A Modified Regularization Method for a Spherically Symmetric Inverse Heat Conduction Problem

Abstract

In this paper, we investigate a spherically symmetric inverse heat conduction problem, which determines the internal surface temperature distribution of the hollow sphere from measured data at the fixed location inside it. This problem is ill-posed, and a conditional stability result of its solution is given. A modified quasi-boundary value method is proposed to solve the ill-posed problem. Two H$\ddot{o}$lder-type error estimates between the approximation solution and its exact solution are obtained under an a priori and an a posteriori regularization parameter selection rule, respectively.

1. Introduction

The inverse heat conduction problem (IHCP) is an essential part of thermal science and technology research, which has a wide application background in nuclear physics, aerospace, metallurgy, and other industrial research fields [1]. For example, the inversion of the surface heat flux is an effective method to determine the re-entry thermal environment of the aircraft during its entry into the atmosphere [2,3].

The IHCP is a severely ill-posed problem in the sense of Hadamard [4]. That is, the solution of the problem (if it exists) does not depend continuously on measured data. Any small error in the measurement can induce an enormous error in computing the unknown solution. Hence, special regularization methods are needed to restore the stability of the solution for the problem [5,6,7].

As we know, the IHCPs have been studied in many works. For instance, conditional stability has been investigated in [8,9]. The numerical methods have been studied in [10,11,12,13,14,15]. Many regularization methods with convergence analysis have been proposed. These methods include the Tikhonov method [16,17], the mollification method [18,19,20], the Fourier method [21,22,23], the wavelet-Galerkin method [24,25], the variational method [26], the wavelet method [27,28,29,30], the method of fundamental solutions [31,32], etc. It is worth noting that Liu and Wei [33] applied a quasi-reversibility method to solve an IHCP without initial data. The quasi-reversibility method was first proposed by Latt$\stackrel{`}{e}$s and Lions in [34]. Nguyen et al. [35] used a modified quasi-reversibility method to compute the exponentially unstable solution of a nonlocal terminal-boundary value parabolic problem. However, to our knowledge, the results of the IHCP in the literature mainly focus on the heat equation in the Cartesian coordinate system. Few works are developed for the spherically symmetric (or axisymmetric) heat equations in the spherical coordinate systems (or cylindrical coordinate systems). In [36], Johansson et al. studied an IHCP in a cylindrical coordinate system by the fundamental solution method. Cheng and Fu [37] used two regularization methods for solving an axisymmetric IHCP in the cylindrical coordinate system. They gave two quite sharp error estimates by using an a priori regularization parameter choice rule. In this paper, we deal with a spherically symmetric IHCP.

The physical model we considered in this paper is a hollow sphere. $r=a$ and $r=b$ denote its inner and outer radius, respectively. Assume that the outer surface of the model is adiabatic, and we install a thermocouple inside the model at the radius $r=r_1(a<r_1<b)$. Let the temperature of the hollow sphere be independent of zenith angle $\phi$ and azimuth angle $\theta$. For simplicity, we suppose that the initial temperature of the hollow sphere is zero. Then we have the correspondingly mathematical model:

$$\begin{array}{cc}\hfill & u_t=u_{rr}+\frac{2}{r}{u}_r,a<r<b,t>0,\hfill \\ \hfill & u_r(b,t)=0,t\ge 0,\hfill \\ \hfill & u(r_1,t)=h\left(t\right),t\ge 0,\hfill \\ \hfill & u(r,0)=0,a\le r\le b,\hfill \end{array}$$

(1)

where r denotes the radial distance. We want to recover the temperature distribution $u(r,·)$ ($a<r<r_1$) from temperature measurement $h^{\delta }(·)$. This problem is ill-posed (the details can be seen in Section 2). In this paper, we will apply a modified quasi-boundary value method to solve the spherically symmetric IHCP (1).

The quasi-boundary value method is a regularization technique replacing the boundary condition or final condition by a new approximate condition. This method has been applied to solve many inverse problems, such as the inverse source problem [38,39], the inverse heat conduction problem [40], the Cauchy problem for Laplace equations [41,42,43], and the final value problem [44,45]. In this paper, we use it to deal with the spherically symmetric IHCP (1) for $r\in (a,r_1)$ and provide convergence analyses with an a priori and an a posteriori regularization parameter choice rule, respectively. It is worth noting that most of the works on the IHCP are absorbed in providing a convergence analysis under the a priori regularization parameter choice rule.

