Existence of a solution to a variational data assimilation method in two-dimensional hydrodynamics

Graduation date: 1992The main result of this paper is a proof of the existence of a solution\ud generated by a method for the variational assimilation of observational data\ud into the two-dimensional, incompressible Euler equations. The data are\ud assumed to be given by linear (measurement) functionals acting on the space\ud of functions representing vorticity. From a practical point of view, the data\ud are considered to be sparse and available on a fixed space-time domain.\ud The objective of the variational assimilation is to obtain an estimate\ud of the vorticity which minimizes a cost functional. The cost functional is\ud the sum of a generalized mean squared error in the dynamics, a generalized\ud mean squared error in the initial condition, and a weighted squared error\ud in the misfit to the observed data. These generalized mean squared errors\ud are computed over the fixed space-time domain containing the data. The\ud estimate then provides a best (generalized) least squares fit between the\ud model, the initial condition, and the data. A necessary condition for the estimate of vorticity to minimize the cost\ud functional is that it must satisfy the corresponding system of Euler-Lagrange\ud equations, which consist of a nonlinear, coupled system of partial differential\ud equations with an initial condition, a final condition, and boundary conditions.\ud Construction of a solution to the Euler-Lagrange equations is possible\ud provided they are linearized through an iterative scheme.\ud Analysis of one such scheme motivates a reformulation of the variational\ud problem in terms of an iterated linearization of the dynamics. This second\ud method results in a slightly different iterated system of Euler-Lagrange equations.\ud The sequence of solutions generated is shown to be bounded in the\ud Sobolev space W[superscrit k,p] (in space-time). It follows from a Sobolev imbedding\ud theorem that the sequence contains a convergent subsequence, the limit of\ud which is a classical solution of the nonlinear, forced Euler equation corresponding\ud to the forward problem of the Euler-Lagrange system.\ud The two schemes mentioned above are compared based on formal applications\ud of Newton's method to the operators defining the systems. We\ud conclude that the two formulations of the assimilation problem are in fact\ud different and provide some intuitive reasons for preferring the second method,\ud beyond the fact that the existence proof is established