(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.