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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.
1. Introduction
Inverse source problem is of great importance in many branches of engineering and science; such as heat source determination [Citation1,Citation2], heat conduction problem [Citation3–5], Stephan design problem [Citation6] 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 [Citation7], the projective method [Citation8], the Green's function method [Citation9], the dual reciprocity boundary element method [Citation10,Citation11] and the method of fundamental solution (MFS) [Citation12–14].
Quasi-reversibility method is originally introduced by Lattes and Lions [Citation15], and later studied by Melnikova and Filinkov [Citation16]. 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 [Citation17] replaced the heat operator
by a perturbed operator
, perturbing the final condition we get an approximate solution from the final boundary value problem with a small parameter ε. The authors [Citation18] take
using logarithmic convexity to obtain well-posed solution as above Lattes and Lions [Citation15]. The final value problem in [Citation19] is considered about perturbing the final conditions to obtain an approximate non-local problem after operator perturbation. In [Citation20], 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 [Citation21] is replaced with a boundary value problem for a fourth order equation by using the method of quasi-reversibility. They consider the wave equation
, constructing Tikhonov functional firstly
, it is equivalent with the abstract Euler equation
, for all
, then through a minimizer
of calculation for above equation, they obtain perturb term to approximate the solution of ill-posed problem with a small parameter ε. In [Citation22], 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
of
as a solution of the weak formulation and
denotes α for small
and
. The method of quasi-reversibility proposed by [Citation23] is a particular case of Tikhonov regularization and
or
, which provides corresponding error estimate with a priori choice for ε as a function of δ. In [Citation24], the article adds
to the left-hand side of the equation
, 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 [Citation25,Citation26], and hyperbolic equations [Citation27–29].
In this article, we consider the following inverse problem:
(1)
(1) to find a pair of function
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 with a noise level δ is merely measured in
, and we give that
(2)
(2) We obtain that the solution of problem (Equation1
(1)
(1) ) using separation of variables has the following form:
(3)
(3) where
We define the operator
, then we have
(4)
(4) The singular values
of K satisfy
correspondingly
i.e.
then,
From [Citation30], the solution does not depend on the data continuously, the problem (Equation1
(1)
(1) ) is ill-posed. Several articles impose regularization method to deal with ill-posed problem (Equation1
(1)
(1) ): for examples, the Tikhonov regularization method [Citation31,Citation32], the super order regularization method [Citation33], the quasi-boundary value regularization method [Citation34,Citation35], the quasi-reversibility method [Citation36], the modified regularization method [Citation30,Citation37], the truncation method [Citation38]. Recently, Boussetila and Rebbani [Citation39] propose a modified quasi-reversibility method, and it is employed by Huang [Citation40] and Fury [Citation41] and Trong and Tuan [Citation42] 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 . (a): heat source for p = 1; (b): heat source for p = 3.
![Figure 1. The exact and approximate solutions with M = 50, N = 5 and various noise level for Example 1 where x∈[0,π]. (a): heat source for p = 1; (b): heat source for p = 3.](/cms/asset/2bfe9453-7911-40d0-9990-28ffa078cdbd/gipe_a_1902516_f0001_oc.jpg)
Figure 2. The exact and approximate solutions with M = 50, N = 5 and various noise level for Example 2 where . (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∈[0,π]. (a): heat source for p = 1; (b): heat source for p = 3.](/cms/asset/522de2ce-b814-4d65-90ce-f396447d9ab4/gipe_a_1902516_f0002_oc.jpg)
Figure 3. The exact and approximate solutions with M = 50, N = 5 and various noise level for Example 2 where . (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∈[0,π]. (a): heat source for p = 1; (b): heat source for p = 3.](/cms/asset/61f2a080-35d9-4db1-9224-71e73b6f5194/gipe_a_1902516_f0003_oc.jpg)
In this article, we will use a modified quasi-reversibility method to deal with identifying the unknown source of the problem (Equation1(1)
(1) ). Before doing that, we need to define an a priori bound on unknown source,
(5)
(5) where E>0 is a constant and
denotes the norm in Sobolev space which is defined as follows [Citation43]:
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 (Equation1
(1)
(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 ,
Lemma 2.2
If ,
,
Proof.
Lemma 2.3
[Citation30]
If ,
, p>0,
Lemma 2.4
If ,
,
Proof.
where
3. A modified regularization method and convergence estimates
We will investigate the following problem:
(6)
(6) By separation of variables, we obtain that
(7)
(7) which is called the modified regularized solution of problem (Equation1
(1)
(1) ), correspondingly
(8)
(8)
Theorem 3.1
Let be measured data at y = 1 satisfying (Equation2
(2)
(2) ) and the a priori condition (Equation5
(5)
(5) ) hold for p>0, if selecting
then we obtain the following error estimate:
Proof.