The organizational structure of this paper is as follows: In Section 2, we give the ill-posedness and a conditional stability of the ill-posed problem (1) for $r\in (a,r_1)$. In Section 3, we propose a modified quasi-boundary value method. Two convergence estimates are obtained under an a priori and an a posteriori regularization parameter choice rule, respectively. The paper ends with a brief conclusion in Section 4.

2. Ill-Posedness and Conditional Stability

First, we analyze the ill-posedness for the problem we considered.

As we consider problem (1) in $L^2\left(\mathbb{R}\right)$ regarding variable t, all functions of variable t are extended to be zero for $t<0$. Assuming these functions are in $L^2\left(\mathbb{R}\right)$, and

$$\widehat{g}\left(\eta \right)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }{e}^{-i\eta t}g\left(t\right)dt,$$

where $\widehat{g}$ denotes the Fourier transform of function g. The notation $\parallel ·\parallel$ represents the norm in $L^2\left(\mathbb{R}\right)$.

Due to the noise in the measurement data of $h\left(t\right)$, we denote the noise measurement of h(t) as $h^{\delta }\left(t\right)$, which satisfies:

$$\parallel h-h^{\delta }\parallel \le \delta ,$$

(2)

here, $\delta >0$ denotes the noise level. It is also assumed that there exists an a priori condition:

$$\parallel u(a,·)\parallel \le E,$$

(3)

here, E is a constant. The problem (1) is changed to the following problem by taking the Fourier transform:

$$\left\{\begin{array}{cc}i\eta \widehat{u}(r,\eta )=\frac{{\partial }^2\widehat{u}(r,\eta )}{\partial r^2}+\frac{2}{r}\frac{\partial \widehat{u}(r,\eta )}{\partial r},\hfill & r\in (a,b),\eta \in \mathbb{R},\hfill \\ {\widehat{u}}_r(b,\eta )=0,\hfill & \eta \in \mathbb{R},\hfill \\ \widehat{u}(r_1,\eta )=\widehat{h}\left(\eta \right),\hfill & \eta \in \mathbb{R}.\hfill \end{array}\right.$$

(4)

According to Lemmas 2.1 and 2.3 in [46], we obtain a formal solution to the problem (4):

$$\widehat{u}(r,\eta )=(r_1/r)\psi (r,\eta )e^{(r_1-r)\sqrt{i\eta }}\widehat{h}\left(\eta \right),r\in [a,b),\eta \in \mathbb{R},$$

(5)

where,

$$\psi (r,\eta )=\frac{(\sqrt{i\eta }b+1)e^{2r\sqrt{i\eta }}+(\sqrt{i\eta }b-1)e^{2b\sqrt{i\eta }}}{(\sqrt{i\eta }b+1)e^{2r_1\sqrt{i\eta }}+(\sqrt{i\eta }b-1)e^{2b\sqrt{i\eta }}},$$

and,

$$c_1\le \left|\psi (r,\eta )\right|\le c_2,r\in [a,r_1),\eta \in \mathbb{R},$$

(6)

where $c_1$ and $c_2$ are positive constants. Applying an inverse Fourier transform to (5), problem (1) has a unique solution:

$$u(r,t)=\frac{r_1}{\sqrt{2\pi }r}{\int }_{-\infty }^{\infty }\psi (r,\eta )e^{i\eta t}{e}^{(r_1-r)\sqrt{i\eta }}\widehat{h}\left(\eta \right)d\eta ,a<r<b.$$

(7)

Using the Parseval formula, there holds:

$${\parallel u(r,·)\parallel }^2={\int }_{-\infty }^{\infty }{\left|(r_1/r)\psi (r,\eta )e^{(r_1-r)\sqrt{i\eta }}\right|}^2{\left|\widehat{h\left(\eta \right)}\right|}^{2}d\eta .$$

We can see that $\left|(r_1/r)\psi (r,\eta )e^{(r_1-r)\sqrt{i\eta }}\right|$ increases rapidly as $\left|\eta \right|\to \infty$ if $a\le r<r_1$. This requires a sharp drop in $\widehat{h}\left(\eta \right)$ at high frequencies for $a\le r<r_1$. However, $h^{\delta }\left(t\right)$ is unlikely to have such a decay. Thus, for $r\in [a,r_1)$, the problem we considered is ill-posed.

Next, we provide conditional stability.

Theorem 1.

Suppose that $u(r,t)$ is the exact solution of Problem (1) in interval $r\in (a,r_1)$ given by (7), and the a priori bound (3) is satisfied. Then, there holds:

$$\parallel u(r,·)\parallel \le C{E}^{\frac{r_1-r}{r_1-a}}{\parallel h\parallel }^{\frac{r-a}{r_1-a}},\forall r\in (a,r_1),$$

(8)

where $C=c_2{\left(c_1\right)}^{\frac{r-r_1}{r_1-a}}{(r_1/a)}^{\frac{r-a}{r_1-a}}$, $c_1$ and $c_2$ are constants given by (6).

