Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
inverse parabolic problem; unknown source; adjoint problem; Fréchet derivative; Lipschitz continuity
inverse parabolic problem; unknown source; adjoint problem; Fréchet derivative; Lipschitz continuity
Summary:
The problem of determining the source term $F(x,t)$ in the linear parabolic equation $u_t=(k(x)u_x(x,t))_x + F(x,t)$ from the measured data at the final time $u(x,T)=\mu (x)$ is formulated. It is proved that the Fréchet derivative of the cost functional $J(F) = \|\mu _T(x)- u(x,T)\|_{0}^2$ can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is proved. An existence result for a quasi solution of the considered inverse problem is proved. A monotone iteration scheme is obtained based on the gradient method. Convergence rate is proved.
References:
[1] Bushuyev, I.: Global uniqueness for inverse parabolic problems with final observation. Inverse Probl. 11 L11--L16 (1995). MR 1345998 | Zbl 0840.35120
[2] Chen, Q., Liu, J.: Solving an inverse parabolic problem by optimization from final measurement data. J. Comput. Appl. Math. 193 183-203 (2006). DOI 10.1016/j.cam.2005.06.003 | MR 2228714 | Zbl 1091.35113
[3] Duchateau, P.: An introduction to inverse problems in partial differential equations for physicists, scientists and engineers. A tutorial. Parameter Identification and Inverse Problems in Hydrology, Geology and Ecology ({K}arlsruhe, 1995) Water Sci. Technol. Libr. 23 Kluwer Acad. Publ., Dordrecht (1996). MR 1434090
[4] Hasanov, A.: Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: weak solution approach. J. Math. Anal. Appl. 330 766-779 (2007). DOI 10.1016/j.jmaa.2006.08.018 | MR 2308406 | Zbl 1120.35083
[5] Hasanov, A.: Simultaneous determination of the source terms in a linear hyperbolic problem from the final overdetermination: Weak solution approach. IMA J. Appl. Math. 74 1-19 (2009). DOI 10.1093/imamat/hxn042 | MR 2471318 | Zbl 1162.35404
[6] Hasanov, A., Duchateau, P., Pektaş, B.: An adjoint problem approach and coarse-fine mesh method for identification of the diffusion coefficient in a linear parabolic equation. J. Inverse Ill-Posed Probl. 14 435-463 (2006). DOI 10.1515/156939406778247615 | MR 2258244 | Zbl 1135.35095
[7] Isakov, V.: Inverse parabolic problems with the final overdetermination. Commun. Pure Appl. Math. 44 185-209 (1991). DOI 10.1002/cpa.3160440203 | MR 1085828 | Zbl 0729.35146
[8] Kabanikhin, S. I.: Conditional stability stopping rule for gradient methods applied to inverse and ill-posed problems. J. Inverse Ill-Posed Probl. 14 805-812 (2006). DOI 10.1515/156939406779768337 | MR 2270701 | Zbl 1115.65061
[9] Kamynin, V. L.: On the unique solvability of an inverse problem for parabolic equations under a final overdetermination condition. Math. Notes 73 (2003), 202-211. Transl. from the Russian. Mat. Zametki {\it 73} (2003), 217-227. DOI 10.1023/A:1022107024916 | MR 1997661 | Zbl 1033.35138
[10] Ladyzhenskaya, O. A.: The Boundary Value Problems of Mathematical Physics. Transl. from the Russian. Applied Mathematical Sciences 49 Springer, New York (1985). DOI 10.1007/978-1-4757-4317-3 | MR 0793735 | Zbl 0588.35003
[11] Prilepko, A. I., Orlovsky, D. G., Vasin, I. A.: Methods for Solving Inverse Problems in Mathematical Physics. Pure and Applied Mathematics 231 Marcel Dekker, New York (2000). MR 1748236 | Zbl 0947.35173
[12] Prilepko, A. I., Tkachenko, D. S.: An inverse problem for a parabolic equation with final overdetermination. V. G. Romanov Ill-posed and Inverse Problems VSP, Utrecht 345-381 (2002). MR 2024593 | Zbl 1059.35173
[13] Zeidler, E.: Nonlinear Functional Analysis and Its Applications. I: Fixed-Point Theorems. Transl. from the German. Springer, New York (1986). MR 0816732 | Zbl 0583.47050
Search
Partner of
