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Benjamin Unger

Dr.

Dynamical Systems
Independent Junior Research Group Leader
[Photo: SimTech/Max Kovalenko]

Contact

+49 711 685 60114
Email
Website

Universitätsstraße 32
70569 Stuttgart
Deutschland
Room: 02.324

Office Hours

Consultation by appointment

Subject

Model order reduction
Data-driven surrogate models
Delay differential-algebraic equations
Port-Hamiltonian systems
Digital Twins and Physics-informed machine learning

Publications

2023
1. V. Mehrmann and B. Unger, “Control of port-Hamiltonian differential-algebraic systems and applications,” Acta Numerica, vol. 32, pp. 395–515, 2023, doi: 10.1017/S0962492922000083.
  - Abstract
  - BibTeX
  Abstract
  We discuss the modelling framework of port-Hamiltonian descriptor systems and their use in numerical simulation and control. The structure is ideal for automated network-based modelling since it is invariant under power-conserving interconnection, congruence transformations and Galerkin projection. Moreover, stability and passivity properties are easily shown. Condensed forms under orthogonal transformations present easy analysis tools for existence, uniqueness, regularity and numerical methods to check these properties. After recalling the concepts for general linear and nonlinear descriptor systems, we demonstrate that many difficulties that arise in general descriptor systems can be easily overcome within the port-Hamiltonian framework. The properties of port-Hamiltonian descriptor systems are analysed, and time discretization and numerical linear algebra techniques are discussed. Structure-preserving regularization procedures for descriptor systems are presented to make them suitable for simulation and control. Model reduction techniques that preserve the structure and stabilization and optimal control techniques are discussed. The properties of port-Hamiltonian descriptor systems and their use in modelling simulation and control methods are illustrated with several examples from different physical domains. The survey concludes with open problems and research topics that deserve further attention.
  BibTeX
  @article{MehU23, abstract = {We discuss the modelling framework of port-Hamiltonian descriptor systems and their use in numerical simulation and control. The structure is ideal for automated network-based modelling since it is invariant under power-conserving interconnection, congruence transformations and Galerkin projection. Moreover, stability and passivity properties are easily shown. Condensed forms under orthogonal transformations present easy analysis tools for existence, uniqueness, regularity and numerical methods to check these properties. After recalling the concepts for general linear and nonlinear descriptor systems, we demonstrate that many difficulties that arise in general descriptor systems can be easily overcome within the port-Hamiltonian framework. The properties of port-Hamiltonian descriptor systems are analysed, and time discretization and numerical linear algebra techniques are discussed. Structure-preserving regularization procedures for descriptor systems are presented to make them suitable for simulation and control. Model reduction techniques that preserve the structure and stabilization and optimal control techniques are discussed. The properties of port-Hamiltonian descriptor systems and their use in modelling simulation and control methods are illustrated with several examples from different physical domains. The survey concludes with open problems and research topics that deserve further attention.}, author = {Mehrmann, Volker and Unger, Benjamin}, booktitle = {Acta Numerica}, doi = {10.1017/S0962492922000083}, issn = {09624929}, pages = {395-515}, publisher = {Cambridge University Press}, title = {Control of port-Hamiltonian differential-algebraic systems and applications}, url = {https://www.cambridge.org/core/article/control-of-porthamiltonian-differentialalgebraic-systems-and-applications/102B84AD17404344FAC406DC2A40D284}, volume = 32, year = 2023 }
2. T. Holicki, J. Nicodemus, P. Schwerdtner, and B. Unger, “Energy matching in reduced passive and port-Hamiltonian systems,” 2023.
  - Abstract
  - BibTeX
  Abstract
  It is well known that any port-Hamiltonian (pH) system is passive, and conversely, any minimal and stable passive system has a pH representation. Nevertheless, this equivalence is only concerned with the input-output mapping but not with the Hamiltonian itself. Thus, we propose to view a pH system either as an enlarged dynamical system with the Hamiltonian as additional output or as two dynamical systems with the input-output and the Hamiltonian dynamic. Our first main result is a structure-preserving Kalman-like decomposition of the enlarged pH system that separates the controllable and zero-state observable parts. Moreover, for further approximations in the context of structure-preserving model-order reduction (MOR), we propose to search for a Hamiltonian in the reduced pH system that minimizes the 2-distance to the full-order Hamiltonian without altering the input-output dynamic, thus discussing a particular aspect of the corresponding multi-objective minimization problem corresponding to $H_2$-optimal MOR for pH systems. We show that this optimization problem is uniquely solvable, can be recast as a standard semidefinite program, and present two numerical approaches for solving it. The results are illustrated with three academic examples.
