Randomised preconditioning for the forcing formulation of weak constraint 4D-Var

TL;DR

This paper introduces randomized low-rank eigenvalue decomposition methods to improve preconditioning in weak constraint 4D-Var data assimilation, leading to more efficient and robust optimization performance.

Contribution

It proposes novel randomized techniques for constructing limited memory preconditioners, enhancing the efficiency of weak constraint 4D-Var in handling model errors.

Findings

Randomized methods outperform existing preconditioners in idealized tests.

Improved preconditioning leads to faster convergence in inner loop minimizations.

The approach enhances robustness of the 4D-Var data assimilation process.

Abstract

There is growing awareness that errors in the model equations cannot be ignored in data assimilation methods such as four-dimensional variational assimilation (4D-Var). If allowed for, more information can be extracted from observations, longer time windows are possible, and the minimisation process is easier, at least in principle. Weak constraint 4D-Var estimates the model error and minimises a series of linear least-squares cost functionsfunctions, which can be achieved using the conjugate gradient (CG) method; minimising each cost function is called an inner loop. CG needs preconditioning to improve its performance. In previous work, limited memory preconditioners (LMPs) have been constructed using approximations of the eigenvalues and eigenvectors of the Hessian in the previous inner loop. If the Hessian changes significantly in consecutive inner loops, the LMP may be of limited usefulness. To circumvent this, we propose using randomised methods for low rank eigenvalue decomposition and use these approximations to cheaply construct LMPs using information from the current inner loop. Three randomised methods are compared. Numerical experiments in idealized systems show that the resulting LMPs perform better than the existing LMPs. Using these methods may allow more efficient and robust implementations of incremental weak constraint 4D-Var.