A quasi-boundary value regularization method for identifying an unknown source in the Poisson equation
© Yang et al.; licensee Springer. 2014
Received: 15 October 2013
Accepted: 7 March 2014
Published: 20 March 2014
In this paper, we consider the problem for identifying the unknown source in the Poisson equation in a half unbounded domain. A conditional stability result is given and a quasi-boundary value regularization method is presented to deal with this problem. For the regularization solution, the Hölder type stability estimate between the regularization solution and the exact solution is given. Numerical results are presented to illustrate the accuracy and efficiency of this method.
MSC:35R25, 47A52, 35R30.
Keywordsill-posed problem unknown source conditional stability quasi-boundary value Poisson equation
Inverse source problems arise in many branches of science and engineering, e.g. heat conduction, crack identification, electromagnetic theory, geophysical prospecting, and pollutant detection. For the heat source identification, there have been a large number of research results for different forms of heat source [1–9]. To the author’s knowledge, there were few papers for identifying an unknown source in the Poisson equation using the regularization method. In , the authors identified the unknown point source with the logarithmic potential. In , the author identified the unknown point source using the projective method. In , the authors identified the unknown point source using the Green’s function. In [13, 14], the authors identified the unknown source dependent only on one variable using the dual reciprocity method. In , the authors identified the unknown source dependent only on one variable using the method of fundamental solution. But by the regularization method, there are a few papers with strict theoretical analysis on identifying the unknown source.
where the constant represents a noise level of input data.
The problem (1.1) is ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. One way to solve an ill-posed problem is by perturbing it into a well-posed one. Many perturbing techniques have been proposed, including a biharmonic regularization developed by Lattés and Lions in , a pseudo-parabolic regularization proposed by Showalter and Ting in , a stabilized quasi-reversibility proposed by Miller in , the method of quasi-reversibility proposed by Mel’nikova in , a hyperbolic regularization proposed by Ames and Cobb in , the Gajewski and Zacharias quasi-reversibility proposed by Huang and Zheng in , a quasi-boundary value method by Denche and Bessila in , and an optimal regularization proposed by Boussetila and Rebbani in . It appears that Showalter in  was the first who used the quasi-boundary value regularization method to consider the backward heat conduction problem. In , the authors used the quasi-boundary-value method to consider the Cauchy problem for elliptic equations with nonhomogeneous Neumann data. In this paper, we use the quasi-boundary value regularization method to identify the unknown source for the Poisson equation.
The outline of the paper is as follows. Section 2 gives an analysis on the ill-posedness of this inverse problem and some auxiliary results. Section 3 gives a regularization solution and error estimation. Section 4 gives some examples to illustrate the accuracy and efficiency of this method. Section 5 puts an end to this paper with a brief conclusion.
2 Ill-posedness of the problem and some auxiliary results
be the Fourier transform of the function .
Now we give some lemmas which are very useful for our main conclusion.
The proof of (2.8) is separated into three cases.
Combining (2.10) with (2.12), (2.13) and (2.14), the inequality (2.8) holds. □
3 The conditional stability result
Since the problem (1.1) is linear, stability estimates can be derived by estimating the size of solutions to the corresponding homogeneous problem. We establish the stability estimate for the problem (1.1).
From (3.3), it is obvious that when . However, this conditional stability result cannot ensure the stability of numerical computation with noisy data. We must use the regularization method to deal with this ill-posed problem.
4 The quasi-boundary value regularization method and the error estimate
Note that for small β, is close to . On the other hand, if becomes large, is bounded. So is considered as an approximation of .
Now we will give an error estimate between the regularization solution and the exact solution by the following theorem.
5 Several numerical examples
In this section, we present three numerical examples to illustrate the usefulness of the proposed method. The numerical examples were constructed in the following way: First we selected the source function , and we obtained the exact data function by solving the direct problem. Then we added a normally distributed perturbation to each data function and obtained vectors . Finally we obtained the regularization solution by solving the inverse problem.
In the following, we first give an example which has the exact expression of the solutions .
Moreover, we need to make the vector periodical  and then we take the discrete Fourier transform for the vector . The approximation of the regularization solution are computed by using the fast Fourier transform algorithm  and the range of the variable x in the numerical experiment is .
In this paper, we identify an unknown source term depending only on one variable in two-dimensional Poisson equation. This problem is ill-posed, i.e., the solution (if it exists) does not depend on the input data. We obtained the stability estimate using the conditional stability. Moreover, using the quasi-boundary value regularization method, we obtain the regularization solution and the Hölder type error estimate between the exact solution and the regularization solution. According to , this Hölder type error estimate is order optimal.
The project is supported by the National Natural Science Foundation of China (No. 11171136, No. 11261032), the Distinguished Young Scholars Fund of Lanzhou University of Technology (Q201015) and the Basic Scientific Research Business Expenses of Gansu Province College.
