right)$.

3. Transformation of the Problem

For given $f\in L^2\left(\mathrm{\Omega }\right)$, the solution $w\in H^1\left(\mathrm{\Omega }\right)$ of the following Cauchy problem for elliptic equation

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathrm{\Delta }w=f\left(x\right),\hfill & x\in \mathrm{\Omega },\hfill \\ w\left(x\right)=0,\hfill & x\in \mathrm{\Gamma },\hfill \\ {\displaystyle \frac{\partial w}{\partial \nu }}\left(x\right)=0,\hfill & x\in \mathrm{\Gamma },\hfill \end{array}\right.\end{array}$$

(4)

is uniquely determined. Inspired by the article [25], we let

$$v(x,t)=u(x,t)+w\left(x\right),$$

(5)

then problems (2) and (3) are reformulated into

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{v}_t-\mathrm{\Delta }v=0,\hfill & (x,t)\in Q,\hfill \\ v(x,0)=r\left(x\right),\hfill & x\in \mathrm{\Omega },\hfill \\ v(x,t)=g(x,t),\hfill & x\in \mathrm{\Gamma },0<t\le t_f,\hfill \\ {\displaystyle \frac{\partial v}{\partial \nu }}(x,t)=h(x,t),\hfill & x\in \mathrm{\Gamma },0<t\le t_f,\hfill \end{array}\right.\end{array}$$

(6)

and

$$v(x,t_0)=\phi \left(x\right)+w\left(x\right).$$

(7)

A combination of the initial condition in (6) and (7), v also satisfies

$$\begin{array}{cc}\hfill & v_t-\mathrm{\Delta }v=0,(x,t)\in Q,\end{array}$$

(8)

$$\begin{array}{cc}& v(x,t_0)-v(x,0)=\phi \left(x\right),x\in \mathrm{\Omega },\end{array}$$

(9)

$$\begin{array}{cc}& v(x,t)=g(x,t),x\in \mathrm{\Gamma },0<t\le t_f,\end{array}$$

(10)

$$\begin{array}{cc}& {\displaystyle \frac{\partial v}{\partial \nu }}(x,t)=h(x,t),x\in \mathrm{\Gamma },0<t\le t_f.\end{array}$$

(11)

In Section 4, the MFS is applied to obtain the discrete ill-conditioned system of equations. To ensure numerical stability, the classical Tikhonov regularization method together with an appropriate parameter choice rule is implemented.

4. The MFS and the Tikhonov Regularization

The fundamental solution of Equation (8) is represented as

$$F(x,t)={\displaystyle \frac{1}{{\left(4\pi t\right)}^{\frac{d}{2}}}}{e}^{-{\displaystyle \frac{{\left|x\right|}^2}{4t}}}H\left(t\right),$$

(12)

in which $H\left(t\right)$ denotes the Heaviside function, i.e.,

$$H\left(t\right)=\left\{\begin{array}{ccc}\hfill & 1,\hfill & \hfill t\ge 0,\\ \hfill & 0,\hfill & \hfill t<0.\end{array}\right.$$

(13)

Let $T>t_f$ be a constant; then, the following function

$$\varphi (x,t)=F(x,t+T),$$

(14)

also satisfies Equation (8).

Assume the measured data are provided at the points ${\left\{(x_j,t_j)\right\}}_{j=1}^m$ in $\mathrm{\Omega }\times \left\{t_0\right\}$. ${\left\{(x_j,t_j)\right\}}_{j=m+1}^{m+n}$ and ${\left\{(x_j,t_j)\right\}}_{j=m+n+1}^{m+2n}$ denote the boundary collocation points on $\mathrm{\Gamma }$ satisfying the boundary condition (10) and (11), respectively. According to the idea of MFS in resolving other heat conduction equation, the solution of problem (8)–(11) is approximated by

$$\overline{v}(x,t)=\sum _{j=1}^{m+2n}{\lambda }_j\varphi (x-x_j,t-t_j),$$

(15)

in which ${\lambda }_j$ are undetermined coefficients and need to be resolved. In the above expression, for $t\ge 0$ we have $t-t_j+T>0$ because of $T>t_f$. Thus, all the singular points are shifted to the outside of $\overline{Q}$.