Proof. 

From expression (5), the Parseval formula and the H$\ddot{o}$lder inequality, there holds:

$$\begin{array}{cc}\hfill & {\parallel u(r,·)\parallel }^2={\parallel \widehat{u}(r,·)\parallel }^2={\int }_{-\infty }^{\infty }{\left|(r_1/r)\psi (r,\eta )e^{(r_1-r)\sqrt{i\eta }}\widehat{h}\left(\eta \right)\right|}^{2}d\eta \hfill \\ \hfill & ={\int }_{-\infty }^{\infty }{\left[{\left|(r_1/r)\psi (r,\eta )\right|}^{\frac{2(r_1-a)}{r_1-r}}{\left|e^{(r_1-a)\sqrt{i\eta }}\widehat{h}\left(\eta \right)\right|}^2\right]}^{\frac{r_1-r}{r_1-a}}{\left[{\left|\widehat{h}\left(\eta \right)\right|}^2\right]}^{\frac{r-a}{r_1-a}}d\eta \hfill \\ \hfill & \le {\left({\int }_{-\infty }^{\infty }{\left|(r_1/r)\psi (r,\eta )\right|}^{\frac{2(r_1-a)}{r_1-r}}{\left|e^{(r_1-a)\sqrt{i\eta }}\widehat{h}\left(\eta \right)\right|}^{2}d\eta \right)}^{\frac{r_1-r}{r_1-a}}{\left({\int }_{-\infty }^{\infty }{\left|\widehat{h}\left(\eta \right)\right|}^{2}d\eta \right)}^{\frac{r-a}{r_1-a}}\hfill \\ \hfill & ={\left({\int }_{-\infty }^{\infty }{\left|(r_1/r)\psi (r,\eta )\right|}^{\frac{2(r_1-a)}{r_1-r}}{\left|(r_1/a)\psi (a,\eta )\right|}^{-2}{\left|\widehat{u}(a,\eta )\right|}^{2}d\eta \right)}^{\frac{r_1-r}{r_1-a}}{\parallel h\parallel }^{\frac{2(r-a)}{r_1-a}}\hfill \\ \hfill & \le \underset{\eta \in \mathbb{R}}{sup}\left[{\left|(r_1/r)\psi (r,\eta )\right|}^2{\left|(r_1/a)\psi (a,\eta )\right|}^{\frac{2(r-r_1)}{r_1-a}}\right]\parallel \widehat{u}{(a,·)\parallel }^{\frac{2(r_1-r)}{r_1-a}}{\parallel h\parallel }^{\frac{2(r-a)}{r_1-a}}.\hfill \end{array}$$

Using inequalities (6), we have:

$${\left|(r_1/r)\psi (r,\eta )\right|}^2{\left|(r_1/a)\psi (a,\eta )\right|}^{\frac{2(r-r_1)}{r_1-a}}\le {\left|(r_1/r)c_2\right|}^2{\left|(r_1/a)c_1\right|}^{\frac{2(r-r_1)}{r_1-a}}.$$

Combining with (3), we obtain:

$$\parallel u(r,·)\parallel \le c_2{\left(c_1\right)}^{\frac{(r-r_1)}{r_1-a}}{(r_1/a)}^{\frac{(r-a)}{r_1-a}}{E}^{\frac{(r_1-r)}{r_1-a}}{\parallel h\parallel }^{\frac{(r-a)}{r_1-a}}.$$

This ends the proof. □

According to the above theorem, we can know that if $u_1$ and $u_2$ are the solutions of problem (1) corresponding to the exact data $h_1$ and $h_2$, respectively, then there holds:

$$\parallel u_1(r,·)-u_2(r,·)\parallel \le C{E}^{\frac{r_1-r}{r_1-a}}{\parallel h_1-h_2\parallel }^{\frac{r-a}{r_1-a}},\forall r\in (a,r_1).$$

(9)

It is obvious that if $\parallel h_1-h_2\parallel \to 0$, then $\parallel u_1(r,·)-u_2(r,·)\parallel \to 0$ for $a<r<r_1$. However, such conditionally stable results cannot guarantee the stability of numerical calculations with noisy data. Hence, an effective regularization method is needed for dealing with the spherically symmetric ill-posed problem (1) for $a<r<r_1$.

