A regularized optimization method for identifying the space-dependent source and the initial value simultaneously in a parabolic equation. (English) Zbl 1350.65107
Summary: In this paper, a regularized optimization method is proposed for identifying the space-dependent source and the initial value simultaneously in an inverse parabolic equation problem from two over-specified measurements at different instants of time. The solvability of the direct problem is presented and then the inverse problem is formulated into a regularized optimization problem for the stable identification of both the source term and the initial value. Based on a sequence of well-posed direct problems solved by the finite element method, a numerical scheme formulated into a linear system is proposed to implement the regularized optimization problem. Numerical results of three examples show that the proposed method is efficient and robust with respect to data noise.
MSC:
| 65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |
| 35R30 | Inverse problems for PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
inverse problem; source inversion; inverse initial value; parabolic equation; regularized optimization; finite element method; numerical resultReferences:
| [1] | Xiong, X.; Yan, Y.; Wang, J., A direct numerical method for solving inverse heat source problems, J. Phys. Conf. Ser., 290, 012017 (2011) |
| [2] | Xiong, X.; Wang, J., A Tikhonov-type method for solving a multidimensional inverse heat source problem in an unbounded domain, J. Comput. Appl. Math., 236, 1766-1774 (2012) · Zbl 1275.65058 |
| [3] | Yan, L.; Yang, F. L.; Fu, C. L., A meshless method for solving an inverse spacewise-dependent heat source problem, J. Comput. Phys., 228, 123-136 (2009) · Zbl 1157.65444 |
| [4] | Yan, L.; Fu, C. L.; Dou, F. F., A computational method for identifying a spacewise-dependent heat source, Int. J. Numer. Methods Biomed. Eng., 26, 597-608 (2010) · Zbl 1190.65145 |
| [5] | Yang, F.; Fu, C. L., A simplified Tikhonov regularization method for determining the heat source, Appl. Math. Model., 34, 3286-3299 (2010) · Zbl 1201.65177 |
| [6] | Dou, F. F.; Fu, C. L.; Yang, F., Identifying an unknown source term in a heat equation, Inverse Probl. Sci. Eng., 17, 901-913 (2009) · Zbl 1183.65116 |
| [7] | Johansson, T.; Lesnic, D., Determination of a spacewise dependent heat source, J. Comput. Appl. Math., 209, 66-80 (2007) · Zbl 1135.35097 |
| [8] | Johansson, T.; Lesnic, D., A variational method for identifying a spacewise dependent heat source, IMA J. Appl. Math., 72, 748-760 (2007) · Zbl 1135.65034 |
| [9] | Ma, Y. J.; Fu, C. L.; Zhang, Y. X., Identification of an unknown source depending on both time and space variables by a variational method, Appl. Math. Model., 36, 5080-5090 (2012) · Zbl 1252.65106 |
| [10] | Wang, Z.; Liu, J., Identification of the pollution source from one-dimensional parabolic equation models, Appl. Math. Comput., 219, 3403-3413 (2012) · Zbl 1311.35333 |
| [11] | Cheng, J.; Liu, J., A quasi Tikhonov regularization for 2-dimensional backward heat problem by fundamental solution, Inverse Problems, 24, 6, 1-18 (2008), 065012 · Zbl 1157.35120 |
| [12] | Liu, J.; Lou, D., On stability and regularization for backward heat equation, Chin. Ann. Math. Ser. B, 24, 1, 35-44 (2003) · Zbl 1038.35150 |
| [13] | Li, J.; Yamamoto, M.; Zou, J., Conditional stability and numerical reconstruction of initial temperature, Commun. Pure Appl. Anal., 8, 361-382 (2009) · Zbl 1181.65121 |
| [14] | Chen, Q.; Liu, J., Solving the backward heat conduction problem by data fitting with multiple regularizing parameters, J. Comput. Math., 30, 4, 418-432 (2012) · Zbl 1274.65263 |
| [15] | Hasanov, A.; Mueller, J. L., A numerical method for backward parabolic problems with non-selfadjoint elliptic operators, Appl. Numer. Math., 37, 55-78 (2001) · Zbl 0976.65080 |
| [16] | Lattes, R.; Lions, J. L., The method of quasi-reversibility, (Applications to Partial Differential Equations (1969), Elsevier: Elsevier New York) · Zbl 1220.65002 |
| [17] | Johansson, B. T.; Lesnic, D., A procedure for determining a spacewise dependent heat source and the initial temperature, Appl. Anal., 87, 3, 265-276 (2008) · Zbl 1133.35436 |
| [18] | Wei, T.; Wang, J. C., Simultaneous determination for a space-dependent heatsource and the initial data by the MFS, Eng. Anal. Bound. Elem., 36, 1848-1855 (2012) · Zbl 1352.65316 |
| [19] | Yamamoto, M.; Zou, J., Simultaneous reconstruction of the initial temperature and heat radiative coefficient, Inverse Problems, 17, 1181-1202 (2001) · Zbl 0987.35166 |
| [20] | Xie, J.; Zou, J., Numerical reconstruction of heat fluxes, SIAM J. Numer. Anal., 43, 1504-1535 (2005) · Zbl 1101.65097 |
| [21] | Keung, Y. L.; Zou, J., Numerical identifications of parameters in parabolic systems, Inverse Problems, 14, 83-100 (1998) · Zbl 0894.35127 |
| [22] | Chen, Q.; Liu, J., Solving an inverse parabolic problem by optimization from final measurement data, J. Comput. Appl. Math., 193, 1, 183-203 (2006) · Zbl 1091.35113 |
| [23] | Prilepko, A. I.; Orlovsky, D. G.; Vasin, I. A., Methods for Solving Inverse Problems in Mathematical Physics (2000), Marcel Dekker: Marcel Dekker New York · Zbl 0947.35173 |
| [24] | Huang, M. Y., Numerical Methods for Evolution Equations (2004), Science Press: Science Press Beijing |
| [25] | Ciarlet, P. G., Basic error estimates for elliptic problems, (Ciarlet, P. G.; Lions, J. L., Handbook of Numerical Analysis, Vol. II (1991), North-Holland: North-Holland Amsterdam), 17-35 · Zbl 0875.65086 |
| [26] | Skeel, R. D.; Berzins, M., A method for the spatial discretization of parabolic equations in one space variable, SIAM J. Sci. Stat. Comput., 11, 1-32 (1990) · Zbl 0701.65065 |
| [27] | Wang, Z.; Liu, J., New model function methods for determining regularization parameters in linear inverse problems, Appl. Numer. Math., 59, 2489-2506 (2009) · Zbl 1167.65367 |
| [28] | Wang, Z.; Xu, D., On the linear model function method for choosing Tikhonov regularization parameters in linear ill-posed problems, Gongcheng Shuxue Xuebao, 30, 3, 451-466 (2013) · Zbl 1299.47023 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