By the triangle inequality, we know
Firstly, we give an estimate for the first term as follows:
According to Lemmas 2.1–2.4 and an a priori bound condition of unknown source, we obtain
Combining above two estimates, we have
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 (Equation4(4)
(4) ), we know that
(9)
(9)
(10)
(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
,
, we get
(11)
(11) where
. Correspondingly, we obtain
(12)
(12) Adding a random distribute perturbation to each data function, we obtain
, i.e.
The total noise level δ can be measured in the sense of root mean square error(RMSE) according to
In order to research the effect of numerical computations, we compute the relative root mean squares error (RRMSE) of
by
(13)
(13) where
is a set of discrete points in internal
.
The numerical examples are constructed in the following way: First we select the exact solution and obtain the exact data function
using (Equation11
(11)
(11) ). Then we add a normally distributed perturbation to each data function giving vector
. Finally we obtain the regularization solutions using (Equation12
(12)
(12) ).
Example 1
We suppose that the solution of equation and the source function
, easily know that the data function
, we choose
in this example.
Table 1. δ, μ, with respect to various values of ε while p = 1, M = 50, N = 5 and for Example 1.
Table 2. δ, μ, (relative error of the source term) with respect to various values of ε while p = 3, M = 50, N = 5 and for Example 1.
Example 2
Consider the reconstruction of a Gaussian normal distribution:
where
,
Table 3. δ, μ, with respect to various values of ε while p = 1, M = 50, N = 5 and the
approaches 0.0093 when
for Example 2.
Table 4. δ, μ, with respect to various values of ε while p = 3, M = 50, N = 5 and the
approaches 0.1053 when
for Example 2.
Example 3
Consider the reconstruction of a piecewise smooth source:
Table 5. δ, μ, with respect to various values of ε while p = 1, M = 50, N = 5 and the
approaches 0.0134 when
for Example 3.
Table 6. δ, μ, with respect to various values of ε while p = 3, M = 50, N = 5 and the
approaches 0.0889 when
for Example 3.
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.
References
- Cannon JR. Determination of an unknown heat source from overspecified boundary data. SIAM J Numer Anal. 1968;5:275–286.
- Shi C, Wang C, Wei T. Numerical reconstruction of a space-dependent heat source term in a multi-dimensional heat equation. CMES Comput Model Eng Sci. 2012;86(2):71–92. doi:https://doi.org/10.1002/qua.23236
- Alifanov OM. Derivation of formulas for the gradient of the error in the iterative solution of inverse problems of heat conduction I. determination of the gradient in terms of the Green's function. Inzh Fiz Zh. 1987;52(3):476–485.
- Arghand M, Amirfakhrian M. A meshless method based on the fundamental solution and radial basis function for solving an inverse heat conduction problem. Adv Math Phys. 2015 Art. ID 256726, 8. doi:https://doi.org/10.1155/2015/256726.
- Dong CF, Li QH. A fundamental solution method based on geodesic distance for anisotropic heat conduction problems. J Zhejiang Univ Sci Ed. 2007;34(4):390–395.400
- Frankel JI. Constraining inverse stefan design problems. Z Angew Math Phys. 1996;47(3):456–466.
- 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(8):446–452.
- Nara T, Ando S. A projective method for an inverse source problem of the poisson equation. Inverse Probl. 2003;19(2):355–369.
- Hon YC, Li M, Melnikov YA. Inverse source identification by Green's function. Eng Anal Bound Elem. 2010;34(4):352–358.
- Farcas A, Elliott L, Ingham DB, et al. A dual reciprovity boundary element method for the regularized numerical solution of the inverse source problem associated to the Poisson equation. Inverse Probl Eng. 2003;11(2):123–139.
- Sun YH, Kagawa Y. Identification of eletric charge distribution using dual reciprocity boundary element models. IEEE Trans Magn. 1997;33(2):1970–1973.
- Jin BT, Marin L. The method of fundamental solutions for inverse source problem associated with the steady-state heat conduction. Int J Numer Methods Eng. 2010;69(8):1570–1589. doi:https://doi.org/10.1002/nme.1826
- Wang FZ, Chen W, Leevan L. Combinations of the method of fundamental solutions for general inverse source identification problems. Appl Math Comput. 2012;219(3):1173–1182.
- Wen J, Cheng JF. The method of fundamental solution for the inverse source problem for the space-fractional diffusion equation. Inverse Probl Sci Eng. 2018;26(7):925–941. doi:https://doi.org/10.1080/17415977.2017.1369537
- Lattès R, Lions JL. The method of quasi-reversibility. Applications to partial differential equations. Translated from the French edition and edited by Richard Bellman. Modern Analytic and Computational Methods in Science and Mathematics, No. 18. American Elsevier Publishing Co., Inc., New York, 1969.
- Mel ′ nikova IV, Filinkov AI. Abstract cauchy problems: three approaches. Boca Raton: Chapman& Hall. Chapman& Hall/CRC; 2001. Monographs and Surveys in Pure and Applied Mathematics 120
- Denche M, Bessila K. Quasi-boundary value method for non-well posed problem for a parabolic equation with integral boundary condition. Math Probl Eng. 2001;7(2):129–145. doi:https://doi.org/10.1155/S1024123X01001570
- Beth M, Hetrick C. Quasi-reversibility for inhomogeneous ill-posed problems in Hilbert spaces. Electron J Differ Equ. 2010;19:37–44.
