Sensitivity of Functionals of Variational Data Assimilation Problems

Abstract

The problem of variational data assimilation for a nonlinear evolutionary model is formulated as an optimal control problem to simultaneously find unknown parameters and the initial state of the model. The response function is treated as a functional of the optimal solution found as a result of assimilation. The sensitivity of the functional to observational data is studied. The gradient of the functional with respect to observations is associated with the solution of a nonstandard problem involving a system of direct and adjoint equations. The solvability of the nonstandard problem is studied using the Hessian of the original cost function. An algorithm for calculating the gradient of the response function with respect to observations is formulated and justified.