  BibTeX
  @article{holicki2023energy, abstract = {It is well known that any port-Hamiltonian (pH) system is passive, and conversely, any minimal and stable passive system has a pH representation. Nevertheless, this equivalence is only concerned with the input-output mapping but not with the Hamiltonian itself. Thus, we propose to view a pH system either as an enlarged dynamical system with the Hamiltonian as additional output or as two dynamical systems with the input-output and the Hamiltonian dynamic. Our first main result is a structure-preserving Kalman-like decomposition of the enlarged pH system that separates the controllable and zero-state observable parts. Moreover, for further approximations in the context of structure-preserving model-order reduction (MOR), we propose to search for a Hamiltonian in the reduced pH system that minimizes the 2-distance to the full-order Hamiltonian without altering the input-output dynamic, thus discussing a particular aspect of the corresponding multi-objective minimization problem corresponding to $\mathcal{H}_2$-optimal MOR for pH systems. We show that this optimization problem is uniquely solvable, can be recast as a standard semidefinite program, and present two numerical approaches for solving it. The results are illustrated with three academic examples.}, archiveprefix = {arXiv}, author = {Holicki, Tobias and Nicodemus, Jonas and Schwerdtner, Paul and Unger, Benjamin}, eprint = {2309.05778}, primaryclass = {math.OC}, title = {Energy matching in reduced passive and port-Hamiltonian systems}, year = 2023 }
3. R. Morandin, J. Nicodemus, and B. Unger, “Port-Hamiltonian Dynamic Mode Decomposition,” SIAM Journal on Scientific Computing, vol. 45, Art. no. 4, Jul. 2023, doi: 10.1137/22m149329x.
  - Abstract
  - BibTeX
  Abstract
  We present a novel physics-informed system identification method to construct a passive linear time-invariant system. In more detail, for a given quadratic energy functional, measurements of the input, state, and output of a system in the time domain, we find a realization that approximates the data well while guaranteeing that the energy functional satisfies a dissipation inequality. To this end, we use the framework of port-Hamiltonian (pH) systems and modify the dynamic mode decomposition, respectively, operator inference, to be feasible for continuous-time pH systems. We propose an iterative numerical method to solve the corresponding least-squares minimization problem. We construct an effective initialization of the algorithm by studying the least-squares problem in a weighted norm, for which we present the analytical minimum-norm solution. The efficiency of the proposed method is demonstrated with several numerical examples.
  BibTeX
  @article{MorNU23, abstract = {We present a novel physics-informed system identification method to construct a passive linear time-invariant system. In more detail, for a given quadratic energy functional, measurements of the input, state, and output of a system in the time domain, we find a realization that approximates the data well while guaranteeing that the energy functional satisfies a dissipation inequality. To this end, we use the framework of port-Hamiltonian (pH) systems and modify the dynamic mode decomposition, respectively, operator inference, to be feasible for continuous-time pH systems. We propose an iterative numerical method to solve the corresponding least-squares minimization problem. We construct an effective initialization of the algorithm by studying the least-squares problem in a weighted norm, for which we present the analytical minimum-norm solution. The efficiency of the proposed method is demonstrated with several numerical examples.}, author = {Morandin, Riccardo and Nicodemus, Jonas and Unger, Benjamin}, doi = {10.1137/22m149329x}, journal = {SIAM Journal on Scientific Computing}, month = {07}, number = 4, pages = {A1690-A1710}, publisher = {Society for Industrial & Applied Mathematics (SIAM)}, title = {Port-Hamiltonian Dynamic Mode Decomposition}, volume = 45, year = 2023 }
2022
1. F. Black, P. Schulze, and B. Unger, “Modal Decomposition of Flow Data via Gradient-Based Transport Optimization,” in Active Flow and Combustion Control 2021, R. King and D. Peitsch, Eds., Cham: Springer International Publishing, 2022, pp. 203–224. doi: 10.1007/978-3-030-90727-3_13.
  - Abstract
  - BibTeX
  Abstract
  In the context of model reduction, we study an optimization problem related to the approximation of given data by a linear combination of transformed modes, called transformed proper orthogonal decomposition (tPOD). In the simplest case, the optimization problem reduces to a minimization problem well-studied in the context of proper orthogonal decomposition. Allowing transformed modes in the approximation renders this approach particularly useful to compress data with transported quantities, which are prevalent in many flow applications. We prove the existence of a solution to the infinite-dimensional optimization problem. Towards a numerical implementation, we compute the gradient of the cost functional and derive a suitable discretization in time and space. We demonstrate the theoretical findings with three numerical examples using a periodic shift operator as transformation.
