Cross-Gramian-based dominant subspaces

Benner, Peter

doi:10.1007/s10444-019-09724-7

Cited by 13 publications

(8 citation statements)

References 43 publications

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“…The dominant subspaces (DS) method, constructs a Galerkin projection by combining the dominant controllability and observability subspaces [41], obtained from the respective (empirical) Gramians; while the variant based on the (empirical) cross Gramian is introduced in [8]. The column-rank of the projection is then determined by the conjoined and orthogonalized singular vectors of the system Gramians, weighted by their associated singular values.…”

Section: Empirical Dominant Subspacesmentioning

confidence: 99%

“…As an orthogonal projection, DS is stability preserving for dissipative systems. Furthermore, a Hardy-2 error bound exists for the controllability and observability Gramianbased DS [50] (in two variants), while a Lebesgue-2 error indicator is introduced in [8] for the cross-Gramian-based DS. To obtain and conjoin the system Gramians' singular vectors, various algorithms are available, here, we use the truncated SVDs and rank-revealing SVDs for this task.…”

Section: Empirical Dominant Subspacesmentioning

confidence: 99%

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Comparing (Empirical-Gramian-Based) Model Order Reduction Algorithms

Himpe¹

2020

Preprint

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Section: Empirical Dominant Subspacesmentioning

confidence: 99%

Section: Empirical Dominant Subspacesmentioning

confidence: 99%

Comparing (Empirical-Gramian-Based) Model Order Reduction Algorithms

Himpe¹

2020

Preprint

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“…A truncated SVD of the cross Gramian also engenders the sought dominant subspaces [15]: The (empirical) cross Gramian's left and right singular vectors (approximately) span the reachability and observability subspaces, respectively, and their orthogonalized concatenation, via a truncated SVD,…”

Section: Cross-gramian-basedmentioning

confidence: 99%

“…A closure is a pair of a function together with its scoped environment. 10 https://mathworks.com/help/matlab/ref/ode23s.html (accessed: 2020-[11][12][13][14][15][16][17][18] …”

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confidence: 99%

Model Order Reduction for Gas and Energy Networks

Himpe,

Grundel,

Benner

2020

Preprint

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To counter the volatile nature of renewable energy sources, gas networks take a vital role. But, to ensure fulfillment of contracts under these new circumstances, a vast number of possible scenarios, incorporating uncertain supply and demand, has to be simulated ahead of time. This many-query task can be accelerated by model order reduction, yet, large-scale, nonlinear, parametric, hyperbolic partial differential(-algebraic) equation systems, modeling gas transport, are a challenging application for model reduction algorithms. For this industrial application, we bring together the scientific computing topics of: mathematical modeling of gas transport networks, numerical simulation of hyperbolic partial differential equation, and model order reduction for nonlinear parametric systems. This research resulted in the morgen (Model Order Reduction for Gas and Energy Networks) software platform, which enables modular testing of various combinations of models, solvers, and model reduction methods. In this work we present the theoretical background on systemic modeling and structured, data-driven, system-theoretic model reduction for gas networks, as well as the implementation of morgen and associated numerical experiments testing model reduction adapted to gas network models.

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“…The model reduction techniques POD, BT and DSPMR, based on computation via (3) and (6) are compared in the resulting relative output model reduction error ε MOR = y − y r L2 / y L2 for different reduced orders. In Figure 1, POD, BT and (cross-Gramian-based) DSPMR [5], computed via matrix equations and empirically, are compared, whereas the empirical system Gramians are computed using the empirical Gramian framework emgr [6]. The reduced order models (ROMs) are tested using a smooth input signal which is not included in the training set for the empirical Gramians, encompassing step functions, and the first order Euler's method with suitable time discretization is used to generate the trajectory data.…”

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confidence: 99%

On Empirical System Gramians

Grundel

Himpe

Saak

2019

Proc Appl Math and Mech

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State-space realizations of input-output systems or control systems are a widely used class of models in engineering, physics, chemistry and biology. For the qualitative and quantitative classification of such systems, the system-theoretic properties of reachability and observability are essential, which are encoded in so-called system Gramian matrices. For linear systems these Gramians are computed as solutions to matrix equations, for nonlinear or parametric systems the data-driven empirical system Gramians approximate the actual system Gramians. These empirical Gramians have manifold applications, for example in model reduction or decentralized control of nonlinear systems, as well as sensitivity analysis, parameter identification and combined state and parameter reduction of parametric systems. Here, we demonstrate that empirical system Gramians are also useful for linear but hyperbolic input-output systems.While empirical system Gramians are designed for nonlinear and parametric systems, a data-driven computation can be advantageous even for linear systems (1). This is demonstrated using a principal application of (empirical) system Gramians: Model reduction. Gramian-based model reduction computes (Petrov-)Galerkin projection operators

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Cross-Gramian-based dominant subspaces

Cited by 13 publications

References 43 publications

Comparing (Empirical-Gramian-Based) Model Order Reduction Algorithms

Comparing (Empirical-Gramian-Based) Model Order Reduction Algorithms

Model Order Reduction for Gas and Energy Networks

On Empirical System Gramians

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