Application of the restrained optimal perturbation method to study the backward heat conduction problem. (English) Zbl 1329.80012
Summary: In this paper, a restrained optimal perturbation method is firstly proposed to solve the backward heat conduction problem, the initial temperature distribution will be identified from the overspecified data, a regularization term is introduced in the objective functional for overcoming the ill-posedness of this problem, spectral projected gradient algorithm is used to solve the optimal problem, and we give the sensitivity analysis of the initial value. The results of numerical experiments are also presented.
MSC:
| 80M25 | Other numerical methods (thermodynamics) (MSC2010) |
| 80A23 | Inverse problems in thermodynamics and heat transfer |
| 80M20 | Finite difference methods applied to problems in thermodynamics and heat transfer |
| 80M22 | Spectral, collocation and related (meshless) methods applied to problems in thermodynamics and heat transfer |
Keywords:
backward heat conduction problem; restrained optimal perturbation method; spectral projected gradient algorithm; finite difference approximation; numerical experimentsReferences:
| [1] | Fatullayev, A. G., Numerical solution of the inverse problem of determining an unknown source term in a heat equation, Math. Comput. Simul., 58, 247-253 (2002) · Zbl 0994.65100 |
| [2] | Karami, G.; Hematiyan, M. R., A boundary element method of inverse nonlinear heat conduction analysis with point and line heat sources, Commun. Numer. Methods Eng., 16, 191-203 (2000) · Zbl 0951.65091 |
| [3] | Cannon, J. R.; Duchateau, P., Structural identification of an unknown source term in a heat equation, Inverse Prob., 14, 535-551 (1998) · Zbl 0917.35156 |
| [4] | Geng, F. Z.; Lin, Y. Z., Application of the variational iteration method to inverse heat source problems, Comput. Math. Appl., 58, 2098-2102 (2009) · Zbl 1189.65216 |
| [5] | Savateev, E. G.; Riganti, R., Inverse problem for the nonlinear heat equation with the final overdetermination, Math. Comput. Model, 22, 29-43 (1995) · Zbl 0835.35157 |
| [6] | Shidfar, A.; Karamali, G. R.; Damirchi, J., An inverse heat conduction problem with a nonlinear source term, Nonlinear Anal., 65, 615-621 (2006) · Zbl 1106.35141 |
| [7] | Chen, Q.; Liu, J. J., Solving an inverse parabolic problem by optimization from final measurement data, J. Comput. Appl. Math., 193, 183-203 (2006) · Zbl 1091.35113 |
| [8] | Mera, N. S.; Elliott, L.; Ingham, D. B.; Lesnic, D., An iterative boundary element method for solving the one-dimensional backward heat conduction problem, J. Heat Mass Transfer, 44, 1937-1946 (2001) · Zbl 0979.80008 |
| [9] | Liu, J. J., Numerical solution of forward and backward problem for 2-D heat conduction equation, J. Comput. Appl. Math., 145, 2, 459-482 (2002) · Zbl 1005.65107 |
| [10] | Wu, Z. K.; Fan, H. M.; Chen, X. R., The numerical study of the inverse problem in reverse process of convection-diffusion equation with adjoint assimilation method, Chin. J. Hydrodyn., 23, 2, 121-125 (2008), (in Chinese) |
| [11] | Mu, M.; Duan, W. S.; Wang, B., Conditional nonlinear optimal perturbation and its applications, Nonlinear Process.s Geophys., 10, 493-501 (2003) |
| [12] | Mu, M.; Duan, W. S., A new approach to studying ENSO predictability: conditional nonlinear optimal perturbation, Chin. Sci. Bull., 48, 1045-1047 (2003) |
| [13] | Mu, M.; Sun, L.; Dijkstra, H. A., The sensitivity and stability of ocean’s thermohaline circulation to finite amplitude perturbations, J. Phy. Ocean., 34, 2305-2315 (2004) |
| [14] | Mu, M.; Wang, B., Nonlinear instability and sensitivity of a theoretical grassland ecosystem to finite-amplitude perturbations, Nonlinear Process. Geophys., 14, 409-423 (2007) |
| [15] | Duan, W. S.; Zhang, R., Is model parameter error related to spring predictability barrier for El Nino events?, Adv. Atmos. Sci., 27, 1003-1013 (2010) |
| [16] | Duan, W. S.; Liu, X. C.; Zhu, K. Y.; Mu, M., Exploring the initial error that causes a significant spring predictability barrier for El Nino events, J. Geophy. Res., 114, C04022 (2009) |
| [17] | Duan, W. S.; Mu, M., Conditional nonlinear optimal perturbation: applications to stability, sensitivity, and predictability, Sci. Chin. (D), 52, 884-906 (2009) |
| [18] | Duan, W. S.; Xu, H.; Mu, M., Decisive role of nonlinear temperature advection in El Nino and La Nina amplitude asymmetry, J. Geophy. Res., 113, C01014 (2008) |
| [19] | Duan, W. S.; Xue, F.; Mu, M., Investigating a nonlinear characteristic of ENSO events by conditional nonlinear optimal perturbation, Atmos. Res. (2008) |
| [20] | Duan, W. S.; Mu, M., 2006, Investigating decadal variability of El Nino-Southern oscillation asymmetry by conditional nonlinear optimal perturbation, J. Geophy. Res., 111, C07015 (2006) |
| [21] | Birgin, E. G.; Martinez, J. M.; Raydan, M., Nonmonotone spectral projected gradient methods on convex sets, SIAM J. Optim., 10, 1196-1211 (2000) · Zbl 1047.90077 |
| [22] | Birgin, E. G.; Martinez, J. M.; Raydan, M., Inexact spectral projected gradient methods on convex sets, SIAM J. Numer. Anal., 23, 539-559 (2003) · Zbl 1047.65042 |
| [23] | Quarteroni, A.; Valli, A., Numerical Approximation of Partial Differential Equations (1994), Springer-verlag · Zbl 0803.65088 |
| [24] | Samarskii, A. A.; Vabishchevich, P. N., Numerical methods for solving inverse problems of mathematical physics (2007), Walter de Gruyter GmbH Co. KG: Walter de Gruyter GmbH Co. KG Berlin · Zbl 1136.65105 |
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