- Ahmadabadi MN, Arab M, Malek Ghaini FM: The method of fundamental solutions for the inverse space-dependent heat source problem. Eng. Anal. Bound. Elem. 2009, 33: 1231-1235.MathSciNetView ArticleGoogle Scholar
- Cannon JR, Duchateau P: Structural identification of an unknown source term in a heat equation. Inverse Probl. 1998, 14: 535-551.MathSciNetView ArticleGoogle Scholar
- Yan L, Fu CL, Yang FL: The method of fundamental solutions for the inverse heat source problem. Eng. Anal. Bound. Elem. 2008, 32: 216-222.View ArticleGoogle Scholar
- Dou FF, Fu CL, Yang FL: Optimal error bound and Fourier regularization for identifying an unknown source in the heat equation. J. Comput. Appl. Math. 2009, 230: 728-737.MathSciNetView ArticleGoogle Scholar
- Wei T, Zhang ZQ: Reconstruction of a time-dependent source term in a time-fractional diffusion equation. Eng. Anal. Bound. Elem. 2013, 37: 23-31.MathSciNetView ArticleGoogle Scholar
- Li GS: Data compatibility and conditional stability for an inverse source problem in the heat equation. Appl. Math. Comput. 2006, 173: 566-581.MathSciNetView ArticleGoogle Scholar
- Ma YJ, Fu CL, Zhang YX: Identification of an unknown source depending on both time and space variables by a variational method. Appl. Math. Model. 2012, 36: 5080-5090.MathSciNetView ArticleGoogle Scholar
- Yang L, Deng ZC, Yu JN, Luo GW: Optimization method for the inverse problem of reconstructing the source term in a parabolic equation. Math. Comput. Simul. 2009, 80: 314-326.MathSciNetView ArticleGoogle Scholar
- Yang F, Fu CL: Two regularization methods to identify time-dependent heat source through an internal measurement of temperature. Math. Comput. Model. 2011, 53: 793-804.MathSciNetView ArticleGoogle Scholar
- Ohe T, Ohnaka K: A precise estimation method for locations in an inverse logarithmic potential problem for point mass models. Appl. Math. Model. 1994, 18: 446-452.MathSciNetView ArticleGoogle Scholar
- Nara T, Ando S: A projective method for an inverse source problem of the Poisson equation. Inverse Probl. 2003, 19: 355-369.MathSciNetView ArticleGoogle Scholar
- Farcas A, Elliott L, Ingham DB, Lesnic D, Mera NS: A dual reciprocity boundary element method for the regularized numerical solution of the inverse source problem associated to the Poisson equation. Inverse Probl. Sci. Eng. 2003,11(2):123-139.View ArticleGoogle Scholar
- Kagawa Y, Sun Y, Matsumoto O: Inverse solution of Poisson equation using DRM boundary element models - identification of space charge distribution. Inverse Probl. Sci. Eng. 1995,1(2):247-265.View ArticleGoogle Scholar
- Sun Y, Kagawa Y: Identification of electric charge distribution using dual reciprocity boundary element models. IEEE Trans. Magn. 1997,33(2):1970-1973.View ArticleGoogle Scholar
- Jin BT, Marin L: The method of fundamental solutions for inverse source problems associated with the steady-state heat conduction. Int. J. Numer. Methods Eng. 2007, 69: 1570-1589.MathSciNetView ArticleGoogle Scholar
- Lattés R, Lions JL: The Method of Quasireversibility: Applications to Partial Differential Equations. Elsevier, New York; 1969.Google Scholar
- Showalter RE, Ting TW: Pseudo-parabolic partial differential equations. SIAM J. Math. Anal. 1970,1(1):1-26.MathSciNetView ArticleGoogle Scholar
- Miller K: Stabilized quasireversibility and other nearly best possible methods for non-well posed problems. Lecture Notes in Math. 316. In Symposium on Non-Well-Posed Problems and Logarithmic Convexity. Springer, Berlin; 1973:161-176.View ArticleGoogle Scholar
- Mel’nikova IV: Regularization of ill-posed differential problems. Sib. Math. J. 1992,33(2):289-298.MathSciNetView ArticleGoogle Scholar
- Ames KA, Cobb SS: Continuous dependence on modeling for related Cauchy problems of a class of evolution equations. J. Math. Anal. Appl. 1997,215(1):15-31.MathSciNetView ArticleGoogle Scholar
- Huang Y, Zheng Q: Regularization for a class of ill-posed Cauchy problems. Proc. Am. Math. Soc. 2005,133(10):3005-3012.MathSciNetView ArticleGoogle Scholar
- Denche M, Bessila K: A modified quasi-boundary value method for ill-posed problems. J. Math. Anal. Appl. 2005,301(2):419-426.MathSciNetView ArticleGoogle Scholar
- Boussetila N, Rebbani F: Optimal regularization method for ill-posed Cauchy problems. Electron. J. Differ. Equ. 2006., 2006: Article ID 147Google Scholar
- Showalter RE: Cauchy problem for hyperparabolic partial differential equations. In Trends in the Theory and Practice of Non-Linear Analysis. Elsevier, Amsterdam; 1983:421-425.Google Scholar
- Feng XL, Eldén L, Fu CL: A quasi-boundary-value method for the Cauchy problem for elliptic equations with nonhomogeneous Neumann data. J. Inverse Ill-Posed Probl. 2010,18(6):617-645.MathSciNetView ArticleGoogle Scholar
- Eldén L, Berntsson F, Regiǹska T: Wavelet and Fourier methods for solving the sideways heat equation. SIAM J. Sci. Comput. 2000, 21: 2187-2205.MathSciNetView ArticleGoogle Scholar
- Kirsch A: An Introduction to the Mathematical Theory of Inverse Problems. Springer, New York; 1996.View ArticleGoogle Scholar
- Engl HW, Hanke M, Neubauer A: Regularization of Inverse Problem. Kluwer Academic, Boston; 1996.View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.