State estimation using model order reduction for unstable systems

Boess, C.; Lawless, Amos S.; Nichols, Nancy K.; Bunse-Gerstner, Angelika

doi:10.1016/j.compfluid.2010.11.033

Cited by 13 publications

(11 citation statements)

References 10 publications

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“…If A has a simple eigenvalue zero, as in our case, there exists an α > 0, such that the matrix A − αI is Hurwitz. For linear systems the shifting can be interpreted as a discounting of the controllability and observability functionals that renders the associated Gramians finite [15].…”

Section: Sparsity Preserving Projectionmentioning

confidence: 99%

Model reduction of controlled Fokker–Planck and Liouville–von Neumann equations

Benner

Breiten

Hartmann

et al. 2020

Journal of Computational Dynamics

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Model reduction methods for bilinear control systems are compared by means of practical examples of Liouville-von Neumann and Fokker-Planck type. Methods based on balancing generalized system Gramians and on minimizing an H 2 -type cost functional are considered. The focus is on the numerical implementation and a thorough comparison of the methods. Structure and stability preservation are investigated, and the competitiveness of the approaches is shown for practically relevant, large-scale examples.

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Section: Sparsity Preserving Projectionmentioning

confidence: 99%

Model reduction of controlled Fokker–Planck and Liouville–von Neumann equations

Benner

Breiten

Hartmann

et al. 2020

Journal of Computational Dynamics

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“…In [71] it is assumed that the time-dependent system underlying the problem has a time-invariant dominant part on which balanced truncation is performed. MOR via balanced truncation was also proposed for incremental 4D-Var in [25,26,124,125]. Model and observation operators are projected as in (34), x is restricted tox = U T x ∈ R r , r ≪ n, and the background error covariance matrix, B, is projected onto U T BU.…”

Section: Model Order Reduction Applied To the Forward Model Operatormentioning

confidence: 99%

Numerical linear algebra in data assimilation

Freitag

2020

GAMM-Mitteilungen

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Data assimilation is a method that combines observations (ie, real world data) of a state of a system with model output for that system in order to improve the estimate of the state of the system and thereby the model output. The model is usually represented by a discretized partial differential equation. The data assimilation problem can be formulated as a large scale Bayesian inverse problem. Based on this interpretation we will derive the most important variational and sequential data assimilation approaches, in particular three‐dimensional and four‐dimensional variational data assimilation (3D‐Var and 4D‐Var) and the Kalman filter. We will then consider more advanced methods which are extensions of the Kalman filter and variational data assimilation and pay particular attention to their advantages and disadvantages. The data assimilation problem usually results in a very large optimization problem and/or a very large linear system to solve (due to inclusion of time and space dimensions). Therefore, the second part of this article aims to review advances and challenges, in particular from the numerical linear algebra perspective, within the various data assimilation approaches.

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“…In this method the probability density function of the observation errors are assumed to be a weighted combination of a standard Gaussian distribution and a flat probability distribution function, with the weights determined by the probability of gross error of the observation. Thus for each single observation y with weight α y , the probability density function of the observation error is assumed to be of the form 14) where P N indicates the appropriate Gaussian probability density function and P F is a flat distribution over a finite interval centred at zero and is equal to zero outside this interval (the size of this interval is taken to be a multiple of the observation error standard deviation). The observation part of the cost function is then taken to be equal to the negative logarithm of P QC .…”

Section: Observation Errorsmentioning

confidence: 99%

“…In this method the snapshot perturbations X are weighted according the sensitivity of the cost function at the time of the snapshot, where the weights are calculated using the adjoint model [30]. The other approach, put forward in the series of papers [76], [75], [14], is to use near-optimal model order reduction methods for linear dynamical systems to derive a reduced order model and observation operator. The inner loop problem of incremental 4D-Var (2.15) is subject to the dynamical system described by the evolution equation (216) and the output equation…”

Section: Reduced Order Approachesmentioning

confidence: 99%

“…The reduction step then calculates the restriction and prolongation operators from 35) where the decay of the Hankel singular values is used to choose the model reduction order r. In idealised models the studies [76], [75], [14] show how this method improves the solution with respect to using low resolution models and how it is important to use information about the assimilation problem in the reduction procedure, including information about the background and observation error covariance matrices. However, whereas reduction methods based on POD can be implemented in large systems, the method of balanced truncation cannot.…”

Section: Reduced Order Approachesmentioning

confidence: 99%

See 1 more Smart Citation

Variational data assimilation for very large environmental problems

Lawless¹

2013

Large Scale Inverse Problems

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Variational data assimilation is commonly used in environmental forecasting to estimate the current state of the system from a model forecast and observational data. The assimilation problem can be written simply in the form of a nonlinear least squares optimization problem. However the practical solution of the problem in large systems requires many careful choices to be made in the implementation. In this article we present the theory of variational data assimilation and then discuss in detail how it is implemented in practice. Current solutions and open questions are discussed.

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State estimation using model order reduction for unstable systems

Cited by 13 publications

References 10 publications

Model reduction of controlled Fokker–Planck and Liouville–von Neumann equations

Model reduction of controlled Fokker–Planck and Liouville–von Neumann equations

Numerical linear algebra in data assimilation

Variational data assimilation for very large environmental problems

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