Identifying an Unknown Source in the Poisson Equation by a Modified Tikhonov Regularization Method

In this paper, we consider the problem for identifying the unknown source in the Poisson equation. A modified Tikhonov regularization method is presented to deal with illposedness of the problem and error estimates are obtained with an a priori strategy and an a posteriori choice rule to find the regularization parameter. Numerical examples show that the proposed method is effective and stable.

• Ou Xie
• Zhenyu Zhao

• Ill-posed problem
• Unknown source
• Poisson equation
• Tikhonov regularization method
• Discrepancy principle

[1] A. Burykin, A. Denisov, Determination of the unknown sources in the heat-conduction equation, Computational Mathematics and Modeling 8 (4) (1997) 309-313.
[2] J. Cannon, P. Duchateau, Structural identification of an unknown source
term in a heat equation, Inverse Problems 14 (3) (1998) 535-551.
[3] J. Cannon, S. Perez-Esteva, Uniqueness and stability of 3d heat sources,
Inverse problems 7 (1991) 57.
[4] M. Choulli, M. Yamamoto, Conditional stability in determining a heat
source, Journal of Inverse and Ill-posed Problems 12 (3) (2004) 233-243.
[5] F. Dou, C. Fu, F. Yang, Optimal error bound and Fourier regularization
for identifying an unknown source in the heat equation, Journal of Computational and Applied Mathematics 230 (2) (2009) 728-737.
[6] A. El Badia, T. Ha-Duong, On an inverse source problem for the heat equation. Application to a pollution detection problem, Journal of Inverse and Ill-posed Problems 10 (6) (2002) 585-600.
[7] A. Farcas, D. Lesnic, The boundary-element method for the determination
of a heat source dependent on one variable, Journal of Engineering Mathematics 54 (4) (2006) 375-388.
[8] A. Kirsch, An introduction to the mathematical theory of inverse
problems, Springer-Verlag,New York, 1996.
[9] G. Li, Data compatibility and conditional stability for an inverse source
problem in the heat equation, Applied Mathematics and Computation
173 (1) (2006) 566-581.
[10] L. Ling, M. Yamamoto, Y. Hon, T. Takeuchi, Identification of source
locations in two-dimensional heat equations, Inverse Problems 22 (4)
(2006) 1289-1305.
[11] H. Park, J. Chung, A sequential method of solving inverse natural
convection problems, Inverse Problems 18 (3) (2002) 529-546.
[12] V. Ryaben'kii, S. Tsynkov, S. Utyuzhnikov, Inverse source problem and
active shielding for composite domains, Applied Mathematics Letters
20 (5) (2007) 511-515.
[13] M. Yamamoto, Conditional stability in determination of force terms of heat equations in a rectangle, Mathematical and Computer Modelling
18 (1) (1993) 79-88.
[14] L. Yan, C. Fu, F. Yang, The method of fundamental solutions for
the inverse heat source problem, Engineering Analysis with Boundary
Elements 32 (3) (2008) 216–222.
[15] L. Yan, F. Yang, C. Fu, A meshless method for solving an inverse
spacewise-dependent heat source problem, Journal of Computational
Physics 228 (1) (2009) 123–136.
[16] F. Yang, The truncation method for identifying an unknown source in
the poisson equation, Applied Mathematics and Computation 22 (2011)
9334–9339.
[17] F. Yang, C. Fu, The modified regularization method for identifying the
unknown source on poisson equation, Applied Mathematical Modelling
2 (2012) 756–763.
[18] Z. Yi, D. Murio, Source term identification in 1-D IHCP, Computers &
Mathematics with Applications 47 (12) (2004) 1921–1933.
[19] Z. Zhao, Z. Meng, A modified tikhonov regularization method for a
backward heat equation, Inverse Problems in Science and Engineering
19 (8) (2011) 1175–1182.