  In meteorological and oceanological studies the classical approach for
finding the numerical solution of the regional model consists in formulating
and solving the Cauchy-Dirichlet problem. The related boundary conditions are
obtained by linear interpolation of data available on a coarse grid (global
data), to the boundary of regional model. Errors, in boundary conditions,
appearing owing to linear interpolation may lead to increasing errors in
numerical solution during integration. The methods developed to reduce these
errors deal with continuous dynamic assimilation of known global data available
inside the regional domain. Essentially, this assimilation procedure performs a
nudging of large-scale component of regional model solution to large-scale
global data component by introducing the relaxation forcing terms into the
regional model equations. As a result, the obtained solution is not a valid
numerical solution of the original regional model. In this work we propose the
optimization approach which is free from the above-mentioned shortcoming. The
formulation of the joint problem of finding the regional model solution and
data assimilation, as a PDE-constrained optimization problem, gives the
possibility to obtain the exact numerical solution of the regional model. Three
simple model examples (ODE Burgers equation, Rossby-Oboukhov equation,
Korteweg-de Vries equation) were considered in this paper. The result of
performed numerical experiments indicates that the optimization approach can
significantly improve the precision of the sought numerical solution, even in
the cases in which the solution of Cauchy-Dirichlet problem is very sensitive
to the errors in the boundary condition.
