Stein-based preconditioners for weak-constraint 4D-var

Abstract

Algorithms for data assimilation try to predict the most likely state of a dynamical system by combining information from observations and prior models. One of the most successful data assimilation frameworks is the linearized weak-constraint four-dimensional variational assimilation problem (4D-Var), that can be ultimately interpreted as a minimization problem. One of the main challenges of such an approach is the solution of large saddle point linear systems arising as an inner linear step within the adopted nonlinear solver. The linear algebraic problem can be solved by means of a Krylov method, like MINRES or GMRES, that needs to be preconditioned to ensure fast convergence in terms of the number of iterations. In this paper we illustrate novel, efficient preconditioning operators which involve the solution of certain Stein matrix equations. In addition to achieving better computational performance, the latter machinery allows us to derive tighter bounds for the eigenvalue distribution of the preconditioned saddle point linear system. A panel of diverse numerical results displays the effectiveness of the proposed methodology compared to current state-of-the-art approaches.