  BibTeX
  @inproceedings{BlaSU22, abstract = {In the context of model reduction, we study an optimization problem related to the approximation of given data by a linear combination of transformed modes, called transformed proper orthogonal decomposition (tPOD). In the simplest case, the optimization problem reduces to a minimization problem well-studied in the context of proper orthogonal decomposition. Allowing transformed modes in the approximation renders this approach particularly useful to compress data with transported quantities, which are prevalent in many flow applications. We prove the existence of a solution to the infinite-dimensional optimization problem. Towards a numerical implementation, we compute the gradient of the cost functional and derive a suitable discretization in time and space. We demonstrate the theoretical findings with three numerical examples using a periodic shift operator as transformation.}, address = {Cham}, author = {Black, Felix and Schulze, Philipp and Unger, Benjamin}, booktitle = {Active Flow and Combustion Control 2021}, doi = {10.1007/978-3-030-90727-3_13}, editor = {King, Rudibert and Peitsch, Dieter}, pages = {203-224}, publisher = {Springer International Publishing}, title = {Modal Decomposition of Flow Data via Gradient-Based Transport Optimization}, year = 2022 }
2. V. Mehrmann and B. Unger, “Control of port-Hamiltonian differential-algebraic systems and applications,” ArXiv e-print 2201.06590, 2022, [Online]. Available: http://arxiv.org/abs/2201.06590
  - Abstract
  - BibTeX
  Abstract
  The modeling framework of port-Hamiltonian descriptor systems and their use in numerical simulation and control are discussed. The structure is ideal for automated network-based modeling since it is invariant under power-conserving interconnection, congruence transformations, and Galerkin projection. Moreover, stability and passivity properties are easily shown. Condensed forms under orthogonal transformations present easy analysis tools for existence, uniqueness, regularity, and numerical methods to check these properties. After recalling the concepts for general linear and nonlinear descriptor systems, we demonstrate that many difficulties that arise in general descriptor systems can be easily overcome within the port-Hamiltonian framework. The properties of port-Hamiltonian descriptor systems are analyzed, time-discretization, and numerical linear algebra techniques are discussed. Structure-preserving regularization procedures for descriptor systems are presented to make them suitable for simulation and control. Model reduction techniques that preserve the structure and stabilization and optimal control techniques are discussed. The properties of port-Hamiltonian descriptor systems and their use in modeling simulation and control methods are illustrated with several examples from different physical domains. The survey concludes with open problems and research topics that deserve further attention.
  BibTeX
  @article{MehU22-ppt, abstract = {The modeling framework of port-Hamiltonian descriptor systems and their use in numerical simulation and control are discussed. The structure is ideal for automated network-based modeling since it is invariant under power-conserving interconnection, congruence transformations, and Galerkin projection. Moreover, stability and passivity properties are easily shown. Condensed forms under orthogonal transformations present easy analysis tools for existence, uniqueness, regularity, and numerical methods to check these properties. After recalling the concepts for general linear and nonlinear descriptor systems, we demonstrate that many difficulties that arise in general descriptor systems can be easily overcome within the port-Hamiltonian framework. The properties of port-Hamiltonian descriptor systems are analyzed, time-discretization, and numerical linear algebra techniques are discussed. Structure-preserving regularization procedures for descriptor systems are presented to make them suitable for simulation and control. Model reduction techniques that preserve the structure and stabilization and optimal control techniques are discussed. The properties of port-Hamiltonian descriptor systems and their use in modeling simulation and control methods are illustrated with several examples from different physical domains. The survey concludes with open problems and research topics that deserve further attention.}, author = {Mehrmann, Volker and Unger, Benjamin}, journal = {ArXiv e-print 2201.06590}, pubdate = {2022-01-19}, title = {Control of port-{H}amiltonian differential-algebraic systems and applications}, url = {http://arxiv.org/abs/2201.06590}, year = 2022 }
3. T. Breiten, D. Hinsen, and B. Unger, “Towards a modeling class for port-Hamiltonian systems with time-delay,” 2022. doi: 10.48550/arXiv.2211.10687.
  - Abstract
  - BibTeX
  Abstract
  The framework of port-Hamiltonian (pH) systems is a powerful and broadly applicable modeling paradigm. In this paper, we extend the scope of pH systems to time-delay systems. Our definition of a delay pH system is motivated by investigating the Kalman-Yakubovich-Popov inequality on the corresponding infinite-dimensional operator equation. Moreover, we show that delay pH systems are passive and closed under interconnection. We describe an explicit way to construct a Lyapunov-Krasovskii functional and discuss implications for delayed feedback.
  BibTeX
  @techreport{breiten2022modeling, abstract = {The framework of port-Hamiltonian (pH) systems is a powerful and broadly applicable modeling paradigm. In this paper, we extend the scope of pH systems to time-delay systems. Our definition of a delay pH system is motivated by investigating the Kalman-Yakubovich-Popov inequality on the corresponding infinite-dimensional operator equation. Moreover, we show that delay pH systems are passive and closed under interconnection. We describe an explicit way to construct a Lyapunov-Krasovskii functional and discuss implications for delayed feedback.}, archiveprefix = {arXiv}, author = {Breiten, Tobias and Hinsen, Dorothea and Unger, Benjamin}, doi = {10.48550/arXiv.2211.10687}, eprint = {2211.10687}, primaryclass = {math.DS}, title = {Towards a modeling class for port-Hamiltonian systems with time-delay}, year = 2022 }
4. D. Frank, D. A. Latif, M. Muehlebach, B. Unger, and S. Staab, “Robust Recurrent Neural Network to Identify Ship Motion in Open Water with Performance Guarantees,” Publication, 2022. doi: 10.48550/arXiv.2212.05781.
