Low rank updates in preconditioning the saddle point systems arising from data assimilation problems

Abstract

The numerical solution of saddle point systems has received a lot of attention over the past few years in a wide variety of applications such as constrained optimization, computational fluid dynamics and optimal control, to name a few. In this paper, we focus on the saddle point formulation of a large-scale variational data assimilation problem, where the computations involving the constraint blocks are supposed to be much more expensive than those related to the (1, 1) block of the saddle point matrix. New low-rank limited memory preconditioners exploiting the particular structure of the problem are proposed and analysed theoretically. Numerical experiments performed within the Object-Oriented Prediction System are presented to highlight the relevance of the proposed preconditioners.

Guaranteeing the convergence of the saddle formulation for weakly constrained 4D‐Var data assimilation

“…For instance, experiments on the QG problem (Fisher et al. , ; Fisher and Gürol, ) reported in Gratton et al. s*() indicate that the original formulation may luckily produce the desired decrease in J , but also that the cost of using the safeguarded version discussed here is marginal.…”

“…The “saddle formulation”, discussed in Fisher et al. , (; ()) Fisher and Gürol (), has recently attracted interest of practioners because it minimizes the number of calls to B −1 and and because of its very appealing potential for parallel computing, while still allowing a wide choice of preconditioners. However it is fair to say that numerical experience with this approach remains scarse so far, prompting a more detailed assessment.…”

BFGS‐like updates of constraint preconditioners for sequences of KKT linear systems in quadratic programming

“…Limited‐memory quasi‐Newton updating techniques have been widely used to build preconditioners for sequences of linear systems, usually with slowly varying matrices (see, e.g., other works). However, to the best of our knowledge, the update of CPs via quasi‐Newton techniques has been considered only in the work of Fisher et al In this article, we present a preconditioner updating technique based on multiple BFGS‐like corrections that is tailored for CPs. The updated preconditioner still belongs to the class of exact CPs and hence allows the use of the CG method.…”

“…14 The idea of using a CP computed for a KKT system to obtain less expensive (inexact) CPs for subsequent KKT systems in a given sequence has been recently investigated in other works. [15][16][17] The procedure proposed by Bellavia et al 15 builds an inexact CP for a KKT system at a certain IP iteration by performing a low-rank update of the factorized Schur complement of the (1,1) block of a "seed" CP, that is, a CP computed at a previous IP iteration. The definition of the update is guided by theoretical results on the spectrum of the preconditioned matrix.…”

“…For KKT systems with nonzero (2,2) block, in the work of Bellavia et al, 16 the previous strategy is combined with a low-cost updating technique, 18,19 which is able to take into account information discarded by the low-rank correction and expressed as a diagonal modification of the preconditioner arising from the first update. Fisher et al 17 focused on sequences of KKT linear systems with varying off-diagonal blocks, where the computations with these blocks are much more expensive than the computations with the (1,1) block. Inexact CPs for the matrices of the sequence are built by applying generalizations of limited-memory quasi-Newton updates (see, e.g., other works 20,21 ) that act only on the off-diagonal blocks of a previously computed inexact CP of the type described in the work of Bergamaschi et al 12 Limited-memory quasi-Newton updating techniques have been widely used to build preconditioners for sequences of linear systems, usually with slowly varying matrices (see, e.g., other works 17,[22][23][24][25][26].…”