Adjoint to the Hessian derivative and error covariances in variational data assimilation

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The optimal solution error is considered through the errors of input data (background and observation errors). The optimal solution error covariance operator is approximated by the inverse Hessian of the auxiliary (linearized) data assimilation problem, which involves the tangent linear model constraints. We show that the derivative of the inverse Hessian with respect to the exact solution may be treated as the measure of nonlinearity for analysis error covariances in variational data assimilation problems

On optimal solution error in variational data assimilation: theoretical aspects

The problem of variational data assimilation for a nonlinear evolution model is considered to identify the initial condition. An equation for the error of the optimal solution through the statistical errors of input data is derived, based on the Hessian of the misfit functional and second-order adjoint techniques. The covariance operator of the optimal solution error is expressed through the covariance operators of input errors. Numerical algorithms are developed for constructing the covariance operator of the optimal solution error using the covariance operators of input errors

Shape Sensitivity of Free-Surface Interfaces Using a Level Set Methodology

In this paper we develop the continuous adjoint methodology to compute shape sensitivities in free-surface hydrodynamic design problems using the incompressible Euler equations and the level set methodology. The identification of the free-surface requires the convection of the level set variable and, in this work, this equation is introduced in the entire shape design methodology. On the other hand, an alternative continuous adjoint formulation based in the jump condition across the interface, and an internal adjoint boundary condition is also presented. It is important to highlight that this new methodology will allow the specific design of the free-surface interface, which has a great potential in problems where the target is to reduce the wave energy (ship design), or increase the size of the wave (surfing wave pools). The complete continuous adjoint derivation, the description of the numerical methods (including a new high order numerical centered scheme), and numerical tests are detailed in this paper. I