- Clark GW, Oppenheimer SF. Quasi-reversibility methods for non-well-posed problems. Electron J Differ Equ. 1994;8:1C9.
- Dardé J, Hannukainen A, et al. An hdiv-based mixed Quasi-Reversibility method for solving elliptic cauchy problems. SIAM J Numer Anal. 2013;51(4):2123–2148.
- Christian C, Klibanov MV. The Quasi-Reversibility method for thermoacoustic tomography in a heterogeneous medium. SIAM J Sci Comput. 2007;30(1):1–23.
- Bourgeois L. A mixed formulation of quasi-reversibility to solve the cauchy problem for Laplace's equation. Inverse Probl. 2013;21(3):1087.
- Bourgeois L, Lunéville E. The method of quasi-reversibility to solve the cauchy problems for elliptic partial differential equations. Pamm. 2013;7(1):1042101–1042102. doi:https://doi.org/10.1002/pamm.200700001
- Li XX, Yang F, Liu J, Wang L. The Quasi-reversibility regularization method for identifying the unknown source for the modified Helmholtz equation. J Appl Math. 2013; 2013(2013):2133–2178. doi:https://doi.org/10.1155/2013/245963.
- Le TT, Nguyen LH. A convergent numerical method to recover the initial condition of nonlinear parabolic equations from lateral cauchy data. J Inverse Ill-Posed Probl. 2020. doi:https://doi.org/10.1515/jiip-2020-0028
- Nguyen PM, Nguyen LH. A numerical method for an inverse source problem for parabolic equations and its application to a coefficient inverse problem. J Inverse and Ill-posed Probl. 2020;28(3):323–339. doi:https://doi.org/10.1515/jiip-2019-0026
- Le TT, Nguyen LH, Nguyen T. The quasi-reversibility method to numerically solve an inverse source problem for hyperbolic equations. 2020. arXiv:2011.04855.
- Nguyen LH. An inverse space-dependent source problem for hyperbolic equations and the Lipschitz-like convergence of the quasi-reversibility method. Inverse Probl. 2019;35:035007.
- Klibanov MV, Kuzhuget AV, Kabanikhin SI, Nechaev DV. A new version of the quasi-reversibility method for the thermoacoustic tomography and a coefficient inverse problem. Appl Anal. 2008;87(10-11):1227–1254.
- Yang F, Fu CL. The modified regularization method for identifying the unknown source on Poisson equation. Appl Math Model. 2012;36(2):756–763.
- Qian AL, Mao JF. Optimal error bound and a generalized tikhonov regularization method for identifying an unknown source in the Poisson equation. Int J Wavelets Multiresolut Inf Process. 2014;12(1):1450004. 12.
- Zhao ZY, Meng ZH, You L, et al. Identifying an unknown source in the Poisson equation by the method of tikhonov regularization in Hilbert scales. Appl Math Model. 2014;38(19-20):4686–4693.
- Li Z, Zhao ZH, Meng ZH, et al. Identifying an unknown source in the poisson equation with a super order regularization method. Int J Comput Methods. 2020;17(7):1950030. 12.
- Denche M, Bessila K. A modified quasi-boundary value method for ill-posed problems. J Math Anal Appl. 2005;301(2):419–426.
- Yang F, Zhang M, Li XX. A quasi-boundary value regularization method for identifying an unknown source in the Poisson equation. J Inequalities Appl. 2004;1:117.
- Mel ′ nikova IV. Regularization of ill-posed differential problems. Sibirsk Mat Zh. 1992;33(2):125–134. 221.
- Li XX, Guo HZ, Wan SM, et al. Inverse source identification by the modified regularization method on Poisson equation. J Appl Math. 2012;2012(2):13. doi:https://doi.org/10.1155/2012/971952
- Yang F. The truncation method for identifying an unknown source in the Poisson equation. Appl Math Comput. 2011;217(22):9334–9339.
- Boussetila N, Rebbani F. A modified quasi-reversibility method for a class of ill-posed cauchy problems. Georgian Math J. 2007;14(4):627–642.
- Huang YZ. Modified quasi-reversibility method for final value problems in banach spaces. J Math Anal Appl. 2008;340(2):757–769.
- Fury MA. Modified quasi-reversibility method for nonautonomous semilinear problems. Electron. J. Differ. Equ. Conf., Proceedings of the Ninth MSU-UAB Conference on Differential Equations and Computational Simulations Texas State Univ., San Marcos. 2013;20:65–78.
- Trong DD, Tuan NH. Stabilized quasi-reversibility method for a class of nonlinear ill-posed problems. Electron J Differ Equ. 2008;84:359–370. doi:https://doi.org/10.1080/14689360802423530
- Kirsch A. An introduction to the mathematical theory of inverse problems. New York: Springer-Verlag; 1996. ISBN:978-1-4419-8473-9.