Erratum to: Inexact constraint preconditioners for linear systems arising in interior point methods

Bergamaschi, Luca; Gondzio, Jacek; Venturin, Manolo; Zilli, Giovanni

doi:10.1007/s10589-009-9298-6

Cited by 6 publications

(8 citation statements)

References 2 publications

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“…P M being the inexact constraint preconditioner suggested in Bergamaschi et al. , (; ()) and used in Fisher and Gürol (), and P B and P T being the triangular and block‐diagonal ones inspired by Benzi and Wathen () (also Wathen ()). These inverses are given by

\begin{matrix} alignleft \\ align-1 {boldP}_{M}^{- 1} & align-2 = (\begin{matrix} bold0 & bold0 & {\overset{L}{false}}^{- 1} \\ bold0 & R^{- 1} & bold0 \\ {\overset{L}{false}}^{- T} & bold0 & - S^{- 1} \end{matrix}), \end{matrix}

\begin{matrix} alignleft \\ align-1 {boldP}_{B}^{- 1} & align-2 = (\begin{matrix} D^{- 1} & bold0 & bold0 \\ bold0 & R^{- 1} & bold0 \\ bold0 & bold0 & S^{- 1} \end{matrix}), \end{matrix}

\begin{matrix} alignleft \\ align-1 {boldP}_{T}^{- 1} & align-2 = (\begin{matrix} D^{- 1} & bold0 & - D^{- 1} \overset{L}{false} S^{- 1} \\ bold0 & R^{- 1} & - R^{- 1} boldH S^{- 1} \\ bold0 & bold0 & S^{- 1} \end{matrix}), \end{matrix}

and therefore both

…”

Section: Problem Formulations and Preconditioningmentioning

confidence: 99%

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

Gratton

Gürol

Simon

et al. 2018

Quart J Royal Meteoro Soc

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This paper discusses convergence issues for the saddle variational formulation of the weakly constrained 4D‐Var method in data assimilation, a method whose main interests are its parallelizable nature and its limited use of the inverse of the correlation matrices. It is shown that the method, in its original form, may produce erratic results or diverge because of the inherent lack of monotonicity of the produced objective function values. Convergent, variationally coherent variants of the algorithm are then proposed which largely retain the desirable features of the original proposal, and the circumstances in which these variants may be preferable to other approaches is briefly discussed.

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\begin{matrix} alignleft \\ align-1 {boldP}_{M}^{- 1} & align-2 = (\begin{matrix} bold0 & bold0 & {\overset{L}{false}}^{- 1} \\ bold0 & R^{- 1} & bold0 \\ {\overset{L}{false}}^{- T} & bold0 & - S^{- 1} \end{matrix}), \end{matrix}

\begin{matrix} alignleft \\ align-1 {boldP}_{B}^{- 1} & align-2 = (\begin{matrix} D^{- 1} & bold0 & bold0 \\ bold0 & R^{- 1} & bold0 \\ bold0 & bold0 & S^{- 1} \end{matrix}), \end{matrix}

\begin{matrix} alignleft \\ align-1 {boldP}_{T}^{- 1} & align-2 = (\begin{matrix} D^{- 1} & bold0 & - D^{- 1} \overset{L}{false} S^{- 1} \\ bold0 & R^{- 1} & - R^{- 1} boldH S^{- 1} \\ bold0 & bold0 & S^{- 1} \end{matrix}), \end{matrix}

and therefore both

…”

Section: Problem Formulations and Preconditioningmentioning

confidence: 99%

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

Gratton

Gürol

Simon

et al. 2018

Quart J Royal Meteoro Soc

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“…One approach that is of interest is the so‐called inexact constraint preconditioner of Bergamaschi et al . (2007) (see also Bergamaschi et al , 2011):

\overset{scriptP}{true} = ((), \begin{matrix} D & 0 & {\tilde{F}}^{- 1} \\ 0 & R & 0 \\ {({\tilde{F}}^{- 1})}^{T} & 0 & 0 \end{matrix}),

in which

\overset{boldF}{false}

approximates F .…”

Section: The Saddle‐point Formulation Of 4d‐varmentioning

confidence: 99%

Parallelization in the time dimension of four‐dimensional variational data assimilation

Fisher

Gürol

2017

Quart J Royal Meteoro Soc

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The current evolution of computer architectures towards increasing parallelism requires a corresponding evolution towards more parallel data assimilation algorithms. In this article, we consider parallelization of weak-constraint four-dimensional variational data assimilation (4D-Var) in the time dimension. We categorize algorithms according to whether or not they admit such parallelization and introduce a new, highly parallel weakconstraint 4D-Var algorithm based on a saddle-point representation of the underlying optimization problem. The potential benefits of the new saddle-point formulation are illustrated with a simple two-level quasi-geostrophic model.

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“…We start by considering two classes of preconditioner: the block diagonal preconditioner and the inexact constraint preconditioner. Although the standard inexact constraint approach [2,3] would include H, an approximation to H, the use of H = 0 is popular in the data assimilation setting [13,15,14]. In this paper we use the same forms that are considered in [15], which are given by…”

Section: Preconditioners For Saddle Point Systems From Data Assimilationmentioning

confidence: 99%

Saddle point preconditioners for weak-constraint 4D-Var

Tabeart¹,

Pearson²

2021

Preprint

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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.

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Erratum to: Inexact constraint preconditioners for linear systems arising in interior point methods

Cited by 6 publications

References 2 publications

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

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

Parallelization in the time dimension of four‐dimensional variational data assimilation

Saddle point preconditioners for weak-constraint 4D-Var

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