Sensitivity of functionals to input data in a variational assimilation problem for the sea thermodynamics model (Q6546264)

There is an investigation on the sensitivity of some functionals to input data for a specific problem to sea thermodynamics models. The paper is strongly connected to the results and methods developed in a previous article of the authors [\textit{V. P. Shutyaev} and \textit{E. I. Parmuzin}, Sib. Zh. Vychisl. Mat. 22, No. 2, 229--242 (2019; Zbl 07607915)].\N\NThe second section of the article is devoted to the presentation of the variation assimilation problem to a sea thermodynamics model. Written in operator form the considered sea thermodynamics model is\N\[\NT_t + LT = F + BQ,\text{ for }t\in(0,\bar{t})\N\]\N\[\NT=T_0\text{ for }t=0,\N\]\Nthe equation being understood in the generalized sense.\N\NThe function $T=T(x,y,z,t)$ models the temperature, the coefficients $L$, $F$, $B$ are known but the heat flux function $Q\in L_2(\Omega x(0,\bar{t}))$ is unknown.\N\NOne introduces an observation data function $T_{obs}$ defined as \N\[\NT_{obs}=m_0T^t|_{t=0}+\xi_{obs},\N\]\Nwhere $T^t$ is the solution, for some $Q=Q^t$, to the equation ${T^t}_t+LT^t=F+BQ^t$, $T^t=T_0$, for $t=0$ and $\xi_{obs}\in Y_{obs}=L_2(\Omega x(0,\bar{t}))$ is understood as an observation error.\N\NThe following problem of variational assimilation of data is stated: find the temperature $T$ and the heat flux $Q$, which satisfy the above system and minimizes the following error functional $J(Q)=\frac{1}{2}\int_{0}^{\bar{t}}\int_{\Omega}(Q-Q^{(0)})\mathcal{B}^{-1}(Q-Q^{0})d\Omega dt +\frac{1}{2}\int_{0}^{\bar{t}}\int_{\Omega}$ $(m_0T|_{z=0}-T_{obs})R^{-1}(m_0T|_{z=0}-T_{obs})d\Omega dt$ with respect to $Q$.\N\N$Q^{0}$ is an initial approximation for the unknown heat flux and $\mathcal{B}$ denotes the covariance operator to the initial approximation error. From the necessary condition of optimality, the optimality system is deduced. The inputs to the optimality system are the observation data $T_{obs}$ and the initial approximation for the heat flux.\N\NFurther, the third section of the article is devoted to the study of sensitivity of functionals to these input data: the observation as well as initial approximation data. The gradient of the functional is introduced and in the fourth section an algorithm for the calculation of the gradient is presented. Numerical experiments using a three-dimensional model of hydrodynamics for the Baltic Sea have been performed and the results are largely discussed in the fifth section. Some comments and conclusions are to be found in the last section of the article.