Abstract
In this paper, a new approach of the restrained optimal perturbation method is firstly proposed to study the inverse problem in the reverse process of the one-dimensional convection diffusion equation, the idea of this method is brand new that in search for the optimal perturbation value by the given initial estimate, for determining the initial distribution based on the overspecified data, and the initial estimates plus optimal perturbation value can be treated as the final initial distribution, in order to overcome the ill-posedness of this problem, a regularization term is introduced in the objective functional. Numerical examples will be given, and the results show that our method is effective.
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References
Oleg, B.: A Positivity-preserving ALE Finite Element Scheme for Convection–diffusion Equations in Moving Domains. J. Comput. Phys. 230, 2896–2914 (2011)
Zhen, F.T., Yu, P.X.: A High-order Exponential Scheme for Solving 1D Unsteady Convection–diffusion Equations. J. Comput. Appl. Math. 235, 2477–2491 (2011)
Paul, D., Marcus, M.: Error Estimates for a Finite Element–finite Volume Discretization of Convection–diffusion Equations. Appl. Numer. Math. 61, 785–801 (2011)
Mueller, J.L., Shores, T.S.: A New Sinc-Galerkin Method for Convection-Diffusion Equations with Mixed Boundary Conditions. Comput. Math. Appl. 47, 803–822 (2004)
Rap, A., Eeeiott, L., Ingham, D., Lesnic, B.D., Wen, X.: The Inverse Source Problem for the Variable Coefficients Convection-diffusion Equation. Inverse Probl. Sci. Eng. 15, 413–440 (2007)
Chen, Q., Liu, J.J.: Solving an Inverse Parabolic Problem by Optimization from Final Measurement Data. J. Comput. Appl. Math. 193, 183–203 (2006)
Victor, I., Stefan, K.: Identification of the Diffusion Coefficient in One-dimensional Parabolic Equation. Inverse Probl. 16, 665–680 (2000)
Lesnic, D., Wake, G.C.: A Mollified Method for the Solution of the Cauchy Problem for the Convection-diffusion Equation. Inverse Probl. Sci. Eng. 15, 293–302 (2007)
Pan, J.F., Min, T., Zhou, X.D., Feng, M.Q.: Stability Analysis and Numerical Solution of Inverse Problem in Reverse Process of Convection Diffusion Equation. Journal of Wuhan University 3, 10–13 (2005) (in Ch)
Liu, J.J.: Numerical Solution of Forward and Backward Problem for 2-D Heat Conduction Equation. J. Comput. Appl. Math. 145, 459–482 (2002)
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. Chinese J. Hydrodynamics 23, 121–125 (2008) (in Ch)
Min, T., Zhou, X.D.: An Iteration Method of the Iinverse Problem for the Dispersion Coefficient in Water Quality Model. Chinese J. Hydrodynamics 18, 547–552 (2003) (in Ch)
Huang, S.X., Han, W., Wu, R.S.: Theoretical Analysis and Numerical Experiments of Variational Assimilation for One-dimensional Ocean Temperature Model with Techniques in Inverse Problems. Science in China 47, 630–638 (2004)
Mu, M., Duan, W.S., Wang, B.: Conditional Nonlinear Optimal Perturbation and Its Applications. Nonlin. Processes Geophys. 10, 493–501 (2003)
Mu, M., Duan, W.S.: A New Approach to Studying ENSO Predictability: Conditional Nonlinear Optimal Perturbation. Chinese Sci. Bull. 48, 1045–1047 (2003)
Mu, M., Sun, L., Dijkstra, H.A.: The Sensitivity and Stability of Oceans Thermohaline Circulation to Finite Amplitude Perturbations. J. Phy. Ocean. 34, 2305–2315 (2004)
Mu, M., Wang, B.: Nonlinear Instability and Sensitivity of a Theoretical Grassland Ecosystem to Finite-amplitude Perturbations. Nonlin. Processes Geophys. 14, 409–423 (2007)
Birgin, E.G., Martinez, J.m., Raydan, M.: Nonmonotone Spectral Projected Gradient Methods on Convex Sets. SIAM J. Optimiz. 10, 1196–1211 (2000)
Birgin, E.G., Martinez, J.m., Raydan, M.: Inexact Spectral Projected Gradient Methods on Convex sets. SIAM J. Numer. Anal. 23, 539–559 (2003)
Thomas, W.: Numerical Partial Differential Equations: Finite Difference Methods. Springer, Heidelberg (1997)
David, F.G., Desmond, J.H.: Numerical Methods for Ordinary Differential Equations. Springer, Heidelberg (2010)
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Wang, B., Zou, Ga., Zhao, P. (2012). A Restrained Optimal Perturbation Method for Solving the Inverse Problem in Reverse Process of Convection Diffusion Equation. In: Huang, DS., Gan, Y., Gupta, P., Gromiha, M.M. (eds) Advanced Intelligent Computing Theories and Applications. With Aspects of Artificial Intelligence. ICIC 2011. Lecture Notes in Computer Science(), vol 6839. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25944-9_20
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DOI: https://doi.org/10.1007/978-3-642-25944-9_20
Publisher Name: Springer, Berlin, Heidelberg
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