Collocating (15) into Equations (9)–(11), we obtain

$$\sum _{j=1}^{m+2n}{\lambda }_j\left[\varphi (x_i-x_j,t_0-t_j)-\varphi (x_i-x_j,0-t_j)\right]=\phi \left(x_i\right),i=1,2,\dots ,m,$$

(16)

$$\sum _{j=1}^{m+2n}{\lambda }_j\varphi (x_k-x_j,t_k-t_j)=g(x_k,t_k),k=m+1,m+2,\dots ,m+n,$$

(17)

and

$$\sum _{j=1}^{m+2n}{\lambda }_j{\displaystyle \frac{\partial \varphi }{\partial \nu }}(x_s-x_j,t_s-t_j)=h(x_s,t_s),s=m+n+1,m+n+2,\dots ,m+2n.$$

(18)

The above equations may be represented as a matrix system

$$A\lambda =b,$$

(19)

in which

$$A=\left(\begin{array}{c}\varphi (x_i-x_j,t_0-t_j)-\varphi (x_i-x_j,0-t_j)\\ \varphi (x_k-x_j,t_k-t_j)\\ {\displaystyle \frac{\partial \varphi }{\partial \nu }}(x_s-x_j,t_s-t_j)\end{array}\right),$$

(20)

$$b=\left(\begin{array}{c}\phi \left(x_i\right)\\ g(x_k,t_k)\\ h(x_s,t_s)\end{array}\right),$$

(21)

and the ranges of $i,k,s$ are the same as in (16)–(18). Due to the ill-posedness of system (19), we adopt the Tikhonov regularization technique to obtain the regularized approximation.

The Tikhonov regularized solution ${\lambda }^{\alpha }$ for (19) is defined as the minimal element of

$$\underset{\lambda }{min}{\{\parallel A\lambda -b\parallel }^2+{\alpha \parallel \lambda \parallel }^2\},$$

(22)

in which $\parallel ·\parallel$ represents the Euclidean norm and $\alpha >0$ is known as the regularization parameter. Note that $\alpha$ will affect the accuracy of the regularized solution, and thus it is crucial to select an appropriate regularization parameter. In this paper, $\alpha$ is selected to be the minimal element of the following GCV function

$$G\left(\alpha \right)={\displaystyle \frac{\parallel A{\lambda }^{\alpha }{-b\parallel }^2}{{\left(trace(I-A{A}^I)\right)}^2}},\alpha >0,$$

(23)

in which I is a $(m+2n)$-dimensional unit matrix and $A^I={(A^{tr}A+\alpha I)}^{-1}{A}^{tr}$. This method of selecting regularization parameters is called the generalized cross validation (GCV).

Denote the regularized approximation of (19) by ${\lambda }^{{\alpha }^{\ast }}$. The approximate solution of (8)–(11) is written as

$$v_{{\alpha }^{\ast }}(x,t)=\sum _{j=1}^{m+2n}{\lambda }_j^{{\alpha }^{\ast }}\varphi (x-x_j,t-t_j),$$

(24)

then

$$r_{{\alpha }^{\ast }}\left(x\right)=\sum _{j=1}^{m+2n}{\lambda }_j^{{\alpha }^{\ast }}\varphi (x-x_j,0-t_j),$$

(25)

and thus

$$f_{{\alpha }^{\ast }}\left(x\right)=\mathrm{\Delta }{r}_{{\alpha }^{\ast }}\left(x\right)=\sum _{j=1}^{m+2n}{\lambda }_j^{{\alpha }^{\ast }}\mathrm{\Delta }\varphi (x-x_j,0-t_j).$$

(26)

5. Numerical Experiments

Without loss of generality, we take $t_f=1$ and the fixed moment of time is $t_0=0.5$. By adding a random distributed perturbation to original data $\phi$, we get a noisy data ${\phi }^{\delta }$, i.e.,

$${\phi }^{\delta }=\phi +\delta rand\left(size\left(\phi \right)\right),$$

(27)

in which $\delta$ indicates the level of measurement data, and ‘$rand(·)$’ is a uniformly distributed random number in the interval (0,1). As a test of accuracy, the following errors are adopted:

(1)

the root-mean square deviation

$$RMS\left(f\right)=\sqrt{{\displaystyle \frac{1}{N}}\sum _{i=1}^N{(f\left({\tilde{x}}_i\right)-f_{\alpha \ast }\left({\tilde{x}}_i\right))}^2},$$

(28)

(2)

the relative root-mean square deviation

$$RES\left(f\right)={\displaystyle \frac{\sqrt{{\sum }_{i=1}^N{(f\left({\tilde{x}}_i\right)-f_{\alpha \ast }\left({\tilde{x}}_i\right))}^2}}{\sqrt{{\sum }_{i=1}^N{\left(f\left({\tilde{x}}_i\right)\right)}^2}}},$$

(29)

in which N indicates the total quantity of uniformly distributed points $\left\{{\tilde{x}}_i\right\}$ of interest on $\overline{\mathrm{\Omega }}$. $RMS\left(u\right)$ and $RES\left(u\right)$ are defined similarly.