3. Regularization Method and Error Estimates

In this part, we first use a modified quasi-boundary value method to derive the approximate solution of the ill-posed problem (1) in interval $r\in (a,r_1)$. Then, two error estimates are provided by choosing an a priori regularization parameter and Morozov${}^{’}$s discrepancy principle, respectively.

The quasi-boundary value method is a regularization method, which is implemented by substituting the boundary condition or final condition with a new approximation condition. Hence, the following boundary condition is considered by adding a perturbation term $\beta u(a,t)$:

$$u(r_1,t)+\beta u(a,t)=h^{\delta }\left(t\right),$$

(10)

where $\beta$ acts as a role of the regularization parameter and depends on the measurement noise $\delta$.

Thus, we consider the following regularized problem:

$$\left\{\begin{array}{cc}\frac{\partial u_{\beta }^{\delta }}{\partial t}=\frac{{\partial }^2{u}_{\beta }^{\delta }}{\partial r^2}+\frac{2}{r}\frac{\partial u_{\beta }^{\delta }}{\partial r},\hfill & a<r<b,t>0,\hfill \\ \frac{\partial u_{\beta }^{\delta }}{\partial r}(b,t)=0,\hfill & t\ge 0,\hfill \\ u_{\beta }^{\delta }(r_1,t)+\beta u_{\beta }^{\delta }(a,t)=h^{\delta }\left(t\right),\hfill & t\ge 0,\hfill \\ u_{\beta }^{\delta }(r,0)=0,\hfill & a\le r\le b.\hfill \end{array}\right.$$

(11)

Similar to the derivation process of Formula (5), we can obtain the formal solution of Problem (11) in the frequency domain as:

$${\widehat{u}}_{\beta }^{\delta }(r,\eta )=\frac{(r_1/r)\psi (r,\eta )e^{(r_1-r)\sqrt{i\eta }}}{1+\beta (r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}}{\widehat{h}}^{\delta }\left(\eta \right),$$

(12)

or,

$$u_{\beta }^{\delta }(r,t)=\frac{r_1}{r\sqrt{2\pi }}{\int }_{-\infty }^{\infty }{e}^{i\eta t}\frac{\psi (r,\eta )e^{(r_1-r)\sqrt{i\eta }}}{1+\beta (r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}}{\widehat{h}}^{\delta }\left(\eta \right)d\eta .$$

(13)

Defining:

$$q(\beta ,\eta )=\frac{1}{1+\beta (r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}},$$

here, function $q(\beta ,\eta )$ is a regularizing filter.

Comparing Formula (7) for the exact solution with Formula (13) for its quasi-boundary value approximation, we can see that the regularization procedure consists of replacing the $\widehat{h}\left(\eta \right)$ with an appropriately filtered Fourier transform of noisy data $h^{\delta }\left(t\right)$. The filter $q(\beta ,\eta )$ filters out the high frequencies part of ${\widehat{h}}^{\delta }\left(\eta \right)$. According to this idea, we replace the original filter $q(\beta ,\eta )$ with another filter:

$$\tilde{q}(\beta ,\eta )=\frac{1}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}.$$

Therefore, the modified regularized solution of Problem (1) in the interval $a<r<r_1$ becomes:

$$u_{\beta ,*}^{\delta }(r,t)=\frac{r_1}{r\sqrt{2\pi }}{\int }_{-\infty }^{\infty }{e}^{i\eta t}\frac{\psi (r,\eta )e^{(r_1-r)\sqrt{i\eta }}}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}{\widehat{h}}^{\delta }\left(\eta \right)d\eta .$$

(14)

Before giving the stability analysis of the regularized solution, we need the following inequality.

Lemma 1.

Let $a<r<r_1,\beta >0$, then there holds:

$$\underset{x\ge 0}{sup}\frac{e^{x(r_1-r)}}{1+\beta e^{x(r_1-a)}}\le {\beta }^{\frac{r-r_1}{r_1-a}}.$$

(15)

The proof of Inequality (15) is similar to that of Lemma 4 in [16].

Next, we provide two error estimates.

3.1. Stability Estimate under an A Priori Regularization Parameter Choice Rule

In this part, we provide the stability estimate for the regularized solution $u_{\beta ,*}^{\delta }(r,t)$.

Theorem 2.