  - Abstract
  - BibTeX
  Abstract
  Recurrent neural networks are capable of learning the dynamics of an unknown nonlinear system purely from input-output measurements. However, the resulting models do not provide any stability guarantees on the input-output mapping. In this work, we represent a recurrent neural network as a linear time-invariant system with nonlinear disturbances. By introducing constraints on the parameters, we can guarantee finite gain stability and incremental finite gain stability. We apply this identification method to learn the motion of a four-degrees-of-freedom ship that is moving in open water and compare it against other purely learning-based approaches with unconstrained parameters. Our analysis shows that the constrained recurrent neural network has a lower prediction accuracy on the test set, but it achieves comparable results on an out-of-distribution set and respects stability conditions.
  BibTeX
  @techreport{frank2022robust, abstract = {Recurrent neural networks are capable of learning the dynamics of an unknown nonlinear system purely from input-output measurements. However, the resulting models do not provide any stability guarantees on the input-output mapping. In this work, we represent a recurrent neural network as a linear time-invariant system with nonlinear disturbances. By introducing constraints on the parameters, we can guarantee finite gain stability and incremental finite gain stability. We apply this identification method to learn the motion of a four-degrees-of-freedom ship that is moving in open water and compare it against other purely learning-based approaches with unconstrained parameters. Our analysis shows that the constrained recurrent neural network has a lower prediction accuracy on the test set, but it achieves comparable results on an out-of-distribution set and respects stability conditions.}, archiveprefix = {arXiv}, author = {Frank, Daniel and Latif, Decky Aspandi and Muehlebach, Michael and Unger, Benjamin and Staab, Steffen}, doi = {10.48550/arXiv.2212.05781}, eprint = {2212.05781}, primaryclass = {cs.LG}, title = {Robust Recurrent Neural Network to Identify Ship Motion in Open Water with Performance Guarantees}, type = {Publication}, year = 2022 }
5. J. Heiland and B. Unger, “Identification of Linear Time-Invariant Systems with Dynamic Mode Decomposition,” Mathematics, vol. 10, Art. no. 3, 2022, doi: 10.3390/math10030418.
  - Abstract
  - BibTeX
  Abstract
  Dynamic mode decomposition (DMD) is a popular data-driven framework to extract linear dynamics from complex high-dimensional systems. In this work, we study the system identification properties of DMD. We first show that DMD is invariant under linear transformations in the image of the data matrix. If, in addition, the data are constructed from a linear time-invariant system, then we prove that DMD can recover the original dynamics under mild conditions. If the linear dynamics are discretized with the Runge–Kutta method, then we further classify the error of the DMD approximation and detail that for one-stage Runge–Kutta methods; even the continuous dynamics can be recovered with DMD. A numerical example illustrates the theoretical findings.
  BibTeX
  @article{HeiU22, abstract = {Dynamic mode decomposition (DMD) is a popular data-driven framework to extract linear dynamics from complex high-dimensional systems. In this work, we study the system identification properties of DMD. We first show that DMD is invariant under linear transformations in the image of the data matrix. If, in addition, the data are constructed from a linear time-invariant system, then we prove that DMD can recover the original dynamics under mild conditions. If the linear dynamics are discretized with the Runge–Kutta method, then we further classify the error of the DMD approximation and detail that for one-stage Runge–Kutta methods; even the continuous dynamics can be recovered with DMD. A numerical example illustrates the theoretical findings.}, author = {Heiland, Jan and Unger, Benjamin}, doi = {10.3390/math10030418}, journal = {Mathematics}, number = 3, pages = 418, publisher = {{MDPI} {AG}}, title = {Identification of Linear Time-Invariant Systems with Dynamic Mode Decomposition}, url = {https://doi.org/10.3390%2Fmath10030418}, volume = 10, year = 2022 }
6. B. Hillebrecht and B. Unger, “Certified machine learning: A posteriori error estimation for physics-informed neural networks,” ArXiv e-print 2203.17055, 2022, [Online]. Available: http://arxiv.org/abs/2203.17055
  - Abstract
  - BibTeX
  Abstract
  Physics-informed neural networks (PINNs) are one popular approach to introduce a priori knowledge about physical systems into the learning framework. PINNs are known to be robust for smaller training sets, derive better generalization problems, and are faster to train. In this paper, we show that using PINNs in comparison with purely data-driven neural networks is not only favorable for training performance but allows us to extract significant information on the quality of the approximated solution. Assuming that the underlying differential equation for the PINN training is an ordinary differential equation, we derive a rigorous upper limit on the PINN prediction error. This bound is applicable even for input data not included in the training phase and without any prior knowledge about the true solution. Therefore, our a posteriori error estimation is an essential step to certify the PINN. We apply our error estimator exemplarily to two academic toy problems, whereof one falls in the category of model-predictive control and thereby shows the practical use of the derived results.
