A series of studies have been made on the development of methods for investigating and numerical solving the quasilinear data assimilation problems, based on the adjoint equation theory, optimal control methods, and perturbation algorithms. The data assimilation problems are formulated as optimal control problems for the models governed by quasilinear evolution equations with the aim to identify the initial data and/or the right-hand-side (source)functions of the original equations. The neccessary optimality condition reduces the problem under consideration to the optimality system involving the original evolution problem, the adjoint problem, and the optimality condition (the last means that the Gateaux derivative of the cost functional equals zero). For linearized problem, by eliminating the state and adjoint variables, the optimality system is reduced to the only equation for the unknown function to be identified (the control function). This control equation has the form Lu=F, where L is a linear operator (called the control operator), u is the sought-for function, and the right-hand side F is determined by the input data. The properties of the control operators were studied, which are often symmetric, non-negative and compact. Based on the properties of the control operators, the solvability of linear and nonlinear data assimilation problems in a specific functional spaces is proved. To study the solvability of nonlinear data assimilation problems the successive approximation method is used. Using the spectral properties of the control operators, various iterative algorithms for solving the data assimilation problems are formulated and justified with optimal choice of iteration parameters. The convergence rate estimates are derived. The main results of this series are published in the author's book "Control operators and iterative algorithms in variational data assimilation problems (Moscow: Nauka, 2001).
Main publications:
Marchuk G. I., Agoshkov V. I., Shutyaev V. I. Adjoint Equations and Perturbation Algorithms in Nonlinear Problems. New York: CRC Press, 1996. 275 p. (izdano v Rossii: M.: Nauka, 1993).
Shutyaev V. P. Operatory upravleniya i iteratsionnye algoritmy v zadachakh variatsionnogo usvoeniya dannykh. M.: Nauka, 2001. 239 s.
Shutyaev V. P. Ob usvoenii dannykh v shkale gilbertovykh prostranstv dlya kvazilineinykh evolyutsionnykh zadach // Differentsialnye uravneniya, 1998, 34(3), 383–389.
Shutyaev V. P. Iteratsionnye metody vosstanovleniya nachalnykh dannykh v singulyarno vozmuschennykh evolyutsionnykh zadachakh // ZhVM i MF, 1997, 37(9), 1078–1086.
Shutyaev V. P. O svoistvakh operatora upravleniya v odnoi zadache ob usvoenii dannykh i algoritmakh ee resheniya // Matematicheskie zametki, 1995, 57(6), 941–944.
