A posteriori truncated regularization method for identifying unknown heat source on a spherical symmetric domain
- Fan Yang^{1}Email author,
- Miao Zhang^{1},
- Xiao-Xiao Li^{1} and
- Yu-Peng Ren^{1}
https://doi.org/10.1186/s13662-017-1276-1
© The Author(s) 2017
Received: 5 January 2017
Accepted: 12 July 2017
Published: 1 September 2017
Abstract
In this paper, we mainly consider the inverse problem for identifying the unknown heat source in spherical symmetric domain. We propose a truncation regularization method combined with an a posteriori regularization parameter choice rule to deal with this problem. The Hölder type convergence estimate is obtained. Numerical results are presented to illustrate the accuracy and efficiency of this method.
Keywords
1 Introduction
Identifying the unknown heat source in a parabolic partial differential equation from the over-specified data plays an important role in applied mathematics, physics and engineering. These problems are widely encountered in the modeling of physical phenomena. A typical example is groundwater pollutant source estimation in cities with large population [1]. Now many scholars have used different methods to identify various types of heat sources. In [2, 3], the authors used the method of fundamental solutions and radial basis functions to identify the unknown heat source. In [4, 5], the authors used the Fourier truncation method and the wavelet dual least squares method to identify the spatial variable heat source. In [6], the authors used the simplified Tikhonov method to identify the spatial variable heat source. In [7, 8], the authors determined the heat source which depends on one variable in a bounded domain using the boundary-element method and an iterative algorithm. In [9], the authors identified the heat source which depends only on time variable using the Lie-group shooting method (LGSM). In [10], the authors used the truncation method based on Hermite expansion to identify the unknown source in a space fractional diffusion equation. In [11], the authors identified the point source with some point measurement data. In [12], the authors proved the existence and uniqueness for identifying the heat source which depends only on time variable. In [13], the authors used the variational method to identify the heat source which has the form \(F(x,t)\). In [14], the authors used the variational method to identify the heat source which has the form of \(F(x,t)=F(x)H(t)\) for the variable coefficient heat conduction equation. As far as we know, most of the researches on heat source identification problem mainly concentrated on one-dimensional case. But for a high dimensional case, there are few research results. In [15], the authors used the spectral method to identify the heat source in a columnar symmetric domain. In [16], the authors used the spectral method to identify the heat source in a spherically symmetric parabolic equation. But the regularization parameters is selected by the a priori rule. There is a defect for any a priori method, i.e., the a priori choice of the regularization parameter depends seriously on the a priori bound E of the unknown solution. However, the a priori bound E cannot be known exactly in practice, and working with a wrong constant E may lead to a badly regularized solution. In this paper, we not only give the a posteriori choice of the regularization parameter which depends only on the measurable data, but also we give some different examples to compare the effectiveness between the posterior choice rule and the priori choice rule. Moreover, we find the truncation regularization method is better than the other regularization methods, such as Tikhonov regularization and the quasi-boundary value regularization method for solving this problem. To the best of the authors’ knowledge, there are few papers to choose the regularization parameter under the a posteriori rule for this problem.
This paper is organized as follows. In Section 2, under the a posteriori parameter choice rule, we give the convergence error estimate. In Section 3, three numerical examples are used to verify the effectiveness for the proposed method. In Section 4, the conclusion of this paper is given.
2 Main result
Lemma 1
- (a)
\(\rho_{N}\) is a continuous function;
- (b)
\(\lim_{N\rightarrow0^{+}}\rho_{N}=\Vert \varphi^{\delta } \Vert \);
- (c)
\(\lim_{N\rightarrow+\infty}\rho_{N}=0\);
- (d)
\(\rho_{N}\) is a strictly decreasing function over \((0,\infty)\).
Lemma 2
Lemma 3
Proof
Lemma 4
Proof
Lemma 5
Proof
Now we give the convergent error estimate between the exact solution and the regularized solution.
Theorem 6
Proof
3 Numerical experiments
Example 1
Consider a smooth heat source: \(f(r)=r\sin r\).
Example 2
Example 3
Numerical results for different ε under an a posteriori choice rule for three regularization methods about Examples 1
ε | 0.05 | 0.01 | 0.005 | 0.001 | 0.0005 | 0.0001 | |
---|---|---|---|---|---|---|---|
Truncate | \(e_{a}\) | 0.0697 | 0.0376 | 0.0243 | 0.0118 | 0.0082 | 0.0035 |
\(e_{r}\) | 0.0593 | 0.0320 | 0.0207 | 0.0101 | 0.0070 | 0.0030 | |
Tikhonov | \(e_{a}\) | 0.1024 | 0.0407 | 0.0277 | 0.0152 | 0.0065 | 0.0034 |
\(e_{r}\) | 0.0871 | 0.0346 | 0.0236 | 0.0129 | 0.0056 | 0.0029 | |
Quasi-boundary | \(e_{a}\) | 0.6363 | 0.1518 | 0.0437 | 0.0160 | 0.0107 | 0.0043 |
\(e_{r}\) | 0.5414 | 0.1291 | 0.0372 | 0.0136 | 0.0091 | 0.0037 |
Numerical results for different ε under an a posteriori choice rule for three regularization methods about Examples 2
ε | 0.05 | 0.01 | 0.005 | 0.001 | 0.0005 | 0.0001 | |
---|---|---|---|---|---|---|---|
Truncate | \(e_{a}\) | 0.0972 | 0.0521 | 0.0304 | 0.0241 | 0.0226 | 0.0139 |
\(e_{r}\) | 0.2392 | 0.1283 | 0.0749 | 0.0594 | 0.0557 | 0.0341 | |
Tikhonov | \(e_{a}\) | 0.1295 | 0.0871 | 0.0361 | 0.0273 | 0.0243 | 0.0378 |
\(e_{r}\) | 0.3178 | 0.2143 | 0.0889 | 0.0672 | 0.0598 | 0.0930 | |
Quasi-boundary | \(e_{a}\) | 0.7800 | 0.2036 | 0.0566 | 0.0539 | 0.0482 | 0.0393 |
\(e_{r}\) | 1.9194 | 0.5011 | 0.1393 | 0.1327 | 0.1186 | 0.0967 |
Tables 1 and 2 gives the comparisons of the numerical results of the truncate regularization method, the Tikhonov regularization method and the quasi-boundary regularization method under the a posteriori choice rule for different ε. From Tables 1 and 2, we can see that the effectiveness of the truncate regularization method in the present paper is better than the other regularization methods.
4 Conclusion
Using the Morozov discrepancy principle, we obtain an a posteriori parameter choice rule which only depends on the measured data. Under the a posteriori choices of the regularization parameter, the Hölder type error estimate which is order optimal is obtained. Meanwhile, several numerical examples verify the efficiency and accuracy of this method.
Declarations
Acknowledgements
The authors would like to thanks the editor and the referees for their valuable comments and suggestions that improve the quality of our paper. The work is supported by the National Natural Science Foundation of China (11561045,11501272) and the Doctor Fund of Lan Zhou University of Technology.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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