A reduced and limited‐memory preconditioned approach for the 4D‐Var data‐assimilation problem

Gratton, Serge; Laloyaux, Patrick; Sartenaer, Annick; Tshimanga, Jean

doi:10.1002/qj.743

Cited by 7 publications

(10 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, in an incremental 4D‐Var assimilation system, an approximate Hessian derived from a similar methodology could be used to precondition the CG minimization of the inner loops. In particular, the LAST_DIAG_CYC preconditioner could be used in the quasi‐Newton LMP approach proposed by Tshimanga et al () and Gratton et al () to further improve the convergence of the inner loops. Another interesting application of the BFGS method in variational data assimilation was suggested by Fisher and Courtier ().…”

Section: Discussionmentioning

confidence: 99%

“…Note that the idea of using information from multiple inversions to improve the inverse Hessian estimate is not new. For example, the LMP described in Tshimanga et al () and Gratton et al (), when used in the context of incremental 4D‐Var data assimilation, can construct a BFGS approximation of the inverse Hessian from the sequence of quadratic CG minimizations of the outer loops, gradually improving the preconditioning of the systems. In the BFGS_HYBRID approach, the ensemble of minimization problems is obtained through random perturbations of the prior and observations in the 4D‐Var cost function.…”

Section: Numerical Experimentsmentioning

confidence: 99%

See 1 more Smart Citation

Improved analysis‐error covariance matrix for high‐dimensional variational inversions: application to source estimation using a 3D atmospheric transport model

Bousserez

Henze

Perkins

et al. 2015

Quart J Royal Meteoro Soc

View full text Add to dashboard Cite

Variational methods are widely used to solve geophysical inverse problems. Although gradient-based minimization algorithms are available for high-dimensional problems (dimension > 10 6 ), they do not provide an estimate of the errors in the optimal solution. In this study, we assess the performance of several numerical methods to approximate the analysis-error covariance matrix, assuming reasonably linear models. The evaluation is performed for a CO 2 flux estimation problem using synthetic remote-sensing observations of CO 2 columns. A low-dimensional experiment is considered in order to compare the analysis error approximations to a full-rank finite-difference inverse Hessian estimate, followed by a realistic high-dimensional application. Two stochastic approaches, a MonteCarlo simulation and a method based on random gradients of the cost function, produced analysis error variances with a relative error < 10%. The long-distance error correlations due to sampling noise are significantly less pronounced for the gradient-based randomization, which is also particularly attractive when implemented in parallel. Deterministic evaluations of the inverse Hessian using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm are also tested. While existing BFGS preconditioning techniques yield poor approximations of the error variances (relative error > 120%), a new preconditioner that efficiently accumulates information on the diagonal of the inverse Hessian dramatically improves the results (relative error < 50%). Furthermore, performing several cycles of the BFGS algorithm using the same gradient and vector pairs enhances its performance (relative error < 30%) and is necessary to obtain convergence. Leveraging those findings, we proposed a BFGS hybrid approach which combines the new preconditioner with several BFGS cycles using information from a few (3-5) Monte-Carlo simulations. Its performance is comparable to the stochastic approximations for the low-dimensional case, while good scalability is obtained for the high-dimensional experiment. Potential applications of these new BFGS methods range from characterizing the information content of high-dimensional inverse problems to improving the convergence rate of current minimization algorithms.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

Improved analysis‐error covariance matrix for high‐dimensional variational inversions: application to source estimation using a 3D atmospheric transport model

Bousserez

Henze

Perkins

et al. 2015

Quart J Royal Meteoro Soc

View full text Add to dashboard Cite

show abstract

“…Gratton et al . () propose, for instance, to compute an approximate initial guess for the inner linear systems in a reduced space given by a spectral decomposition of the initial Hessian operator.…”

Section: Discussionmentioning

confidence: 99%

Full waveform inversion and the truncated Newton method: quantitative imaging of complex subsurface structures

