Selime Gürol

Serge Gratton, Selime Gürol, Ehouarn Simon et al.

The effect of preconditioning linear weighted least-squares using an approximation of the model matrix is analyzed, showing the interplay of the eigenstructures of both the model and weighting matrices. A small example is given illustrating the resulting potential inefficiency of such preconditioners. Consequences of these results in the context of the weakly-constrained 4D-Var data assimilation problem are finally discussed.

Jemima M. Tabeart, Selime Gürol, John W. Pearson et al.

Covariance matrices are central to data assimilation and inverse methods derived from statistical estimation theory. Previous work has considered the application of an all-at-once diffusion-based representation of a covariance matrix operator in order to exploit inherent parallelism in the underlying problem. In this paper, we provide practical methods to apply block $α$-circulant preconditioners to the all-at-once system for the case where the main diffusion operation matrix cannot be readily diagonalized using a discrete Fourier transform. Our new framework applies the block $α$-circulant preconditioner approximately by solving an inner block diagonal problem via a choice of inner iterative approaches. Our first method applies Chebyshev semi-iteration to a symmetric positive definite matrix, shifted by a complex scaling of the identity. We extend theoretical results for Chebyshev semi-iteration in the symmetric positive definite setting, to obtain computable bounds on the asymptotic convergence factor for each of the complex sub-problems. The second approach transforms the complex sub-problem into a (generalized) saddle point system with real coefficients. Numerical experiments reveal that in the case of unlimited computational resources, both methods can match the iteration counts of the `best-case' block $α$-circulant preconditioner. We also provide a practical adaptation to the nested Chebyshev approach, which improves performance in the case of a limited computational budget. Using an appropriate choice of $α$ our new approaches are robust and efficient in terms of outer iterations and matrix--vector products.

Mathis Peyron, Anthony Fillion, Selime Gürol et al.

Performing Data Assimilation (DA) at a low cost is of prime concern in Earth system modeling, particularly at the time of big data where huge quantities of observations are available. Capitalizing on the ability of Neural Networks techniques for approximating the solution of PDE's, we incorporate Deep Learning (DL) methods into a DA framework. More precisely, we exploit the latent structure provided by autoencoders (AEs) to design an Ensemble Transform Kalman Filter with model error (ETKF-Q) in the latent space. Model dynamics are also propagated within the latent space via a surrogate neural network. This novel ETKF-Q-Latent (thereafter referred to as ETKF-Q-L) algorithm is tested on a tailored instructional version of Lorenz 96 equations, named the augmented Lorenz 96 system: it possesses a latent structure that accurately represents the observed dynamics. Numerical experiments based on this particular system evidence that the ETKF-Q-L approach both reduces the computational cost and provides better accuracy than state of the art algorithms, such as the ETKF-Q.

Pierre Boudier, Anthony Fillion, Serge Gratton et al.

Data assimilation (DA) aims at forecasting the state of a dynamical system by combining a mathematical representation of the system with noisy observations taking into account their uncertainties. State of the art methods are based on the Gaussian error statistics and the linearization of the non-linear dynamics which may lead to sub-optimal methods. In this respect, there are still open questions how to improve these methods. In this paper, we propose a fully data driven deep learning architecture generalizing recurrent Elman networks and data assimilation algorithms which approximate a sequence of prior and posterior densities conditioned on noisy observations. By construction our approach can be used for general nonlinear dynamics and non-Gaussian densities. On numerical experiments based on the well-known Lorenz-95 system and with Gaussian error statistics, our architecture achieves comparable performance to EnKF on both the analysis and the propagation of probability density functions of the system state at a given time without using any explicit regularization technique.