Positive operators and stable truncation

Benner, Peter; Damm, Tobias; Redmann, Martin; Cruz, Yolanda Rocio Rodriguez

doi:10.1016/j.laa.2014.12.005

Cited by 16 publications

(26 citation statements)

References 10 publications

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“…To prove the asymptotic mean square stability of the uncontrolled reduced order model it remains to show that the Kronecker matrix above has no eigenvalues on the imaginary axis. This was shown in [6]. Hence, we know that balanced truncation preserves asymptotic mean square stability, also in the stochastic case.…”

Section: Proposition 41 Letỹ Y 0 Be the Homogeneous Solution Satisfymentioning

confidence: 68%

“…The homogeneous equation of the truncated system is still asymptotically mean square stable due to [6]. Hence, the matrices…”

Section: Error Bound For Balanced Truncationmentioning

confidence: 98%

“…6 The finite reachability Gramians P t , t > 0, have the same image as the infinite reachability Gramian P, i.e.,…”

Section: Theorem 33mentioning

confidence: 99%

“…We can choose an orthonormal basis of U consisting of eigenvectors (u k ) k∈N of Q. 6 The corresponding eigenvalues we denote by (μ k ) k∈N such that…”

Section: Finite Dimensional Approximations For Stochastic Evolution Ementioning

confidence: 99%

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Model reduction for stochastic systems

Benner

Redmann

2015

Stoch PDE: Anal Comp

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To solve a stochastic linear evolution equation numerically, finite dimensional approximations are commonly used. If one uses the well-known Galerkin scheme, one might end up with a sequence of ordinary stochastic linear equations of high order. To reduce the high dimension for practical computations we consider balanced truncation as a model order reduction technique. This approach is well-known from deterministic control theory and successfully employed in practice for decades. So, we generalize balanced truncation for controlled linear systems with Levy noise, discuss properties of the reduced order model, provide an error bound, and give some examples.

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Section: Proposition 41 Letỹ Y 0 Be the Homogeneous Solution Satisfymentioning

confidence: 68%

“…The homogeneous equation of the truncated system is still asymptotically mean square stable due to [6]. Hence, the matrices…”

Section: Error Bound For Balanced Truncationmentioning

confidence: 98%

“…6 The finite reachability Gramians P t , t > 0, have the same image as the infinite reachability Gramian P, i.e.,…”

Section: Theorem 33mentioning

confidence: 99%

“…We can choose an orthonormal basis of U consisting of eigenvectors (u k ) k∈N of Q. 6 The corresponding eigenvalues we denote by (μ k ) k∈N such that…”

Section: Finite Dimensional Approximations For Stochastic Evolution Ementioning

confidence: 99%

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Model reduction for stochastic systems

Benner

Redmann

2015

Stoch PDE: Anal Comp

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“…We suppose that Assumptions 1-5 hold for all > 0. As we will show in Appendix A, the results in [11] can be modified to show that the matricesÃ 11 andÃ 22 are Hurwitz, and that their eigenvalues are bounded away from the imaginary axis. In this case, BIBO stability of the system together with the assumptions on the admissible controls imply that z 2 → 0 pointwise for all t > 0 as → 0.…”

Section: 2mentioning

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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Full state approximation by Galerkin projection reduced order models for stochastic and bilinear systems

Redmann

Duff

2022

Applied Mathematics and Computation

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Positive operators and stable truncation

Cited by 16 publications

References 10 publications

Model reduction for stochastic systems

Model reduction for stochastic systems

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

Full state approximation by Galerkin projection reduced order models for stochastic and bilinear systems

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