5.1. One-Dimensional Examples

We set $\mathrm{\Omega }=[0,1]$ and boundary data $g,h$ are known at $x=1$ in the following three examples.

Example 1.

For $g(1,t)=0,h(1,t)=-\pi (1-e^{-{\pi }^{2}t})$, the following solution pair solve problem (3):

$$\begin{array}{ccc}& & u(x,t)=(1-e^{-{\pi }^{2}t})sin\left(\pi x\right),0\le x\le 1,0\le t\le 1,\hfill \\ & & f\left(x\right)={\pi }^{2}sin\left(\pi x\right),0\le x\le 1.\hfill \end{array}$$

The measurement data $\phi \left(x\right)=(1-e^{-{\displaystyle \frac{{\pi }^2}{2}}})sin\left(\pi x\right)$.

In order to make a direct comparison with the result obtained by Yan et al. [25], the numbers are taken to be $m=n=11$ and $N=21$ for Example 1. Firstly, we want to know the impact of parameter T on the numerical result. In Figure 1, the errors for Example 1 with regard to $\delta =0.01$ and different T are presented. We see that the errors are relatively small for $T<2.5$, while for Example 1 in [25], the accuracy is almost the same if $T<4$. This shows that the dependence of the result on T in the present paper is relatively large. Such a phenomenon might be related to the missing part of the boundary conditions, which leads to the severely ill-posedness of the problem. The influence of T is also reflected in the following several examples. It is a critical problem to be studied in the future about how to select an appropriate additional parameter T. In this paper, we set $T=1.5$ for other tests in Example 1.

Secondly, the influence of $t_0$ on the result is also considered. In Figure 2, we give the errors for different $t_0$ with $\delta =0.01$ for Example 1. It is easy to see that the accuracy of the approximate solutions is acceptable if $t_0\ge 0.2$, and it has little influence on the result when $0.2<t_0<1$. Meanwhile for Example 1 in [25], the errors are related to $t_0$ to some extent.

Figure 3 shows the choice rule of regularization parameter $\alpha$, i.e., $\alpha$ is taken to be the minimum element of the GCV function $G\left(\alpha \right)$. Some comparison of the analytical solution f and its numerical approximation $f_{\alpha \ast }$ with different $\delta =0.001,0.01,0.02,0.05$ are shown in Figure 4. From this figure, we see that the algorithm performs better when x is closer to 1. The main reason is that the data on $x=1$ are sufficient. Compared with Figure 3 in [25], the accuracy is a little higher in the present paper. It indicates that the MFS combining with the Tikhonov regularization and the GCV is effective in dealing with such inverse source problems concerned with incomplete boundary conditions.

Example 2.

We consider a Guassian normal distribution

$$f\left(x\right)={\displaystyle \frac{1}{\sigma \sqrt{2\pi }}}{e}^{-{\displaystyle \frac{{(x-\mu )}^2}{2{\sigma }^2}}},$$

(30)

in which $\mu =0.5$ is the mean and $\sigma =0.1$ is the standard deviation.

Example 3.

We consider a piece-wise continuous but not smooth source term

$$f\left(x\right)=\left\{\begin{array}{cc}0,\hfill & \hfill 0\le x\le 1/4,\\ 4x-1,\hfill & \hfill 1/4<x\le 1/2,\\ -4x+3,\hfill & \hfill 1/2\le x\le 3/4,\\ 0,\hfill & \hfill 3/4<x\le 1.\end{array}\right.$$

(31)

