Behavior of the iterative ensemble-based variational method in nonlinear problems

Mapping Intimacies ◽

10.5194/npg-2020-9 ◽

2020 ◽

Author(s):

Shin'ya Nakano

Keyword(s):

Data Assimilation ◽

Variational Method ◽

Objective Function ◽

Iterative Algorithm ◽

Linear Approximation ◽

Local Maximum ◽

Nonlinear Problems ◽

Variational Data Assimilation ◽

Monotonic Convergence ◽

System Nonlinearity

Abstract. The behavior of the iterative ensemble-based data assimilation algorithm is discussed. The ensemble-based method for variational data assimilation problems, referred to as the 4-dimensional ensemble variational method (4DEnVar), is a useful tool for data assimilation problems. Although the 4DEnVar is derived based on a linear approximation, highly uncertain problems, where system nonlinearity is significant, are solved by applying this method iteratively. However, it is not necessarily trivial how the algorithm works in highly uncertain problems where nonlinearity is not negligible. In the present study, an ensemble-based iterative algorithm is reformulated to allow us to analyze its behavior in nonlinear problems. The conditions for monotonic convergence to a local maximum of the objective function are discussed in nonlinear context. The findings as the results of the present study were also experimentally supported.

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Behavior of the iterative ensemble-based variational method in nonlinear problems

Nonlinear Processes in Geophysics ◽

10.5194/npg-28-93-2021 ◽

2021 ◽

Vol 28 (1) ◽

pp. 93-109

Author(s):

Shin'ya Nakano

Keyword(s):

Data Assimilation ◽

Variational Method ◽

Local Maximum ◽

Nonlinear Problems ◽

Dimensional Subspace ◽

High Dimensional ◽

Dimensional Approximation ◽

Monotonic Convergence ◽

System Nonlinearity ◽

Lower Dimensional

Abstract. The behavior of the iterative ensemble-based data assimilation algorithm is discussed. The ensemble-based method for variational data assimilation problems, referred to as the 4D ensemble variational method (4DEnVar), is a useful tool for data assimilation problems. Although the 4DEnVar is derived based on a linear approximation, highly uncertain problems, in which system nonlinearity is significant, are solved by applying this method iteratively. However, the ensemble-based methods basically seek the solution within a lower-dimensional subspace spanned by the ensemble members. It is not necessarily trivial how high-dimensional problems can be solved with the ensemble-based algorithm which employs the lower-dimensional approximation based on the ensemble. In the present study, an ensemble-based iterative algorithm is reformulated to allow us to analyze its behavior in high-dimensional nonlinear problems. The conditions for monotonic convergence to a local maximum of the objective function are discussed in a high-dimensional context. It is shown that the ensemble-based algorithm can solve high-dimensional problems by distributing the ensemble in different subspace at each iteration. The findings as the results of the present study were also experimentally supported.

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Optimal solution error covariance in highly nonlinear problems of variational data assimilation

Nonlinear Processes in Geophysics ◽

10.5194/npg-19-177-2012 ◽

2012 ◽

Vol 19 (2) ◽

pp. 177-184 ◽

Cited By ~ 8

Author(s):

V. Shutyaev ◽

I. Gejadze ◽

G. J. M. Copeland ◽

F.-X. Le Dimet

Keyword(s):

Nonlinear Dynamics ◽

Data Assimilation ◽

Optimal Solution ◽

Nonlinear Problems ◽

Variational Data Assimilation ◽

Model Parameters ◽

Efficient Computation ◽

Viscous Term ◽

Highly Nonlinear ◽

The Cost

Abstract. The problem of variational data assimilation (DA) for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition, boundary conditions and/or model parameters. The input data contain observation and background errors, hence there is an error in the optimal solution. For mildly nonlinear dynamics, the covariance matrix of the optimal solution error can be approximated by the inverse Hessian of the cost function. For problems with strongly nonlinear dynamics, a new statistical method based on the computation of a sample of inverse Hessians is suggested. This method relies on the efficient computation of the inverse Hessian by means of iterative methods (Lanczos and quasi-Newton BFGS) with preconditioning. Numerical examples are presented for the model governed by the Burgers equation with a nonlinear viscous term.

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