  BibTeX
  @article{HilU22, abstract = {Physics-informed neural networks (PINNs) are one popular approach to introduce a priori knowledge about physical systems into the learning framework. PINNs are known to be robust for smaller training sets, derive better generalization problems, and are faster to train. In this paper, we show that using PINNs in comparison with purely data-driven neural networks is not only favorable for training performance but allows us to extract significant information on the quality of the approximated solution. Assuming that the underlying differential equation for the PINN training is an ordinary differential equation, we derive a rigorous upper limit on the PINN prediction error. This bound is applicable even for input data not included in the training phase and without any prior knowledge about the true solution. Therefore, our a posteriori error estimation is an essential step to certify the PINN. We apply our error estimator exemplarily to two academic toy problems, whereof one falls in the category of model-predictive control and thereby shows the practical use of the derived results.}, author = {Hillebrecht, Birgit and Unger, Benjamin}, journal = {ArXiv e-print 2203.17055}, pubdate = {2022-03-31}, title = {Certified machine learning: A posteriori error estimation for physics-informed neural networks}, url = {http://arxiv.org/abs/2203.17055}, year = 2022 }
7. B. Hillebrecht and B. Unger, “Certified machine learning: Rigorous a posteriori error bounds for PDE defined PINNs,” 2022.
  - Abstract
  - BibTeX
  Abstract
  Prediction error quantification in machine learning has been left out of most methodological investigations of neural networks, for both purely data-driven and physics-informed approaches. Beyond statistical investigations and generic results on the approximation capabilities of neural networks, we present a rigorous upper bound on the prediction error of physics-informed neural networks. This bound can be calculated without the knowledge of the true solution and only with a priori available information about the characteristics of the underlying dynamical system governed by a partial differential equation. We apply this a posteriori error bound exemplarily to four problems: the transport equation, the heat equation, the Navier-Stokes equation and the Klein-Gordon equation.
  BibTeX
  @techreport{hillebrecht2022certified, abstract = {Prediction error quantification in machine learning has been left out of most methodological investigations of neural networks, for both purely data-driven and physics-informed approaches. Beyond statistical investigations and generic results on the approximation capabilities of neural networks, we present a rigorous upper bound on the prediction error of physics-informed neural networks. This bound can be calculated without the knowledge of the true solution and only with a priori available information about the characteristics of the underlying dynamical system governed by a partial differential equation. We apply this a posteriori error bound exemplarily to four problems: the transport equation, the heat equation, the Navier-Stokes equation and the Klein-Gordon equation.}, archiveprefix = {arXiv}, author = {Hillebrecht, Birgit and Unger, Benjamin}, eprint = {2210.03426}, primaryclass = {cs.LG}, title = {Certified machine learning: Rigorous a posteriori error bounds for PDE defined PINNs}, year = 2022 }
8. T. Breiten and B. Unger, “Passivity preserving model reduction via spectral factorization,” Automatica, vol. 142, p. 110368, Aug. 2022, doi: 10.1016/j.automatica.2022.110368.
  - Abstract
  - BibTeX
  Abstract
  We present a novel model-order reduction (MOR) method for linear time-invariant systems that preserves passivity and is thus suited for structure-preserving MOR for port-Hamiltonian (pH) systems. Our algorithm exploits the well-known spectral factorization of the Popov function by a solution of the Kalman–Yakubovich–Popov (KYP) inequality. It performs MOR directly on the spectral factor inheriting the original system’s sparsity enabling MOR in a large-scale context. Our analysis reveals that the spectral factorization corresponding to the minimal solution of an associated algebraic Riccati equation is preferable from a model reduction perspective and benefits pH-preserving MOR methods such as a modified version of the iterative rational Krylov algorithm (IRKA). Numerical examples demonstrate that our approach can produce high-fidelity reduced-order models close to (unstructured) $H_2$-optimal reduced-order models.
  BibTeX
  @article{BreU22, abstract = {We present a novel model-order reduction (MOR) method for linear time-invariant systems that preserves passivity and is thus suited for structure-preserving MOR for port-Hamiltonian (pH) systems. Our algorithm exploits the well-known spectral factorization of the Popov function by a solution of the Kalman–Yakubovich–Popov (KYP) inequality. It performs MOR directly on the spectral factor inheriting the original system’s sparsity enabling MOR in a large-scale context. Our analysis reveals that the spectral factorization corresponding to the minimal solution of an associated algebraic Riccati equation is preferable from a model reduction perspective and benefits pH-preserving MOR methods such as a modified version of the iterative rational Krylov algorithm (IRKA). Numerical examples demonstrate that our approach can produce high-fidelity reduced-order models close to (unstructured) $\mathcal{H}_2$-optimal reduced-order models.}, author = {Breiten, Tobias and Unger, Benjamin}, doi = {10.1016/j.automatica.2022.110368}, journal = {Automatica}, month = {08}, pages = 110368, preprinturl = {https://doi.org/10.48550/arXiv.2103.13194}, publisher = {Elsevier}, title = {Passivity preserving model reduction via spectral factorization}, url = {https://doi.org/10.1016/j.automatica.2022.110368}, volume = 142, year = 2022 }
9. B. Hillebrecht and B. Unger, “Certified machine learning: A posteriori error estimation for physics-informed neural networks,” in 2022 International Joint Conference on Neural Networks (IJCNN), IEEE, Jul. 2022, pp. 1–8. doi: 10.1109/ijcnn55064.2022.9892569.
