(Non)Uniqueness of critical points in variational data assimilation

•Bayesian formulation of variational assimilation for quasilinear equations.•Uniqueness of minimizers for small observational times.•Uniqueness of minimizers for small prior covariance.•Existence of critical points with large Morse index.

In this paper we apply the 4D-Var data assimilation scheme to the initialization problem for a family of quasilinear evolution equations. The resulting variational problem is non-convex, so it need not have a unique minimizer. We comment on the implications of non-uniqueness for numerical applications, then prove uniqueness results in the following situations: (1) the observational times are sufficiently small; (2) the prior covariance is sufficiently small. We also give an example of a data set where the cost functional has a critical point of arbitrarily large Morse index, thus demonstrating that the geometry can be highly nonconvex even for a relatively mild nonlinearity.