693
Views
4
CrossRef citations to date
0
Altmetric
Research Article

A modified quasi-reversibility method for inverse source problem of Poisson equation

ORCID Icon, &
Pages 2098-2109 | Received 09 Nov 2020, Accepted 24 Feb 2021, Published online: 22 Mar 2021

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.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.