Saddle point preconditioners for weak-constraint 4D-Var

Abstract

Data assimilation algorithms combine information from observations and prior model information to obtain the most likely state of a dynamical system. The linearised weak-constraint four-dimensional variational assimilation problem can be reformulated as a saddle point problem, in order to exploit highly-parallel modern computer architectures. In this setting, the choice of preconditioner is crucial to ensure fast convergence and retain the inherent parallelism of the saddle point formulation. We propose new preconditioning approaches for the model term and observation error covariance term which lead to fast convergence of preconditioned Krylov subspace methods, and many of these suggested approximations are highly parallelisable. In particular our novel approach includes model information in the model term within the preconditioner, which to our knowledge has not previously been considered for data assimilation problems. We develop new theory demonstrating the effectiveness of the new preconditioners for a specific class of problems. Linear and non-linear numerical experiments reveal that our new approach leads to faster convergence than existing state-of-the-art preconditioners for a broader range of problems than indicated by the theory alone. We present a range of numerical experiments performed in serial, with further improvements expected if the highly parallelisable nature of the preconditioners is exploited.

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

“…In the 4D-Var context, many authors considered approximations S of the form S := L T D −1 L thus neglecting the second term H T R −1 H in the definition of S. See, e.g., Freitag & Green (2018); Gratton et al (2018); Tabeart & Pearson (2021a). This leads to an easier-to-invert preconditioning operator P D as S −1 = L −1 DL −T .…”

“…One of the main strategies to overcome this issue is the introduction of a further layer of approximation related to employing an operator L ≈ L in the definition of S such that multiplication of a vector by S −1 = L −1 D L −T can be distributed over multiple processors. Different options for the selection of L can be found in, e.g., Freitag & Green (2018); Tabeart & Pearson (2021a). 6 of 26 D. PALITTA AND J. M. TABEART Similar considerations have led to the design of the following block triangular preconditioner…”

“…The performance of our strategy is also compared to the one attained by different state-of-the-art preconditioners designed for saddle-point linear systems stemming from data assimilation problems. In particular, we consider the scheme from Tabeart & Pearson (2021a) where a user-specified parameter k defines the approximation L −1 which is also used in the Schur complement approximation S = L T D −1 L; see (Tabeart & Pearson, 2021a, Section 4) for further details 2 .…”

“…As previously mentioned, we employ a matrix-oriented implementation of GMRES (and CG) whereas the standard vector form of GMRES is adopted whenever a preconditioning strategy different from the one introduced in this paper is considered. This is mainly due to the possibility of using existing code for the preconditioners in Freitag & Green (2018); Gratton et al (2018); Tabeart & Pearson (2021a). Indeed, these routines have been designed for standard GMRES and not for its matrix-oriented counterpart.…”

“…A variety of preconditioners for the saddle point 4D-Var problem have been proposed in the literature; see, e.g., Freitag & Green (2018); Gratton et al (2018); Tabeart & Pearson (2021a). While these operators guarantee parallisability, they also neglect important features of the original saddle point linear system to achieve affordable computational costs.…”