Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-12-05T02:39:25.706Z Has data issue: false hasContentIssue false
  • English
  • Français

A quasi-Newton approach to identification of a parabolic system

Published online by Cambridge University Press:  17 February 2009

Wenhuan Yu
Wenhuan Yu
Affiliation:
Department of Mathematics, Tianjin University, Tianjin 300072, People's Republic of China.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A quasi-Newton method (QNM) in infinite-dimensional spaces for identifying parameters involved in distributed parameter systems is presented in this paper. Next, the linear convergence of a sequence generated by the QNM algorithm is also proved. We apply the QNM algorithm to an identification problem for a nonlinear parabolic partial differential equation to illustrate the efficiency of the QNM algorithm.

Type
Research Article
Information
The ANZIAM Journal , Volume 40 , Issue 1 , July 1998 , pp. 1 - 22
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Banks, H. T. and Kunisch, K., Estimation Techniques for Distributed Parameter Systems (Birkhäuser, Boston, 1989).CrossRefGoogle Scholar
[2]Banks, H. T., Reich, S. and Rosen, I. G., “An approximation theory for the identification of nonlinear distributed parameter systems”, SIAM J. Control and Optim. 28 (1990) 523–569.Google Scholar
[3]Broyden, C. G., Dennis, J. E. and Moore, J. J., “On the local and superlinear convergence of quasi-Newton methods”, J. Inst. Math. Appl. 12 (1973) 223–245.CrossRefGoogle Scholar
[4]Chavent, G., “Identification of distributed parameter systems: About the output least squares method, its implementation and identifiability”, in: Identification and System Parameter Estimation (ed. Isermann, R.) (Pergamon, New York, 1982) 85–97.Google Scholar
[5]Chen, Y. M. and Zhang, F. G., “Hierarchical multigrid strategy for efficiency improvement of the GPST inversion algorithm”, Applied Numerical Mathematics 6 (1990) 431–446.Google Scholar
[6]Coleman, T. F. and Conn, A. R., “On the local convergence of a quasi-Newton method for the nonlinear programming problem”, SIAM J. Numer. Anal. 21 (1984) 755–769.CrossRefGoogle Scholar
[7]Dunford, N. and Schwartz, J. T., Linear Operators, Part II: Spectral Theory (Interscience, New York, 1963).Google Scholar
[8]Griewank, A., “The local convergence of Broyden-like methods on Lipchitzian problems in Hilbert spaces”, SIAM J. Numer. Anal. 24 (1987) 684–705.CrossRefGoogle Scholar
[9]Hwang, D. M. and Kelley, C. T., “Sequential quadratic programming for parameter identification problems”, in: Fifth Symp. on Control of Distributed Parameter Systems, (ed. Jai, A. E. and Amouroux, M.) (1989) 105–109.Google Scholar
[10]Ito, K. and Kunisch, K., “The augmented Lagrangian method for equality and inequality constrained problems in Hilbert spaces”, Math. Programming 46 (1990) 341–360.Google Scholar
[11]Ito, K. and Kunisch, K., “The augmented Lagrangian method for parameter estimation in elliptic systems”, SIAM J. Control and Optim. 28 (1990) 113–136.CrossRefGoogle Scholar
[12]Kelley, C. T. and Sachs, E. W., “Quasi-Newton methods and unconstrained optimal control problems”, SIAM J. Control and Optim. 23 (1987) 1503–1516.CrossRefGoogle Scholar
[13]Kunisch, C. K. and Sachs, E. W., “Reduced SQP methods for parameter identification problems”, SIAM J. Numer. Anal. 29 (1992) 1793–1820.CrossRefGoogle Scholar
[14]Ladyzenskaja, O. A. et al. , Linear and Quasi-linear Equations of Parabolic Type, (AMS, Providence, 1968).CrossRefGoogle Scholar
[15]Lapidus, L. and Pinder, G. F., Numerical Solution of Partial Differential Equations in Sciences and Engineering (John Wiley and Sons, New York, 1982).Google Scholar
[16]Lee, Tai-Yong and Seinfeld, John H., “Estimation of two-phase petroleum reservoir properties by regularization”, J. Comput. Phys. 69 (1987) 397–419.CrossRefGoogle Scholar
[17]Seinfeld, J. H. and Chen, W. H., “Identification of petroleum reservoir properties”, in: Distributed Parameter Systems: Identification, Estimation and Control (ed. Ray, W. H. amd Lainiotis, D. G.) (Marcel Dekker, New York, 1978).Google Scholar
[18]Yu, W. H., “A Rapid convergent iteration method for identifying distributed parameters in partial differential equations”, Preprints 1988 ’IFAC Symp. Identification and System Parameter Estimation (Beijing, 1988).Google Scholar
[19]Yu, W. H. and Seinfeld, J. H., “Identification of distributed paremeter systems by regularization with differential operator”, J. Math. Anal. Appl. 132 (1988) 365–398.Google Scholar
[20]Yu, W. H. and Seinfeld, J. H., “Identification of distributed parameter systems with pointwise constrains on the parameters”, J. Math. Anal. Appl. 136 (1988) 497–520.Google Scholar
[21]Yu, W. H., “Solving inverse problems for hyperbolic equations via the regularization method”, J. Computational Mathematics 11 (1993) 142–153.Google Scholar
[22]Zhu, Jianping and Chen, Yung Ming, “Multilevel grid method for history matching multi-dimensional multi-phase reservoir models”, Applied Numer. Math. 10 (1992) 159–174.Google Scholar