Suppose $u(r,t)$ is the exact solution of Problem (1) for $r\in (a,r_1)$, and $u_{\beta ,*}^{\delta }(r,t)$ given by (14) is the regularized solution of $u(r,t)$. Let noise assumption (2) and a priori condition (3) be valid. If taking the regularization parameter $\beta =\frac{\delta }{E}$, we have the following error estimate:

$$\parallel u(r,·)-u_{\beta ,*}^{\delta }(r,·)\parallel \le C_1{E}^{1-\frac{r-a}{r_1-a}}{\delta }^{\frac{r-a}{r_1-a}},$$

(16)

where $C_1=\frac{2c_2{r}_1}{min\{a,c_1{r}_1\}}$.

Proof. 

Using the Parseval Formula with (7) and (14), we get:

$$\begin{array}{cc}\hfill & \parallel u(r,·)-u_{\beta ,*}^{\delta }(r,·)\parallel =\parallel \widehat{u}(r,·)-{\widehat{u}}_{\beta ,*}^{\delta }(r,·)\parallel \hfill \\ \hfill =& ∥(r_1/r)\psi (r,\eta )e^{(r_1-r)\sqrt{i\eta }}\widehat{h}-\frac{(r_1/r)\psi (r,\eta )e^{(r_1-r)\sqrt{i\eta }}{\widehat{h}}^{\delta }}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a),\sqrt{i\eta }}\right|}∥,\hfill \end{array}$$

combining this with the triangle inequality, there holds:

$$\begin{array}{cc}\hfill & \parallel u(r,·)-u_{\beta ,*}^{\delta }(r,·)\parallel \le ∥\frac{(r_1/r)\psi (r,\eta )e^{(r_1-r)\sqrt{i\eta }}\left(\widehat{h}-{\widehat{h}}^{\delta }\right)}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}∥\hfill \\ \hfill +& ∥\frac{\beta (r_1/r)\psi (r,\eta )e^{(r_1-r)\sqrt{i\eta }}\left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|\widehat{h}}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}∥\hfill \\ \hfill \le & \underset{\eta \in \mathbb{R}}{sup}\left|\frac{(r_1/r)\psi (r,\eta )e^{(r_1-r)\sqrt{i\eta }}}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}\right|∥\widehat{h}-{\widehat{h}}^{\delta }∥\hfill \\ \hfill +& \underset{\eta \in \mathbb{R}}{sup}\left|\frac{\beta (r_1/r)\psi (r,\eta )e^{(r_1-r)\sqrt{i\eta }}}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}\right|∥\left|(r_1/a)e^{(r_1-a)\sqrt{i\eta }}\psi (a,\eta )\right|\widehat{h}∥\hfill \\ \hfill =& \underset{\eta \in \mathbb{R}}{sup}\frac{(r_1/r)\left|\psi (r,\eta )\right|e^{(r_1-r)\sqrt{\left|\eta \right|/2}}}{1+\beta (r_1/r_0)\left|\psi (a,\eta )\right|e^{(r_1-a)\sqrt{\left|\eta \right|/2}}}\parallel h-h^{\delta }\parallel \hfill \\ \hfill & +\beta \underset{\eta \in \mathbb{R}}{sup}\frac{(r_1/r)\left|\psi (r,\eta )\right|e^{(r_1-r)\sqrt{\left|\eta \right|/2}}}{1+\beta (r_1/a)\left|\psi (a,\eta )\right|e^{(r_1-a)\sqrt{\left|\eta \right|/2}}}\parallel \widehat{u}(a,·)\parallel ,\hfill \end{array}$$

using (6), we have:

$$\begin{array}{cc}\hfill & \parallel u(r,·)-u_{\beta ,*}^{\delta }(r,·)\parallel \hfill \\ \hfill \le & \frac{c_2(r_1/r)}{min\{1,(c_1{r}_1/a)\}}\underset{\eta \in \mathbb{R}}{sup}\frac{e^{(r_1-r)\sqrt{\left|\eta \right|/2}}}{1+\beta e^{(r_1-a)\sqrt{\left|\eta \right|/2}}}\parallel h-h^{\delta }\parallel \hfill \\ \hfill & +\frac{\beta (c_2{r}_1/r)}{min\{1,(c_1{r}_1/a)\}}\underset{\eta \in \mathbb{R}}{sup}\frac{e^{(r_1-r)\sqrt{\left|\eta \right|/2}}}{1+\beta e^{(r_1-a)\sqrt{\left|\eta \right|/2}}}\parallel u(a,·)\parallel .\hfill \end{array}$$

