On data assimilation for quasilinear parabolic problems

Russian Journal of Numerical Analysis and Mathematical Modelling

Volumn 16, Issue 3, 2001, Pages 247-259

  Le Dimet, F X ^a   Shutyaev, V ^b  

^a UNIV GRENOBLE ALPES   (France)

^b INSTITUTE OF NUMERICAL MATHEMATICS   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 17444378020     PISSN: 09276467     EISSN: 15693988     Source Type: Journal    
DOI: 10.1515/rnam-2001-0305     Document Type: Article

References (44)

1. V.I. Agoshkov and G.I. Marchuk, On solvability and numerical solution of data assimilation problems. Russ. J. Numer. Anal. Math. Modelling (1993) 8, No. 1, 1-16.
2. H. W. Alt and S. Luckaus, Quasilinear elliptic-parabolic differential equations. Math. Zeitschrift (1983) 183, 311-341.
3. E. Blayo, J. Blum, and J. Verron, Assimilation variationnelle de données en océanographie et réduction de la dimension de l'espace de contrôle. In: Équations aux Dérivées Partielles et Applications. Elsevier, Paris, 1998, pp. 205-219.
4. G. S. Campbell, A simple method for determining unsaturated conductivity from moisture retention data. Soil Sci. (1974) 117, 311-314.
5. R. B. Clapp and G. M. Hornberger, Empirical equations for some soil hydraulic properties. Water Resour. Res. (1978) 14, 611-614.
6. A. Eden, B. Michiux, and J. M. Rakotson, Doubly nonlinear parabolic-type equations as dynamical systems. J. Dyn. Dijf. Equations (1991) 3, No. 1, 87-131.
7. R. Glovinski and J.-L. Lions, Exact and approximate controllability for distributed parameter systems. Acta Numerica (1994) 1, 269-378.
8. P. H. Groenevelt and J.-Y. Parlange, Thermodynamics stability of swelling soils. Soil Sci. (1974) 118, 1-5.
9. R. Haverkamp, F. Bouraoui, C. Zammit, and R. Angulo-Jaramillo, Soil properties and moisture movement in the unsaturated zone. The Handbook of Groundwater Engineering. Chap. 5, 1998.
10. D. Henry, Geometric Theory of Semilinear Parabolic Equations. Springer, New York, 1981.
11. L. A. Kozdova, The Methods for Solving Nonlinear Heat Conduction Problems. Nauka, Moscow, 1975 (in Russian).
12. A. B. Kurzhanskii and A. Yu. Khapalov, An observation theory for distributed-parameter systems. J. Math. Syst. Estimât. Control. (1991) 1, No. 4, 389-440.
13. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uraltseva, Linear and Quasilinear Parabolic Equations. Nauka, Moscow, 1967 (in Russian).
14. O. A. Ladyzhenskaya and N. N. Uraltseva, A survey on solvability of boundary value problems for uniformly elliptic and parabolic equations of the second order. Uspekhi Mat. Nauk (1986) 41, No. 5, 59-83 (in Russian).
15. M. M. Lavrentiev, A priori Estimates and Existence Theorems for Nonlinear Parabolic Equations. Nauka, Novosibirsk, 1982 (in Russian).
16. F.-X. Le Dimet, A general formalism of variational analysis. Report No. OK-73091-22-1, CIMMS, Norman, 1982.
17. F.-X. Le Dimet and R Ngnepieba, Comparison of codes to obtain the first order information in data assimilation: an application to hydrology parameters determination. INRIA Report, 2000.
18. F.-X. Le Dimet and V.P. Shutyaev, On Newton methods in data assimilation. Russ. J. Numer. Anal. Math. Modelling (2000) 15, No. 5, 419-434.
19. F.-X. Le Dimet, V. P. Shutyaev, M. Mu, and J. Wang, The problem of data assimilation for soil water movement. (Submitted to ESAIM: Control, Optimization and Calculus of Variations, 2000).
20. F.-X. Le Dimet and O. Talagrand, Variational algorithms for analysis and assimlation of meteorological observations: theoretical aspects. Tellus (1986) 38A, 97-110.
21. Zh. Lei and Sh. Yang, The Dynamics of Soil Water. Tsinghua, Tsinghua University Press, 1986.
