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

TL;DR

This paper introduces Stein-based preconditioners for weak-constraint 4D-Var data assimilation, improving the efficiency of solving large linear systems by leveraging Stein matrix equations, with demonstrated superior performance.

Contribution

The paper proposes novel preconditioning operators using Stein matrix equations for weak-constraint 4D-Var, enhancing computational efficiency and eigenvalue bounds over existing methods.

Findings

Preconditioners improve convergence speed of Krylov methods.

Eigenvalue bounds are tighter with Stein-based preconditioning.

Numerical results outperform current state-of-the-art approaches.

Abstract

Algorithms for data assimilation try to predict the most likely state of a dynamical system by combining information from observations and prior models. Variational approaches, such as the weak-constraint four-dimensional variational data assimilation formulation considered in this problem, can ultimately be interpreted as a minimization problem. One of the main challenges of such a formulation is the solution of large linear systems of equations which arise within the inner linear step of the adopted nonlinear solver. Depending on the adopted approach, these linear algebraic problems amount to either a saddle point linear system or a symmetric positive definite (SPD) one. Both formulations can be solved by means of a Krylov method, like GMRES or CG, 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 linear system for certain problem settings. A panel of diverse numerical results displays the effectiveness of the proposed methodology compared to current state-of-the-art approaches.