  - Abstract
  - BibTeX
  Abstract
  Physics-informed neural networks (PINNs) are one popular approach to incorporate a priori knowledge about physical systems into the learning framework. PINNs are known to be robust for smaller training sets, derive better generalization problems, and are faster to train. In this paper, we show that using PINNs in comparison with purely data-driven neural networks is not only favorable for training performance but allows us to extract significant information on the quality of the approximated solution. Assuming that the underlying differential equation for the PINN training is an ordinary differential equation, we derive a rigorous upper limit on the PINN prediction error. This bound is applicable even for input data not included in the training phase and without any prior knowledge about the true solution. Therefore, our a posteriori error estimation is an essential step to certify the PINN. We apply our error estimator exemplarily to two academic toy problems, whereof one falls in the category of model-predictive control and thereby shows the practical use of the derived results.
  BibTeX
  @inproceedings{HillU22a, abstract = {Physics-informed neural networks (PINNs) are one popular approach to incorporate a priori knowledge about physical systems into the learning framework. PINNs are known to be robust for smaller training sets, derive better generalization problems, and are faster to train. In this paper, we show that using PINNs in comparison with purely data-driven neural networks is not only favorable for training performance but allows us to extract significant information on the quality of the approximated solution. Assuming that the underlying differential equation for the PINN training is an ordinary differential equation, we derive a rigorous upper limit on the PINN prediction error. This bound is applicable even for input data not included in the training phase and without any prior knowledge about the true solution. Therefore, our a posteriori error estimation is an essential step to certify the PINN. We apply our error estimator exemplarily to two academic toy problems, whereof one falls in the category of model-predictive control and thereby shows the practical use of the derived results.}, author = {Hillebrecht, Birgit and Unger, Benjamin}, booktitle = {2022 International Joint Conference on Neural Networks (IJCNN)}, doi = {10.1109/ijcnn55064.2022.9892569}, month = {07}, pages = {1-8}, preprinturl = {https://doi.org/10.48550/arXiv.2203.17055}, publisher = {IEEE}, title = {Certified machine learning: A posteriori error estimation for physics-informed neural networks}, url = {https://doi.org/10.1109/ijcnn55064.2022.9892569}, year = 2022 }
10. J. Nicodemus, J. Kneifl, J. Fehr, and B. Unger, “Physics-informed Neural Networks-based Model Predictive Control for Multi-link Manipulators,” IFAC-PapersOnLine, vol. 55, Art. no. 20, 2022, doi: 10.1016/j.ifacol.2022.09.117.
  - Abstract
  - BibTeX
  Abstract
  We discuss nonlinear model predictive control (MPC) for multi-body dynamics via physics-informed machine learning methods. In more detail, we use a physics-informed neural networks (PINNs)-based MPC to solve a tracking problem for a complex mechanical system, a multi-link manipulator. PINNs are a promising tool to approximate (partial) differential equations but are not suited for control tasks in their original form since they are not designed to handle variable control actions or variable initial values. We thus follow the strategy of Antonelo et al. (arXiv:2104.02556, 2021) by enhancing PINNs with adding control actions and initial conditions as additional network inputs. Subsequently, the high-dimensional input space is reduced via a sampling strategy and a zero-hold assumption. This strategy enables the controller design based on a PINN as an approximation of the underlying system dynamics. The additional benefit is that the sensitivities are easily computed via automatic differentiation, thus leading to efficient gradient-based algorithms for the underlying optimal control problem.
  BibTeX
  @article{NicKFU22, abstract = {We discuss nonlinear model predictive control (MPC) for multi-body dynamics via physics-informed machine learning methods. In more detail, we use a physics-informed neural networks (PINNs)-based MPC to solve a tracking problem for a complex mechanical system, a multi-link manipulator. PINNs are a promising tool to approximate (partial) differential equations but are not suited for control tasks in their original form since they are not designed to handle variable control actions or variable initial values. We thus follow the strategy of Antonelo et al. (arXiv:2104.02556, 2021) by enhancing PINNs with adding control actions and initial conditions as additional network inputs. Subsequently, the high-dimensional input space is reduced via a sampling strategy and a zero-hold assumption. This strategy enables the controller design based on a PINN as an approximation of the underlying system dynamics. The additional benefit is that the sensitivities are easily computed via automatic differentiation, thus leading to efficient gradient-based algorithms for the underlying optimal control problem.}, author = {Nicodemus, Jonas and Kneifl, Jonas and Fehr, Jörg and Unger, Benjamin}, doi = {10.1016/j.ifacol.2022.09.117}, journal = {IFAC-PapersOnLine}, number = 20, pages = {331--336}, preprinturl = {https://doi.org/10.48550/arXiv.2109.10793}, publisher = {Elsevier}, title = {Physics-informed Neural Networks-based Model Predictive Control for Multi-link Manipulators}, url = {https://doi.org/10.1016/j.ifacol.2022.09.117}, volume = 55, year = 2022 }
11. R. Altmann, R. Maier, and B. Unger, “Semi-explicit integration of second order for weakly coupled poroelasticity,” ArXiv e-print 2203.16664, 2022, [Online]. Available: http://arxiv.org/abs/2203.16664
  - Abstract
  - BibTeX
  Abstract
  We introduce a semi-explicit time-stepping scheme of second order for linear poroelasticity satisfying a weak coupling condition. Here, semi-explicit means that the system, which needs to be solved in each step, decouples and hence improves the computational efficiency. The construction and the convergence proof are based on the connection to a differential equation with two time delays, namely one and two times the step size. Numerical experiments confirm the theoretical results and indicate the applicability to higher-order schemes.