Métivier

Bretaudeau

Brossier

et al. 2014

Geophysical Prospecting

147

View full text Add to dashboard Cite

Full waveform inversion is a powerful tool for quantitative seismic imaging from wide‐azimuth seismic data. The method is based on the minimization of the misfit between observed and simulated data. This amounts to the solution of a large‐scale nonlinear minimization problem. The inverse Hessian operator plays a crucial role in this reconstruction process. Accounting accurately for the effect of this operator within the minimization scheme should correct for illumination deficits, restore the amplitude of the subsurface parameters, and help to remove artefacts generated by energetic multiple reflections. Conventional minimization methods (nonlinear conjugate gradient, quasi‐Newton methods) only roughly approximate the effect of this operator. In this study, we are interested in the truncated Newton minimization method. These methods are based on the computation of the model update through a matrix‐free conjugate gradient solution of the Newton linear system. We present a feasible implementation of this method for the full waveform inversion problem, based on a second‐order adjoint state formulation for the computation of Hessian‐vector products. We compare this method with conventional methods within the context of 2D acoustic frequency full waveform inversion for the reconstruction of P‐wave velocity models. Two test cases are investigated. The first is the synthetic BP 2004 model, representative of the Gulf of Mexico geology with high velocity contrasts associated with the presence of salt structures. The second is a 2D real data‐set from the Valhall oil field in North sea. Although, from a computational cost point of view, the truncated Newton method appears to be more expensive than conventional optimization algorithms, the results emphasize its increased robustness. A better reconstruction of the P‐wave velocity model is provided when energetic multiple reflections make it difficult to interpret the seismic data. A better trade‐off between regularization and resolution is obtained when noise contamination of the data requires one to regularize the solution of the inverse problem.

show abstract

“…With a linear operator H , solving Equation is equivalent to solving the Kalman filter equation (Gratton et al. , ) and the solution of Equation is given by

x^{a} = x^{b} + boldB H^{T} {((), boldH boldB H^{T} + boldR)}^{- 1} ((), boldy - boldH x^{b}) .

…”

Section: Background‐error Covariance In Data Assimilationmentioning

confidence: 99%

Implicit and explicit cross‐correlations in coupled data assimilation

Laloyaux

Фролов

Ménétrier

et al. 2018

Quart J Royal Meteoro Soc

View full text Add to dashboard Cite

The European Centre for Medium-Range Weather Forecasts has recently developed an implicit coupling approach in the coupled atmosphere-ocean assimilation (CERA) system, where atmospheric 4D-Var and ocean 3D-Var are synchronized using multiple outer iterations in the incremental variational formulation. Since this original work on outer-loop coupling, it has been unclear just how closely the CERA system with implicit coupled covariances approximates a strongly coupled data assimilation with ensemble-based coupled covariances. A series of single-observation experiments in a perfect twin framework has been conducted to evaluate the effectiveness and efficiency of outer-loop coupling and the amplitude of the implicit cross-correlations between atmospheric and ocean temperature. They are compared against a coupled assimilation system that specifies cross-correlations explicitly from an ensemble of coupled forecasts. We find that both the implicit outer-loop and explicit ensemble-based methods produce equally skilful estimates of the coupled state in the middle of a 24 hr assimilation window. Our estimates of the rate of outer-loop convergence suggest that atmospheric and ocean states synchronize within the first 6-10 hr of the assimilation window. We conclude that outer-loop coupling is effective when the window is long enough that original imbalances in the atmospheric and ocean increments can synchronize within the length of the assimilation window. On the other hand, we suggest that explicit coupling is preferable for data assimilation systems with short assimilation windows (e.g. 6 hr or less).

show abstract

A reduced and limited‐memory preconditioned approach for the 4D‐Var data‐assimilation problem

Cited by 7 publications

References 37 publications

Improved analysis‐error covariance matrix for high‐dimensional variational inversions: application to source estimation using a 3D atmospheric transport model

Improved analysis‐error covariance matrix for high‐dimensional variational inversions: application to source estimation using a 3D atmospheric transport model

Full waveform inversion and the truncated Newton method: quantitative imaging of complex subsurface structures

Implicit and explicit cross‐correlations in coupled data assimilation

Contact Info

Product

Resources

About