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

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

doi:10.1002/qj.3355

Cited by 14 publications

(44 citation statements)

References 35 publications

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“…An active research topic in this area is the weak constraint four-dimensional variational (4D-Var) data assimilation method. [8][9][10][11][12][13][14] It is employed in the search for states of the system over a time period, called the assimilation window. This method uses a cost function that is formulated under the assumption that the numerical model is not perfect and penalizes the weighted discrepancy between the analysis and the observations, the analysis and the background state, and the difference between the analysis and the trajectory given by integrating the dynamical model.…”

Section: Introductionmentioning

confidence: 99%

Spectral estimates for saddle point matrices arising in weak constraint four‐dimensional variational data assimilation

Daužickaitė

Lawless

Scott

et al. 2020

Numerical Linear Algebra App

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Summary We consider the large sparse symmetric linear systems of equations that arise in the solution of weak constraint four‐dimensional variational data assimilation, a method of high interest for numerical weather prediction. These systems can be written as saddle point systems with a 3 × 3 block structure but block eliminations can be performed to reduce them to saddle point systems with a 2 × 2 block structure, or further to symmetric positive definite systems. In this article, we analyse how sensitive the spectra of these matrices are to the number of observations of the underlying dynamical system. We also obtain bounds on the eigenvalues of the matrices. Numerical experiments are used to confirm the theoretical analysis and bounds.

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Section: Introductionmentioning

confidence: 99%

Spectral estimates for saddle point matrices arising in weak constraint four‐dimensional variational data assimilation

Daužickaitė

Lawless

Scott

et al. 2020

Numerical Linear Algebra App

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“…One should also remember that our analysis merely provides bounds on the conditioning, which are pessimistic by nature, and that the observation term H T R −1 H (which we ignored here) may not always be negligible. The situation is therefore often problem-dependent, as has been demonstrated in Gratton et al (2017b) where very different behaviours (good and bad) were observed for two contrasting data assimilation problems.…”

Section: Application To Weakly Constrained Data Assimilationmentioning

confidence: 96%

“…While practitioners have been aware of the difficulty for some time (e.g. Fisher and Gürol, ; ; Gratton et al ; T. Janjić and Y. Zhang, Personal communication, 2017), a formal analysis, and hence a complete understanding, has been missing so far. A first step in this direction was made by Braess and Peisker (), where they showed (in a slighly different context) that, if A is square, symmetric and positive‐definite, and if W is the identity matrix, then preconditioning A 2 (which corresponds to unweighted symmetric least‐squares) using the square of an approximation of A as a preconditioner might lead to a situation worse than not preconditioning at all, unless A and its preconditioner commute.…”

Section: Introductionmentioning

confidence: 99%

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

Gratton

Gürol

Simon

et al. 2018

Quart J Royal Meteoro Soc

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The effect of preconditioning linear weighted least‐squares using an approximation of the model matrix is analyzed. The aim is to investigate from a theoretical point of view the inefficiencies of this approach as observed in the application of the weakly constrained 4D‐Var algorithm in geosciences. Bounds on the eigenvalues of the preconditioned system matrix are provided. It highlights the interplay of the eigenstructures of both the model and weighting matrices: maintaining a low bound on the eigenvalues of the preconditioned system matrix requires an approximation error of the model matrix which compensates for the condition number of the weighting matrix. A low‐dimension analytical example is given illustrating the resulting potential inefficiency of such preconditioners. The consequences of these results in the context of the state formulation of the weakly constrained 4D‐Var data assimilation problem are discussed. It is shown that the common approximations of the tangent linear model which maintain parallelization‐in‐time properties (identity or null matrix) can result in large bounds on the eigenvalues of the preconditioned matrix system.

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“…Spatial parallelization of variational DA techniques has long been considered (e.g., Trémolet and Dimet, ; Rantakokko, ; Elbern and Schmidt, ). Moreover, methods have been developed to exploit time parallelism in 4D‐Var through subdivision of the assimilation time window (Rao and Sandu, ) or through the saddle‐point formulation in the weak‐constraint algorithm (Fisher and Gürol, ), although the convergence of the latter cannot always be guaranteed (Gratton et al ., ). Most recently, Mercier et al .…”

Section: Introductionmentioning

confidence: 97%

Enhanced parallelization of the incremental 4D‐Var data assimilation algorithm using the Randomized Incremental Optimal Technique

Bousserez

Guerrette

Henze

2020

Quart J Royal Meteoro Soc

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Incremental 4D-Var is a data assimilation algorithm used routinely at operational numerical weather prediction (NWP) centres worldwide. The algorithm solves a series of quadratic minimization problems (inner-loops) obtained from linear approximations of the forward model around nonlinear trajectories (outer-loops). Since most of the computational burden is associated with the inner-loops, many studies have focused on developing computationally efficient algorithms to solve the least-square quadratic minimization problem, in particular through time parallelization. This paper presents the first implementation and testing of a recently proposed method for parallelizing incremental 4D-Var, the Randomized Incremental Optimal Technique (RIOT), which replaces the traditional sequential conjugate gradient (CG) iterations in the inner-loop of the minimization with fully parallel randomized singular value decomposition (RSVD) of the preconditioned Hessian of the cost function. RIOT is tested using the standard Lorenz-96 model (L-96) as well as two realistic high-dimensional atmospheric source inversion problems based on aircraft observations of black carbon concentrations. A new outer-loop preconditioning technique tailored to RSVD was introduced to improve convergence stability and performance. Results obtained with the L-96 system show that the performance improvement from RIOT compared to standard CG algorithms increases significantly with nonlinearities. Overall, in the realistic black carbon source inversion experiments, RIOT reduces the wall-clock time of the 4D-Var minimization by a factor of 2 to 3, at the cost of a factor of 4 to 10 increase in energy cost due to the large number of parallel cores used. Furthermore, RIOT enables reduction of the wall-clock time computation of the analysis-error covariance matrix by a factor of 40 compared to a standard iterative Lanczos approach. Finally, as evidenced in this study, implementation of RIOT in an operational NWP system will require a better understanding of its convergence properties as a function of the Hessian characteristics and, in particular, the degree of freedom for signal of the inverse problem.

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

Cited by 14 publications

References 35 publications

Spectral estimates for saddle point matrices arising in weak constraint four‐dimensional variational data assimilation

Spectral estimates for saddle point matrices arising in weak constraint four‐dimensional variational data assimilation

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

Enhanced parallelization of the incremental 4D‐Var data assimilation algorithm using the Randomized Incremental Optimal Technique

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