  BibTeX
  @article{AltMU22, abstract = {We introduce a semi-explicit time-stepping scheme of second order for linear poroelasticity satisfying a weak coupling condition. Here, semi-explicit means that the system, which needs to be solved in each step, decouples and hence improves the computational efficiency. The construction and the convergence proof are based on the connection to a differential equation with two time delays, namely one and two times the step size. Numerical experiments confirm the theoretical results and indicate the applicability to higher-order schemes.}, author = {Altmann, Robert and Maier, Roland and Unger, Benjamin}, journal = {ArXiv e-print 2203.16664}, pubdate = {2022-03-31}, title = {Semi-explicit integration of second order for weakly coupled poroelasticity}, url = {http://arxiv.org/abs/2203.16664}, year = 2022 }
12. R. Morandin, J. Nicodemus, and B. Unger, “Port-Hamiltonian Dynamic Mode Decomposition,” ArXiv e-print 2204.13474, 2022, doi: 10.48550/arXiv.2204.13474.
  - Abstract
  - BibTeX
  Abstract
  We present a novel physics-informed system identification method to construct a passive linear time-invariant system. In more detail, for a given quadratic energy functional, measurements of the input, state, and output of a system in the time domain, we find a realization that approximates the data well while guaranteeing that the energy functional satisfies a dissipation inequality. To this end, we use the framework of port-Hamiltonian (pH) systems and modify the dynamic mode decomposition to be feasible for continuous-time pH systems. We propose an iterative numerical method to solve the corresponding least-squares minimization problem. We construct an effective initialization of the algorithm by studying the least-squares problem in a weighted norm, for which we present the analytical minimum-norm solution. The efficiency of the proposed method is demonstrated with several numerical examples.
  BibTeX
  @article{MorNU22, abstract = {We present a novel physics-informed system identification method to construct a passive linear time-invariant system. In more detail, for a given quadratic energy functional, measurements of the input, state, and output of a system in the time domain, we find a realization that approximates the data well while guaranteeing that the energy functional satisfies a dissipation inequality. To this end, we use the framework of port-Hamiltonian (pH) systems and modify the dynamic mode decomposition to be feasible for continuous-time pH systems. We propose an iterative numerical method to solve the corresponding least-squares minimization problem. We construct an effective initialization of the algorithm by studying the least-squares problem in a weighted norm, for which we present the analytical minimum-norm solution. The efficiency of the proposed method is demonstrated with several numerical examples.}, author = {Morandin, Riccardo and Nicodemus, Jonas and Unger, Benjamin}, doi = {10.48550/arXiv.2204.13474}, journal = {ArXiv e-print 2204.13474}, pubdate = {2022-04-28}, title = {Port-Hamiltonian Dynamic Mode Decomposition}, year = 2022 }
2021
1. R. Altmann, R. Maier, and B. Unger, “Semi-explicit discretization schemes for weakly coupled elliptic-parabolic problems,” Mathematics of Computation, vol. 90, Art. no. 329, 2021, doi: 10.1090/mcom/3608.
  - Abstract
  - BibTeX
  Abstract
  We prove first-order convergence of the semi-explicit Euler scheme combined with a finite element discretization in space for elliptic-parabolic problems which are weakly coupled. This setting includes poroelasticity, thermoelasticity, as well as multiple-network models used in medical applications. The semi-explicit approach decouples the system such that each time step requires the solution of two small and well-structured linear systems rather than the solution of one large system. The decoupling improves the computational efficiency without decreasing the convergence rates. The presented convergence proof is based on an interpretation of the scheme as an implicit method applied to a constrained partial differential equation with delay term. Here, the delay time equals the used step size. This connection also allows a deeper understanding of the weak coupling condition, which we accomplish to quantify explicitly.
  BibTeX
  @article{AltMU21b, abstract = {We prove first-order convergence of the semi-explicit Euler scheme combined with a finite element discretization in space for elliptic-parabolic problems which are weakly coupled. This setting includes poroelasticity, thermoelasticity, as well as multiple-network models used in medical applications. The semi-explicit approach decouples the system such that each time step requires the solution of two small and well-structured linear systems rather than the solution of one large system. The decoupling improves the computational efficiency without decreasing the convergence rates. The presented convergence proof is based on an interpretation of the scheme as an implicit method applied to a constrained partial differential equation with delay term. Here, the delay time equals the used step size. This connection also allows a deeper understanding of the weak coupling condition, which we accomplish to quantify explicitly.}, author = {Altmann, R. and Maier, R. and Unger, B.}, doi = {10.1090/mcom/3608}, journal = {Mathematics of Computation}, number = 329, pages = {1089--1118}, publisher = {American Mathematical Society (AMS)}, title = {Semi-explicit discretization schemes for weakly coupled elliptic-parabolic problems}, url = {https://doi.org/10.1090/mcom/3608}, volume = 90, year = 2021 }
2. I. Ahrens and B. Unger, “A simple success check for delay differential-algebraic equations,” in Proceedings in Applied Mathematics and Mechanics 2020, Wiley, 2021, p. e202000270. doi: 10.1002/pamm.202000270.
