Physical Oceanography, Vol.
MATHEMATICAL MODELING OF MARINE SYSTEMS
VARIATIONAL APPROACH TO THE PROBLEMS OF PLANNING OF
EXPERIMENTS AND IDENTIFICATION OF THE INPUT PARAMETERS OF
HYDRODYNAMIC MODELS ACCORDING TO THE DATA OF MEASUREMENTS
V. N. Eremeev and S. V. Kochergin
We discuss the computational properties of variational algorithms of mastering the data of meas-
urements and convergency of iterative procedures of search for an optimum distribution of the
input parameters of a model and consider the model of transfer of a passive admixture. The al-
gorithm of identification is based on the variational principles, solving the conjugate problem,
and minimization of the functional of quality of a prediction. We carry out the analytic verifica-
tion of the efficiency of the algorithm and discuss the problems of optimum planning of experi-
ments from the viewpoint of conditionality of the problem of identification that is being solved.
In dynamic oceanology in the problem of mastering the data of observations, two approaches are clearly
distinguished: the probabilistic  and variational [2
5] ones. The latter possesses some interesting prospects
for use in problems of four-dimensional analysis  and identification of input parameters on numerical simula-
tion . This approach is based on a realization of the direct and conjugate models of the process under study.
The solution of the conjugate problem emerges as an influence function. The informational unity of a model and
measurements is attained at the expense of varying the input parameters of the model (initial data and the coeffi-
cients of boundary conditions and the very equations of the model of transfer) that ensure an extremum for a cer-
tain functional of quality of a prediction, whose form depends on the purposes of the simulation. By computing
the required functions with the help of the conjugate problem, we can get answers to many practically important
questions including estimates of the influence of various regions of the initial fields, the coefficients of a model,
etc. on a model prediction. We then find the required solution by using gradient iterative methods. Questions
related to the application of such algorithms for solving problems of identification in the study of thermophysical
processes are clarified most fully in . On the first iterations, iterative processes have a certain stability against
errors in the initial data. Therefore, a stable approximate solution can be obtained by interrupting the iterative
process at some iteration index, which is consistent with these errors. This demonstrates the important regulariz-
ing properties of gradient methods, in which the iteration index plays the role of a parameter of regularization.
While realizing such algorithms, one meets certain difficulties in solving even linear problems due to the nonlin-
ear character of methods of minimization such as the methods of steepest descent and conjugate gradients. The
iterative regularization for linear problems has been studied rather well. Moreover, the results of computational
experiments indicate that iterative algorithms of solution of nonlinear ill-posed problems turn out to be quite ef-
ficient [7, 3
4]. But prior to testing a model on noise-distorted or full-scale data, it is necessary to ensure its ef-
ficiency with the use of exact information, which is a necessary condition.
Marine Hydrophysical Institute, Ukrainian Academy of Sciences, Sevastopol. Translated from Morskoi Gidrofizicheskii Zhurnal,
46, January–February, 2002. Original article submitted February 29, 2000; revision submitted May 22, 2000.
32 0928–5105/02/1201–0032 $27.00 © 2002 Plenum Publishing Corporation