A note on preconditioning weighted linear least‐squares, with consequences for weakly constrained variational data assimilation

Gratton, Serge; Gürol, Selime; Simon, Ehouarn; Toint, Philippe L.

doi:10.1002/qj.3262

Cited by 9 publications

(13 citation statements)

References 16 publications

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“…M i and H i are the model and observation operators linearised at x i ; they are known as the tangent linear model and tangent linear observation operator, respectively. We define the following matrices (following Gratton et al, 2018a)…”

Section: Incremental Weak Constraint 4d-varmentioning

confidence: 99%

“…Fisher and Gürol could not find a suitable approximation that would guarantee good convergence. Gratton et al, (2018aGratton et al, ( ,2018b…”

Section: Preconditioningmentioning

confidence: 99%

“…

M_{i}

and

H_{i}

are the model and observation operators linearised at

x_{i}

; they are known as the tangent linear model and tangent linear observation operator, respectively. We define the following matrices (following Gratton et al ., 2018a)

\begin{align} {boldL}^{false(j false)} & = ((), \begin{matrix} center center center center centersubarray \\ array I \\ array - M_{0}^{(j)} & array I \\ array - M_{1}^{(j)} & array I \\ array ⋱ & array ⋱ \\ array - M_{N - 1}^{(j)} & array I \end{matrix}) \\ \in ℝ^{false(N + 1 false) n prefix\times false(N + 1 false) n}, \end{align}

\begin{align} {boldH}^{false(j false)} & = diag false({boldH}_{0}^{false(j false)}, {boldH}_{1}^{false(j false)}, \dots, {boldH}_{N}^{false(j false)} false) \in ℝ^{p prefix\times false(N + 1 false) n}, \end{align}

…”

Section: Incremental Weak Constraint 4d‐varmentioning

confidence: 99%

See 2 more Smart Citations

On time‐parallel preconditioning for the state formulation of incremental weak constraint 4D‐Var

Daužickaitė

Lawless

Scott

et al. 2021

Quart J Royal Meteoro Soc

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Using a high degree of parallelism is essential for the efficient performance of data assimilation. The state formulation of the incremental weak constraint four-dimensional variational data assimilation method allows parallel calculations in the time dimension. In this approach, the solution is approximated by minimising a series of quadratic cost functions using the conjugate gradient method. To use this method in practice, effective preconditioning strategies that maintain the potential for parallel calculations are needed. We examine approximations to the control variable transform (CVT) technique when the latter is beneficial. The new strategy employs a randomised singular value decomposition and retains the potential for parallelism in the time domain. Numerical results for the Lorenz '96 model show that this approach accelerates the minimisation in the first few iterations, with better results when CVT performs well. K E Y W O R D Sconjugate gradients, data assimilation, preconditioning, randomised methods, sparse symmetric positive definite systems, time-parallel 4D-Var, weak constraint 4D-VarThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Section: Incremental Weak Constraint 4d-varmentioning

confidence: 99%

“…Fisher and Gürol could not find a suitable approximation that would guarantee good convergence. Gratton et al, (2018aGratton et al, ( ,2018b…”

Section: Preconditioningmentioning

confidence: 99%

“…

M_{i}

and

H_{i}

are the model and observation operators linearised at

x_{i}

; they are known as the tangent linear model and tangent linear observation operator, respectively. We define the following matrices (following Gratton et al ., 2018a)

\begin{align} {boldL}^{false(j false)} & = ((), \begin{matrix} center center center center centersubarray \\ array I \\ array - M_{0}^{(j)} & array I \\ array - M_{1}^{(j)} & array I \\ array ⋱ & array ⋱ \\ array - M_{N - 1}^{(j)} & array I \end{matrix}) \\ \in ℝ^{false(N + 1 false) n prefix\times false(N + 1 false) n}, \end{align}

\begin{align} {boldH}^{false(j false)} & = diag false({boldH}_{0}^{false(j false)}, {boldH}_{1}^{false(j false)}, \dots, {boldH}_{N}^{false(j false)} false) \in ℝ^{p prefix\times false(N + 1 false) n}, \end{align}

…”

Section: Incremental Weak Constraint 4d‐varmentioning

confidence: 99%

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On time‐parallel preconditioning for the state formulation of incremental weak constraint 4D‐Var

Daužickaitė

Lawless

Scott

et al. 2021

Quart J Royal Meteoro Soc

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“…, R N ) ∈ R q ×q , where q = Σ N i =0 q i . We use the notation (following Gratton et al (2018)) +1) , where H (j ) i is the linearised observation operator, and…”

Section: Accepted Articlementioning

confidence: 99%

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

Daužickaitė

Lawless

Scott

et al. 2021

Quart J Royal Meteoro Soc

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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 quadratic cost functions, 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 construct LMPs cheaply 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.

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“…A vast amount of literature on saddle point problems and their solution via Krylov methods and preconditioners is available, see for example [21] and references therein. For this particular saddle point problem low-rank limited memory preconditioners exploiting the structure of the saddle point problem were proposed and analyzed in [73] (see also [91]). In the work [80,97] the Kronecker structure of the saddle point problem was used in order to compute low-rank solutions to GMRES.…”

Section: Weak Constraint 4d-var and Saddle Point Formulation Of The Imentioning

confidence: 99%

Numerical linear algebra in data assimilation

Freitag

2020

GAMM-Mitteilungen

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Data assimilation is a method that combines observations (ie, real world data) of a state of a system with model output for that system in order to improve the estimate of the state of the system and thereby the model output. The model is usually represented by a discretized partial differential equation. The data assimilation problem can be formulated as a large scale Bayesian inverse problem. Based on this interpretation we will derive the most important variational and sequential data assimilation approaches, in particular three‐dimensional and four‐dimensional variational data assimilation (3D‐Var and 4D‐Var) and the Kalman filter. We will then consider more advanced methods which are extensions of the Kalman filter and variational data assimilation and pay particular attention to their advantages and disadvantages. The data assimilation problem usually results in a very large optimization problem and/or a very large linear system to solve (due to inclusion of time and space dimensions). Therefore, the second part of this article aims to review advances and challenges, in particular from the numerical linear algebra perspective, within the various data assimilation approaches.

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A note on preconditioning weighted linear least‐squares, with consequences for weakly constrained variational data assimilation

Cited by 9 publications

References 16 publications

On time‐parallel preconditioning for the state formulation of incremental weak constraint 4D‐Var

On time‐parallel preconditioning for the state formulation of incremental weak constraint 4D‐Var

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

Numerical linear algebra in data assimilation

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