Structured interpolation for multivariate transfer functions of quadratic-bilinear systems
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866911794056921088 |
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| author | Benner, Peter Gugercin, Serkan Werner, Steffen W. R. |
| author_facet | Benner, Peter Gugercin, Serkan Werner, Steffen W. R. |
| contents | High-dimensional/high-fidelity nonlinear dynamical systems appear naturally when the goal is to accurately model real-world phenomena. Many physical properties are thereby encoded in the internal differential structure of these resulting large-scale nonlinear systems. The high-dimensionality of the dynamics causes computational bottlenecks, especially when these large-scale systems need to be simulated for a variety of situations such as different forcing terms. This motivates model reduction where the goal is to replace the full-order dynamics with accurate reduced-order surrogates. Interpolation-based model reduction has been proven to be an effective tool for the construction of cheap-to-evaluate surrogate models that preserve the internal structure in the case of weak nonlinearities. In this paper, we consider the construction of multivariate interpolants in frequency domain for structured quadratic-bilinear systems. We propose definitions for structured variants of the symmetric subsystem and generalized transfer functions of quadratic-bilinear systems and provide conditions for structure-preserving interpolation by projection. The theoretical results are illustrated using two numerical examples including the simulation of molecular dynamics in crystal structures. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_14292 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Structured interpolation for multivariate transfer functions of quadratic-bilinear systems Benner, Peter Gugercin, Serkan Werner, Steffen W. R. Numerical Analysis Systems and Control Dynamical Systems Optimization and Control 30E05, 34K17, 65D05, 93C10, 93A15 High-dimensional/high-fidelity nonlinear dynamical systems appear naturally when the goal is to accurately model real-world phenomena. Many physical properties are thereby encoded in the internal differential structure of these resulting large-scale nonlinear systems. The high-dimensionality of the dynamics causes computational bottlenecks, especially when these large-scale systems need to be simulated for a variety of situations such as different forcing terms. This motivates model reduction where the goal is to replace the full-order dynamics with accurate reduced-order surrogates. Interpolation-based model reduction has been proven to be an effective tool for the construction of cheap-to-evaluate surrogate models that preserve the internal structure in the case of weak nonlinearities. In this paper, we consider the construction of multivariate interpolants in frequency domain for structured quadratic-bilinear systems. We propose definitions for structured variants of the symmetric subsystem and generalized transfer functions of quadratic-bilinear systems and provide conditions for structure-preserving interpolation by projection. The theoretical results are illustrated using two numerical examples including the simulation of molecular dynamics in crystal structures. |
| title | Structured interpolation for multivariate transfer functions of quadratic-bilinear systems |
| topic | Numerical Analysis Systems and Control Dynamical Systems Optimization and Control 30E05, 34K17, 65D05, 93C10, 93A15 |
| url | https://arxiv.org/abs/2304.14292 |