Control operators and iterative algorithms in variational data assimilation problems. (Q2757603)

The monograph presents the author's results on data assimilation problems for quasilinear evolution equations. The data assimilation problem is stated in the following way. The author considers the evolution problem NEWLINE\[NEWLINE\frac{d\varphi}{dt}+A\varphi+\tau F(\varphi)=f, \quad t\in(0,T); \quad \varphi(0)=u NEWLINE\]NEWLINE and the functional NEWLINE\[NEWLINES(\varphi)=\frac{\alpha}{2}\|\varphi(0)\|^2_\gamma+ \frac{\beta}{2}\int\limits_{0}^{T}\|\varphi-\hat{\varphi}\|_\gamma^2 dt.NEWLINE\]NEWLINE The data assimilation problem consists in finding two functions \(u\in X^\gamma, \varphi\in W^\gamma\) such that they satisfy problem (1) and the functional (2) takes the smallest value on a set of solutions of the equation (1). A necessary optimality condition reduces the problem to the search of functions \(\varphi,\varphi^*\in W^\gamma, u\in X^\gamma\). By eliminating \(\varphi, \varphi^*\) from the corresponding equations, the equation for \(u\) is obtained: \(Lu=p\). The problem solution depends on the properties of the operator \(L\) called the control operator. Chapter 1 is devoted to the principles for constructing the conjugate operators and the perturbation algorithms in nonlinear problems. Chapter 2 is the main one. In it the data assimilation problem for reconstructing the initial states for quasilinear evolution equations is investigated (for the above-mentioned statement, \(u\) is unknown). In the following two chapters the data assimilation problems are considered when the source (function \(f\)) is unknown (Chapter 3) and simultaneously the source \(f\) and the initial state \(u\) are unknown (Chapter 4). In Chapter 5 numerical algorithms for a solution to the problems studied are considered. The proof and optimization of iterative processes are given, and estimations of the degree of convergence are obtained. An application of the results obtained to the investigation and numerical solution of problems of mathematical physics is given in the appendix. NEWLINENEWLINENEWLINERewiever's remark. The problems considered are inverse problems of control theory. But the terminology and results of inverse control problems (Kalman criterion, extended observation vector,\dots) are not used in this monograph. It would be interesting to show their connection with optimization methods, to choose a meaningful functional and to establish the properties of solutions and algorithms depending on it.