Let $\sqrt{\left|\eta \right|/2}=x$, with (2) and (3), we have:

$$\begin{array}{cc}\hfill & \parallel \parallel u(r,·)-u_{\beta ,*}^{\delta }(r,·)\parallel \hfill \\ \hfill & \le \frac{c_2{r}_1}{min\{a,c_1{r}_1\}}\left[\underset{x\ge 0}{sup}\frac{e^{x(r_1-r)}}{1+\beta e^{x(r_1-a)}}\delta +\underset{x\ge 0}{sup}\frac{\beta e^{x(r_1-r)}}{1+\beta e^{x(r_1-a)}}E\right],\hfill \end{array}$$

using inequality (15) and the choice of $\beta =\delta /E$, there holds:

$$\parallel \parallel u(r,·)-u_{\beta ,*}^{\delta }(r,·)\parallel \le \frac{2c_2{r}_1}{min\{a,c_1{r}_1\}}{E}^{1-\frac{r-a}{r_1-a}}{\delta }^{\frac{r-a}{r_1-a}}.$$

This proves the estimate (16). □

According to the general theory of regularization, error estimate (16) is order optimal.

3.2. Stability Estimate Based on an A Posteriori Parameter Selection Rule

In this part, Morozov${}^{’}$s discrepancy principle is used as an a posteriori parameter selection rule. Morozov${}^{’}$s discrepancy principle is first seen in [47]. Then, the discrepancy principle has been used for the Tikhonov regularization for the ill-posed problem and chosen as the regularization parameter $\alpha >0$ such that [48]:

$$\parallel F\left(x_{\alpha }^{\delta }\right)-y^{\delta }\parallel =c\delta ,$$

(17)

here, $c>1$ is a constant. Feng et al. [49] extended Morozov’s discrepancy principle and chose the regularization parameter $\alpha >0$ such that:

$$\parallel u_{{\alpha }_i,MTik}^{\delta }(0,·)-{\phi }_i^{\delta }(·)\parallel =\tau \delta ,i=1,2,$$

(18)

where $\tau >1$ is a constant, $u_{{\alpha }_i,MTik}^{\delta }(x,·)$ and ${\phi }_i^{\delta }(·)$ denote the regularized solution and measured data, respectively. Similar to the above a posteriori parameter selection strategy, Liu and Feng [50] and Cheng et al. [51] solved a time-fractional inverse diffusion problem and a Riesz–Feller space-fractional backward diffusion problem, respectively. In this paper, we choose the regularization parameter $\beta >0$ such that:

$$\parallel u_{\beta ,*}^{\delta }(r_1,·)-h^{\delta }\parallel =\tau \delta ,$$

(19)

here, $\tau >1$ is a constant. If the following results are true, the above equation has a unique solution.

Lemma 2.

Suppose $d\left(\beta \right)=\parallel u_{\beta ,*}^{\delta }(r_1,·)-h^{\delta }\parallel$. If $0<\delta <\parallel h^{\delta }\parallel$, then there holds:

(1)

The function $d\left(\beta \right)$ is continuous;

(2)

$d\left(\beta \right)\to \parallel h^{\delta }\parallel$ for $\beta \to +\infty$;

(3)

$d\left(\beta \right)\to 0$ for $\beta \to 0$;

(4)

The function $d\left(\beta \right)$ is a strictly increasing function in the interval $(0,\infty )$.

The proof of Lemma 2 is similar to that of Theorem 2.16 in reference [6] and hence is omitted.

Theorem 3.

Suppose $u(r,t)$ is the exact solution of Problem (1) for $r\in (a,r_1)$, and $u_{\beta ,*}^{\delta }(r,t)$ given by (14) is the regularized solution of $u(r,t)$. Let conditions (2) and (3) be valid. If the regularization parameter $\beta >0$ is chosen by Morozov${}^{’}$s discrepancy principle (19), then we have the following error estimate:

$$\parallel u(r,·)-u_{\beta ,*}^{\delta }(r,·)\parallel \le \left[C_2{(\tau +1)}^{\frac{r-a}{r_1-a}}+(C_1/2){(\tau -1)}^{\frac{r-r_1}{r_1-a}}\right]E^{1-\frac{r-a}{r_1-a}}{\delta }^{\frac{r-a}{r_1-a}},$$

(20)

where $C_2={(r_1/a)}^{\frac{r-a}{r_1-a}}{c}_2{c}_1^{\frac{r-r_1}{r_1-a}}.$

Proof. 

Applying the triangle inequality, we get:

$$\parallel u(r,·)-u_{\beta ,*}^{\delta }(r,·)\parallel \le \parallel u(r,·)-u_{\beta ,*}(r,·)\parallel +\parallel u_{\beta ,*}(r,·)-u_{\beta ,*}^{\delta }(r,·)\parallel :=I_1+I_2.$$

(21)

We give the estimates for $I_1$ and $I_2$, respectively.

