A time-parallel approach to strong-constraint four-dimensional variational data assimilation

Rao, Vishwas; Sandu, Adrian

doi:10.1016/j.jcp.2016.02.040

Cited by 24 publications

(17 citation statements)

References 33 publications

(69 reference statements)

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“…The simultaneous usage of SMDEIM Jacobians and DEIM nonlinear terms has to be investigated in the context of adjoint models and optimization framework, because some inconsistencies may arise. Coupling SMDEIM models with a time‐parallel approach may lead to a very efficient reduced strong‐constraint four‐dimensional variational data assimilation system.…”

Section: Discussionmentioning

confidence: 99%

Efficient approximation of Sparse Jacobians for time‐implicit reduced order models

Ştefanescu

Sandu

2016

Numerical Methods in Fluids

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Summary This paper introduces a sparse matrix discrete interpolation method to effectively compute matrix approximations in the reduced order modeling framework. The sparse algorithm developed herein relies on the discrete empirical interpolation method and uses only samples of the nonzero entries of the matrix series. The proposed approach can approximate very large matrices, unlike the current matrix discrete empirical interpolation method, which is limited by its large computational memory requirements. The empirical interpolation indices obtained by the sparse algorithm slightly differ from the ones computed by the matrix discrete empirical interpolation method as a consequence of the singular vectors round‐off errors introduced by the economy or full singular value decomposition (SVD) algorithms when applied to the full matrix snapshots. When appropriately padded with zeros, the economy SVD factorization of the nonzero elements of the snapshots matrix is a valid economy SVD for the full snapshots matrix. Numerical experiments are performed with the 1D Burgers and 2D shallow water equations test problems where the quadratic reduced nonlinearities are computed via tensorial calculus. The sparse matrix approximation strategy is compared against five existing methods for computing reduced Jacobians: (i) matrix discrete empirical interpolation method, (ii) discrete empirical interpolation method, (iii) tensorial calculus, (iv) full Jacobian projection onto the reduced basis subspace, and (v) directional derivatives of the model along the reduced basis functions. The sparse matrix method outperforms all other algorithms. The use of traditional matrix discrete empirical interpolation method is not possible for very large dimensions because of its excessive memory requirements. Copyright © 2016 John Wiley & Sons, Ltd.

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Section: Discussionmentioning

confidence: 99%

Efficient approximation of Sparse Jacobians for time‐implicit reduced order models

Ştefanescu

Sandu

2016

Numerical Methods in Fluids

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“…We should mention that the block-diagonal matrices (14) and (15) are not the only possible way to extend error covariances to the combined spaces R np and R mp . For instance, the off-block elements can be nonzero, which enables possibility to account for cross-time correlations for the errors within data assimilation window.…”

Section: Parallel Filtering Taskmentioning

confidence: 99%

“…

Kalman filter is a sequential estimation scheme that combines predicted and observed data to reduce the uncertainty of the next prediction. Another algorithm, suggested in [14], extends numerical solution of the strong-constraint 4D-VAR formulation effectively enabling possibilities for parallel implementation. In this paper, we attempt to address pitfalls of the earlier low-memory approach described in and extend it for parallel implementation.

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confidence: 99%

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Parallel implementation of data assimilation

Bibov

Haario

2016

Numerical Methods in Fluids

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Summary Kalman filter is a sequential estimation scheme that combines predicted and observed data to reduce the uncertainty of the next prediction. Because of its sequential nature, the algorithm cannot be efficiently implemented on modern parallel compute hardware nor can it be practically implemented on large‐scale dynamical systems because of memory issues. In this paper, we attempt to address pitfalls of the earlier low‐memory approach described in and extend it for parallel implementation. First, we describe a low‐memory method that enables one to pack covariance matrix data employed by the Kalman filter into a low‐memory form by means of certain quasi‐Newton approximation. Second, we derive parallel formulation of the filtering task, which allows to compute several filter iterations independently. Furthermore, this leads to an improvement of estimation quality as the method takes into account the cross‐correlations between consequent system states. We experimentally demonstrate this improvement by comparing the suggested algorithm with the other data assimilation methods that can benefit from parallel implementation. Copyright © 2016 John Wiley & Sons, Ltd.

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“…In particular, we show how the implicit diffusion problem can be reformulated in a way that allows the costly matrix–vector products involved in the M ‐step (pseudo‐time) diffusion processes to be performed in parallel on each iteration of the CI solver. The approach falls into the general class of time‐parallel algorithms, whose development is an active area of research in numerical modelling ((Gander, ) gives a review) and more recently in four‐dimensional data assimilation (Rao and Sandu, ; Fisher and Gürol, ). The time‐parallel formulation of implicit diffusion presented in this paper builds on an idea described in Zhu () (chapter 5), which first involves recasting the sequential M ‐step IDO in terms of a single non‐symmetric positive‐definite (NSPD) linear system.…”

Section: Introductionmentioning

confidence: 99%

“Time”‐Parallel diffusion‐based correlation operators

Weaver

Gürol

Tshimanga

et al. 2018

Quart J Royal Meteoro Soc

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Correlation operators based on the solution of an implicitly formulated diffusion equation can be implemented numerically using the Chebyshev iteration method. The attractive properties of the algorithm for modelling correlation functions on high‐performance computers have been discussed in a recent paper. The current paper describes a straightforward variant of that algorithm that allows the matrix–vector products involved in the sequential pseudo‐time diffusion process to be performed in parallel. Contrary to the original algorithm, which requires solving, in sequence, linear systems involving a symmetric positive‐definite (SPD) matrix, the “time”‐parallel algorithm requires solving a single linear system involving a non‐symmetric positive‐definite (NSPD) matrix. The key information required by the Chebyshev iteration for solving the NSPD problem is an estimate of the extreme eigenvalues of the NSPD matrix. For the problem under consideration, the extreme eigenvalues of the NSPD matrix are the same as those of the original SPD matrix, and can be pre‐computed using a Lanczos algorithm applied to the latter. The convergence properties of the algorithm are studied from a theoretical perspective and using numerical experiments with a diffusion‐based covariance model in a variational data assimilation system for the global ocean. Results suggest that time‐parallelization can reduce the run‐time of an implicit diffusion‐based correlation operator by greater than a factor of two. It can be implemented practically using a hybrid parallelization approach that combines Message Passing Interface tasks in the spatial domain with Open Multi‐Processing threads spanning the pseudo‐time dimension. The sensitivity of the results to preconditioning, to the choice of first guess and to the stopping criterion is discussed.

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A time-parallel approach to strong-constraint four-dimensional variational data assimilation

Cited by 24 publications

References 33 publications

Efficient approximation of Sparse Jacobians for time‐implicit reduced order models

Efficient approximation of Sparse Jacobians for time‐implicit reduced order models

Parallel implementation of data assimilation

“Time”‐Parallel diffusion‐based correlation operators

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