Different from Example 1, there are no analytical solutions for Examples 2 and 3. Suppose that $u_x(0,t)=u_x(1,t)=0$, the Crank–Nicholson algorithm is used to solve the corresponding direct problem, and thus g and $\phi$ are obtained. The discrete form of this direct problem is as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{u_i^{k+1}-u_i^k}{\mathrm{\Delta }t}}={\displaystyle \frac{1}{2{\left(\mathrm{\Delta }x\right)}^2}}(u_{i-1}^k-2u_i^k+u_{i+1}^k+u_{i-1}^{k+1}-2u_i^{k+1}+u_{i+1}^{k+1}),\hfill & 2\le i\le m-1,\hfill \\ u_i^1=0,\hfill & 1\le i\le m,\hfill \\ {\displaystyle \frac{u_2^k-u_1^k+u_2^{k+1}-u_1^{k+1}}{\mathrm{\Delta }x}}-{\displaystyle \frac{\mathrm{\Delta }x}{\mathrm{\Delta }t}}(u_1^{k+1}-u_1^k)+\mathrm{\Delta }x{f}_1=0,\hfill & 1\le k\le n-1,\hfill \\ {\displaystyle \frac{u_m^k-u_{m-1}^k+u_m^{k+1}-u_{m-1}^{k+1}}{\mathrm{\Delta }x}}+{\displaystyle \frac{\mathrm{\Delta }x}{\mathrm{\Delta }t}}(u_m^{k+1}-u_m^k)-\mathrm{\Delta }x{f}_m=0,\hfill & 1\le k\le n-1,\hfill \end{array}\right.\end{array}$$

in which $\mathrm{\Delta }x={\displaystyle \frac{1}{m-1}}$, $\mathrm{\Delta }t={\displaystyle \frac{1}{n-1}}$ and $u_i^k$ is the approximate value of $u\left(\right(i-1)\mathrm{\Delta }x,(k-1\left)\mathrm{\Delta }t\right)$. In [28], the authors used a modified Crank–Nicolson method to obtain a solution of the complex fuzzy heat equation.

In Examples 2 and 3, T is taken to be $1.01$. For Example 2, if we continue to select $m=n=11$, the error is relatively large as shown in

Table 1. Comparison of $RES\left(f\right)$ for Example 2 with $\delta =0.001$.

$\mathit{m}(=\mathit{n})$	11	21	31	41	51	61
RES(f)	0.3116	0.0930	0.0677	0.0577	0.1579	0.1469

. In order to achieve a better reconstruction result, we set $m=n=21$ and $N=31$ in Examples 2 and 3.

Figure 5 and Figure 6 present the comparison of exact source f and its approximations $f_{{\alpha }^{\ast }}$ with $\delta =0.001,0.01$. From Figure 5, the result is satisfactory for $x\ge 0.1$. If the noise $\delta$ increases to 0.02 and 0.05, the errors $RES\left(f\right)$ are 0.2709 and 0.4599, and thus the comparison figures are not given here. From Figure 6, the result is better for $\delta =0.001$. While for $\delta =0.01$, the approximate solution is evidently diverge from the exact solution near the non-smooth points. The effect is not as good as Figure 6 in [25]. In view of the severely ill-posedness of this problem, the effectiveness of the suggested algorithm is also acceptable.

5.2. Two-Dimensional Example

Example 4.

It is easy to verify

$$u(x_1,x_2,t)=(1-e^{-t})(cos{x}_1+cos{x}_2),(x_1,x_2,t)\in [0,1]\times [0,1]\times [0,1],$$

(32)

is an analytic solution of problem (3) with

$$f(x_1,x_2)=cos{x}_1+cos{x}_2,(x_1,x_2)\in [0,1]\times [0,1].$$

(33)

For simplicity, the data are only given on $\mathrm{\Gamma }=\left\{(x_1,x_2)\right|x_1=0,x_2\in [0,1]\}$.

For this example, we set $m=21\times 21,n=21\times 5$ and $N=31\times 31$. The numerical results are presented in Figure 7, where $T=8$ and $\delta =0,0.01,0.02,0.05$, respectively. Compared with Figure 8 and 9 in [25], although the accuracy in the present paper is not higher than that in [25] for exact measurement data, the errors in other cases are smaller. This figure shows the effectiveness of the suggested method in dealing with the severely ill-posed problem related to incomplete boundary conditions in the two-dimensional case.

From the comparison of the three examples with [25], we see that the approximate effect is better for the case where there are analytic solutions, while for the example without analytic solutions, the deviation is relatively large for larger noise.

6. Conclusions

In this paper, the MFS together with the Tikhonov regularization technique and the GCV parameter choice rule is successfully used to reconstruct a spacewise dependent source term from just part of the boundary conditions. Several numerical examples in one- and two-dimensions are presented to show the effectiveness of the suggested method. However, there are still some problems that need to be resolved urgently. Firstly, the dependence of the result on T is relatively large in the present paper; this might be related to the missing of part of the boundary conditions which leads to the severely ill-posedness of the problem. Secondly, it is not possible to obtain general theoretical results regarding the assessment of the accuracy of the approximate solution at present. These are both critical problems to be studied in the future.