Analytical Jacobian-vector products for the matrix-free time integration of partial differential equations

Glandon, S. Ross; Sarshar, Arash; Sandu, Adrian

doi:10.1016/j.cam.2016.05.002

Cited by 12 publications

(14 citation statements)

References 13 publications

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“…In this section we test the several implementations of the new BOROK methods on a set of problems. As the base ROK methods (and other K-methods) have been previously compared against other standard methods (see [27,[47][48][49][50]), we restrict ourselves to comparing BOROK only against the base ROK methods and compatible variations.…”

Section: Numerical Resultsmentioning

confidence: 99%

Biorthogonal Rosenbrock-Krylov time discretization methods

Glandon

Sandu

2020

Applied Numerical Mathematics

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Many scientific applications require the solution of large initial-value problems, such as those produced by the method of lines after semi-discretization in space of partial differential equations. The computational cost of implicit time discretizations is dominated by the solution of nonlinear systems of equations at each time step. In order to decrease this cost, the recently developed Rosenbrock-Krylov (ROK) time integration methods extend the classical linearly-implicit Rosenbrock(-W) methods, and make use of a Krylov subspace approximation to the Jacobian computed via an Arnoldi process. Since the ROK order conditions rely on the construction of a single Krylov space, no restarting of the Arnoldi process is allowed, and the iterations quickly become expensive with increasing subspace dimensions. This work extends the ROK framework to make use of the Lanczos biorthogonalization procedure for constructing Jacobian approximations. The resulting new family of methods is named biorthogonal ROK (BOROK). The Lanczos procedure's short two-term recurrence allows BOROK methods to utilize larger subspaces for the Jacobian approximation, resulting in increased numerical stability of the time integration at a reduced computational cost. Adaptive subspace size selection and basis extension procedures are also developed for the new schemes. Numerical experiments show that for stiff problems, where a large subspace used to approximate the Jacobian is required for stability, the BOROK methods outperform the original ROK methods.

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Section: Numerical Resultsmentioning

confidence: 99%

Biorthogonal Rosenbrock-Krylov time discretization methods

Glandon

Sandu

2020

Applied Numerical Mathematics

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“…Similar to some 4-D ensemble approaches, the gradient computations can be approximated by the tangent linear model with the adjoint not being required. In fact all that is required is matrix vector products which can be approximated with finite differences (Tranquilli et al, 2017).…”

Section: Adaptive Localization Via a Time-distributed Cost Functionmentioning

confidence: 99%

A Bayesian approach to multivariate adaptive localization in ensemble-based data assimilation with time-dependent extensions

Popov¹,

Sandu²

2019

Nonlin. Processes Geophys.

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Abstract. Ever since its inception, the ensemble Kalman filter (EnKF) has elicited many heuristic approaches that sought to improve it. One such method is covariance localization, which alleviates spurious correlations due to finite ensemble sizes by using relevant spatial correlation information. Adaptive localization techniques account for how correlations change in time and space, in order to obtain improved covariance estimates. This work develops a Bayesian approach to adaptive Schur-product localization for the deterministic ensemble Kalman filter (DEnKF) and extends it to support multiple radii of influence. We test the proposed adaptive localization using the toy Lorenz'96 problem and a more realistic 1.5-layer quasi-geostrophic model. Results with the toy problem show that the multivariate approach informs us that strongly observed variables can tolerate larger localization radii. The univariate approach leads to markedly improved filter performance for the realistic geophysical model, with a reduction in error by as much as 33 %.

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“…Similar to all other 4D ensemble approaches, this requires only the computation of the tangent linear model in order to derive the gradient; an adjoint model is not required. In fact all that is required is matrix vector products which can be approximated with finite differences (Tranquilli et al, 2017).…”

Section: Adaptive Localization Via a Time-distributed Cost Functionmentioning

confidence: 99%

“…The inflation factor is kept constant, and we test α values from 1.02 to 1.16 (represented on the x-axis). The constant localization radius varies in increments of 5 over the range[5,45]. Only the best results are plotted, and are used as the mean seeds for the adaptive algorithm.…”

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confidence: 99%

A Bayesian Approach to Multivariate Adaptive Localization in Ensemble-Based Data Assimilation with Time-Dependent Extensions

Popov¹,

Sandu²

2018

Preprint

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Abstract. Ever since its inception, the Ensemble Kalman Filter has elicited many heuristic methods that sought to correct it. One such method is localization – the thought that nearby variables should be highly correlated with far away variable not. Recognizing that correlation is a time-dependent property, adaptive localization is a natural extension to these heuristics. We propose a Bayesian approach to adaptive Schur-product localization for the DEnKF, and extend it to support multiple radii of influence. We test both the empirical validity of (multivariate) adaptive localization, and of our approach. We test a simple toy problem (Lorenz '96), extending it to a multivariate model, and a more realistic geophysical problem (1.5 Layer Quasi-Geostrophic). We show that the multivariate approach has great promise on the toy problem, and that the univariate approach leads to improved filter performance for the realistic geophysical problem.

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Analytical Jacobian-vector products for the matrix-free time integration of partial differential equations

Cited by 12 publications

References 13 publications

Biorthogonal Rosenbrock-Krylov time discretization methods

Biorthogonal Rosenbrock-Krylov time discretization methods

A Bayesian approach to multivariate adaptive localization in ensemble-based data assimilation with time-dependent extensions

A Bayesian Approach to Multivariate Adaptive Localization in Ensemble-Based Data Assimilation with Time-Dependent Extensions

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