From (7), (14) and the H$\ddot{o}$lder inequality, there holds:

$$\begin{array}{cc}\hfill I_1^2& =\parallel \widehat{u}(r,·)-{\widehat{u}}_{\beta ,*}{(r,·)\parallel }^2\hfill \\ \hfill =& ∥(r_1/r)\psi (r,\eta )e^{(r_1-r)\sqrt{i\eta }}\widehat{h}-\frac{(r_1/r)\psi (r,\eta )e^{(r_1-r)\sqrt{i\eta }}\widehat{h}}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}∥\hfill \\ \hfill =& ∥\frac{\beta (r_1/r)\psi (r,\eta )e^{(r_1-r)\sqrt{i\eta }}\left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|\widehat{h}}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}∥\hfill \\ \hfill =& {\int }_{-\infty }^{\infty }{\left|\frac{\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}\right|}^2{\left|(r_1/r)\psi (r,\eta )e^{(r_1-r)\sqrt{i\eta }}\right|}^2{\left|\widehat{h}\right|}^{2}d\eta \hfill \\ \hfill \le & {\int }_{-\infty }^{\infty }{\left|\frac{\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|\widehat{h}}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}\right|}^{\frac{2(r-a)}{r_1-a}}\left({\left|\frac{r_1}{r}\psi (r,\eta )e^{(r_1-r)\sqrt{i\eta }}\right|}^2{\left|\widehat{h}\right|}^{\frac{2(r_1-r)}{r_1-a}}\right)d\eta \hfill \\ \hfill \le & {\left[{\int }_{-\infty }^{\infty }{\left|\frac{\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|\widehat{h}}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}\right|}^{2}d\eta \right]}^{\frac{r-a}{r_1-a}}\hfill \\ \hfill & ·{\left[{\int }_{-\infty }^{\infty }{\left({\left|\frac{r_1}{r}\psi (r,\eta )e^{(r_1-r)\sqrt{i\eta }}\right|}^2{\left|\widehat{h}\right|}^{\frac{2(r_1-r)}{r_1-a}}\right)}^{\frac{r_1-a}{r_1-r}}d\xi \right]}^{\frac{r_1-r}{r_1-a}}\hfill \\ \hfill \le & {∥\frac{\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|\widehat{h}}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}∥}^{\frac{2(r-a)}{r_1-a}}\hfill \\ \hfill & ·{\left[{\int }_{-\infty }^{\infty }\left({\left|\frac{r_1}{r}\psi (r,\eta )\right|}^{\frac{2(r_1-a)}{r_1-r}}{\left|\frac{r_1}{a}\psi (a,\eta )\right|}^{-2}\right){\left|\frac{r_1}{a}\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\widehat{h}\right|}^{2}d\eta \right]}^{\frac{r_1-r}{r_1-a}}\hfill \\ \hfill \le & {∥\frac{\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|\widehat{h}}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}∥}^{\frac{2(r-a)}{r_1-a}}\hfill \\ \hfill & ·\underset{\eta \in \mathbb{R}}{sup}\left({\left|\frac{r_1}{r}\psi (r,\eta )\right|}^2{\left|\frac{r_1}{a}\psi (a,\eta )\right|}^{\frac{2(r-r_1)}{r_1-a}}\right){\parallel \widehat{u}(a,·)\parallel }^{\frac{2(r_1-r)}{r_1-a}}\hfill \\ \hfill \le & {∥\frac{\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|\widehat{h}}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}∥}^{\frac{2(r-a)}{r_1-a}}{\left(\frac{r_1}{a}\right)}^{\frac{2(r-a)}{r_1-a}}{c}_2^2{c}_1^{\frac{2(r-r_1)}{r_1-a}}{\parallel u(a,·)\parallel }^{\frac{2(r_1-r)}{r_1-a}},\hfill \end{array}$$

combining this with the a priori condition (3), the following estimate holds:

$$I_1\le {\left(\frac{r_1}{a}\right)}^{\frac{r-a}{r_1-a}}{c}_2{c}_1^{\frac{r-r_1}{r_1-a}}{E}^{\frac{r_1-r}{r_1-a}}{∥\frac{\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|\widehat{h}}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}∥}^{\frac{r-a}{r_1-a}}.$$

(22)