22. J. Lewis and J. Derber, The use of adjoint equations to solve a variational adjustment problem with advective constraints. Tellus (1985) 37A, 309-322.
23. J.-L. Lions, Contrôle Optimal des Systèmes Gouvernés par des Equations aux Dérivées Partielles. Dunod, Paris, 1968.
24. J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogenes et Applications. Dunod, Paris, 1968.
25. G. I. Marchuk and V. V. Penenko, Application of optimization methods to the problem of mathematical simulation of atmospheric processes and environment. Modelling and Optimization of Complex Systems: Proc. of the IFIP-TC7 Working conference. Springer, New York, 1978, pp. 240-252.
26. G. I. Marchuk, V. I. Agoshkov, and V. P. Shutyaev, Adjoint Equations and Perturbation Algorithms in Nonlinear Problems. CRC Press Inc., New York, 1996.
27. G. I. Marchuk and V. P. Shutyaev, Iteration methods for solving a data assimilation problem. Russ. J. Numer. Anal. Math. Modelling (1994) 9, No. 3, 265-279.
28. I. M. Navon, A review of variational and optimization methods in meteorology. Variational Methods in Geosciences. Elsevier, New York, 1986, pp. 29-34.
29. I. M. Navon, Practical and theoretical aspects of adjoint parameter estimation and identifiability in meteorology and oceanography. Dynamics Atmosph. Oceans (1998) 27, No. 1-4, 55-79.
30. O. A. Oleinik and E. V. Radkevich, Method of introducing a parameter for study of evolution equations. Uspekhi Mat. Nauk (1978) 33, No. 5(203), 7-76 (in Russian).
31. J.-Y. Parlange, Theory of water movement in soils: 1. One-dimensional absorption. Soil Sci. (1971) 111, No. 2, 134-137.
32. V. Penenko and N. N. Obraztsov, A variational initialization method for the fields of meteorological elements. Sov. Meteorol. Hydrology (1976) No. 11, 1-11 (in Russian).
33. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko, The Mathematical Theory of Optimal Processes. John Wiley, New York, 1962.
34. Y. K. Sasaki, Some basic formalisms in numerical variational analysis. Monthly Weather Review (1970) 98, 857-883.
35. V. P. Shutyaev, On a class of insensitive control problems. Control and Cybernetics (1994) 23, 257-266.
36. V. P. Shutyaev, Some properties of the control operator in the problem of data assimilation and iterative algorithms. Russ. J. Numer. Anal. Math. Modelling (1995) 10, No. 4, 357-371.
37. D. E. Smiles and M. J. Rosenthal, The movement of water in swelling material. Aust. J. Soil Res. (1968) 6, 237-248.
38. M. T. Van Genuchten, A closed form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Amer. J. (1980) 44, 892-898.
39. M. T. Van Genuchten and D. R. Nielsen, On describing and predicting the hydraulic properties of unsaturated soils. Ann. Geophys. (1985) 3, 615-628.
40. Zhi Wang, K. Droegemeier, and L. White, The adjoint Newton algorithm for large-scale unconstrained optimization in meteorology applications. Comp. Opt. Appl. (1998) 10, 283-320.
41. Zhi Wang, I. M. Navon, X. Zou, and F.-X. Le Dimet, A truncated Newton optimization algorithm in meteorology applications with analytic Hessian/vector products. Comp. Opt. Appl. (1995) 4, No. 3, 241-262.
42. T. I. Zelenyak, M. M. Lavrentiev, and M. P. Vishnevsky Qualitative Theory of Parabolic Equations. VSP Publishers, Utrecht, 1997.
43. T. I. Zelenyak and V. P. Mikhailov, Asymptotical behaviour of solutions of mathematical physics. In: Partial Differential Equations. Nauka, Moscow, 1970, pp. 96-110 (in Russian).
44. X. Zou, I. M. Navon, and F.-X. LeDimet, Incomplete observations and control of gravity waves in variational data assimilation. Tellus (1992) 44A, 273-296.