Using the Parseval formula with (19) and (14), we have:

$$\tau \delta =\parallel {\widehat{u}}_{\beta ,*}^{\delta }(r_1,·)-{\widehat{h}}^{\delta }\parallel =∥\frac{\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}{\widehat{h}}^{\delta }∥,$$

then, we get:

$$\begin{array}{cc}\hfill & ∥\frac{\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|\widehat{h}}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}∥\le ∥\frac{\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}(\widehat{h}-{\widehat{h}}^{\delta })∥\hfill \\ \hfill & +∥\frac{\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}{\widehat{h}}^{\delta }∥\le \parallel h-h^{\delta }\parallel +\tau \delta =(\tau +1)\delta .\hfill \end{array}$$

(23)

Substituting (23) into (22), we obtain:

$$I_1\le {\left(\frac{r_1}{a}\right)}^{\frac{r-a}{r_1-a}}{c}_2{c}_1^{\frac{r-r_1}{r_1-a}}{E}^{1-\frac{r-a}{r_1-a}}{\left[(\tau +1)\delta \right]}^{\frac{r-a}{r_1-a}}.$$

(24)

Next, we give the bound for the $I_2$. With inequality (15) and the noise assumption (2), we have:

$$\begin{array}{cc}\hfill I_2& =\parallel {\widehat{u}}_{\beta ,*}(r,·)-{\widehat{u}}_{\beta ,*}^{\delta }(r,·)\parallel =∥\frac{(r_1/r)\psi (r,\eta )e^{(r_1-r)\sqrt{i\eta }}}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}(\widehat{h}-{\widehat{h}}^{\delta })∥\hfill \\ \hfill & \le \underset{\eta \in \mathbb{R}}{sup}\left|\frac{(r_1/r)\psi (r,\eta )e^{(r_1-r)\sqrt{i\eta }}}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}\right|∥\widehat{h}-{\widehat{h}}^{\delta }∥\hfill \\ \hfill & \le \frac{c_2(r_1/r)}{min\{1,(c_1{r}_1/a)\}}\underset{\eta \in \mathbb{R}}{sup}\frac{e^{(r_1-r)\sqrt{\left|\eta \right|/2}}}{1+\beta e^{(r_1-a)\sqrt{\left|\eta \right|/2}}}\parallel h-h^{\delta }\parallel \hfill \\ \hfill & \le \frac{c_2{r}_1}{min\{a,c_1{r}_1\}}\underset{x\ge 0}{sup}\frac{e^{x(r_1-r)}}{1+\beta e^{x(r_1-a)}}\delta \le \frac{c_2{r}_1}{min\{a,c_1{r}_1\}}{\beta }^{\frac{r-r_1}{r_1-a}}\delta .\hfill \end{array}$$

(25)

On the other hand,

$$\begin{array}{cc}\hfill \tau \delta & =∥\frac{\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}{\widehat{h}}^{\delta }∥\hfill \\ \hfill & \le ∥\frac{\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}({\widehat{h}}^{\delta }-\widehat{h})∥+∥\frac{\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}{1+\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|}\widehat{h}∥\hfill \\ \hfill & \le \parallel h-h^{\delta }\parallel +∥\beta \left|(r_1/a)\psi (a,\eta )e^{(r_1-a)\sqrt{i\eta }}\right|\widehat{h}∥\hfill \\ \hfill & \le \delta +\beta ∥u(a,·)∥\le \delta +\beta E,\hfill \end{array}$$

then this yields,

$$\beta \ge \frac{\delta }{E}(\tau -1).$$

(26)

Substituting (26) into (25), there holds:

$$I_2\le \frac{c_2{r}_1}{min\{a,c_1{r}_1\}}{\left[\frac{\delta }{E}(\tau -1)\right]}^{\frac{r-r_1}{r_1-a}}\delta =\frac{c_2{r}_1{(\tau -1)}^{\frac{r-r_1}{r_1-a}}}{min\{a,c_1{r}_1\}}{E}^{1-\frac{r-a}{r_1-a}}{\delta }^{\frac{r-a}{r_1-a}}.$$

(27)

Combining (24) and (27) with (21), we obtain the error estimate (20). □

Remark 1.

From Theorems 2 and 3, we can see that the a priori and a posteriori regularization parameter selection rules have the same convergence rate. However, in practice, this kind of a priori information is rarely obtained. Therefore, we prefer to use the a posteriori choice rule to get the regularization parameter.

4. Conclusions

In this paper, we consider the spherically symmetric IHCP that determines the internal surface temperature distribution of the hollow sphere from measured data at the fixed location inside it. The conditional stability is given. We propose a modified quasi-boundary value method to formulate a regularized solution. Two H$\ddot{o}$lder-type error estimates between the approximate solution and its exact solution are obtained under Morozov${}^{’}$s discrepancy principle and an a priori regularization parameter choice rule, respectively.