  - Abstract
  - BibTeX
  Abstract
  Solutions of delay differential‐algebraic equations (DDAEs) may depend on derivatives and future evaluations of some of its equations. Structural analysis can be used to determine how often each equation needs to be differentiated and shifted. Unfortunately, these numbers are not always correct, such that a post‐processing step is required to validate the result. In this contribution, we present the first step towards a success check for DDAEs.
  BibTeX
  @inproceedings{AhrU21a, abstract = {Solutions of delay differential‐algebraic equations (DDAEs) may depend on derivatives and future evaluations of some of its equations. Structural analysis can be used to determine how often each equation needs to be differentiated and shifted. Unfortunately, these numbers are not always correct, such that a post‐processing step is required to validate the result. In this contribution, we present the first step towards a success check for DDAEs.}, author = {Ahrens, Ines and Unger, Benjamin}, booktitle = {Proceedings in Applied Mathematics and Mechanics 2020}, doi = {10.1002/pamm.202000270}, number = 1, pages = {e202000270}, publisher = {Wiley}, title = {A simple success check for delay differential-algebraic equations}, url = {https://doi.org/10.1002/pamm.202000270}, volume = 20, year = 2021 }
3. F. Black, P. Schulze, and B. Unger, “Efficient Wildland Fire Simulation via Nonlinear Model Order Reduction,” Fluids, vol. 6, Art. no. 8, 2021, doi: 10.3390/fluids6080280.
  - Abstract
  - BibTeX
  Abstract
  We propose a new hyper-reduction method for a recently introduced nonlinear model reduction framework based on dynamically transformed basis functions and especially well-suited for transport-dominated systems. Furthermore, we discuss applying this new method to a wildland fire model whose dynamics feature traveling combustion waves and local ignition and is thus challenging for classical model reduction schemes based on linear subspaces. The new hyper-reduction framework allows us to construct parameter-dependent reduced-order models (ROMs) with efficient offline/online decomposition. The numerical experiments demonstrate that the ROMs obtained by the novel method outperform those obtained by a classical approach using the proper orthogonal decomposition and the discrete empirical interpolation method in terms of run time and accuracy.
  BibTeX
  @article{BlaSU21b, abstract = {We propose a new hyper-reduction method for a recently introduced nonlinear model reduction framework based on dynamically transformed basis functions and especially well-suited for transport-dominated systems. Furthermore, we discuss applying this new method to a wildland fire model whose dynamics feature traveling combustion waves and local ignition and is thus challenging for classical model reduction schemes based on linear subspaces. The new hyper-reduction framework allows us to construct parameter-dependent reduced-order models (ROMs) with efficient offline/online decomposition. The numerical experiments demonstrate that the ROMs obtained by the novel method outperform those obtained by a classical approach using the proper orthogonal decomposition and the discrete empirical interpolation method in terms of run time and accuracy.}, author = {Black, Felix and Schulze, Philipp and Unger, Benjamin}, doi = {10.3390/fluids6080280}, journal = {Fluids}, number = 8, pages = 280, publisher = {{MDPI} {AG}}, title = {Efficient Wildland Fire Simulation via Nonlinear Model Order Reduction}, volume = 6, year = 2021 }
4. I. V. Gosea, S. Gugercin, and B. Unger, “Parametric model reduction via rational interpolation along parameters,” in 2021 60th IEEE Conference on Decision and Control (CDC), 2021, pp. 6895–6900. doi: 10.1109/CDC45484.2021.9682841.
  - Abstract
  - BibTeX
  Abstract
  We present a novel projection-based model reduction framework for parametric linear time-invariant systems that allows interpolating the transfer function at a given frequency point along parameter-dependent curves as opposed to the standard approach where transfer function interpolation is achieved for a discrete set of parameter and frequency samples. We accomplish this goal by using parameter-dependent projection spaces. Our main result shows that for holomorphic system matrices, the corresponding interpolatory projection spaces are also holomorphic. The coefficients of the power series representation of the projection spaces can be computed iteratively using standard methods. We illustrate the analysis on three numerical examples.
  BibTeX
  @inproceedings{9682841, abstract = {We present a novel projection-based model reduction framework for parametric linear time-invariant systems that allows interpolating the transfer function at a given frequency point along parameter-dependent curves as opposed to the standard approach where transfer function interpolation is achieved for a discrete set of parameter and frequency samples. We accomplish this goal by using parameter-dependent projection spaces. Our main result shows that for holomorphic system matrices, the corresponding interpolatory projection spaces are also holomorphic. The coefficients of the power series representation of the projection spaces can be computed iteratively using standard methods. We illustrate the analysis on three numerical examples.}, author = {Gosea, Ion Victor and Gugercin, Serkan and Unger, Benjamin}, booktitle = {2021 60th IEEE Conference on Decision and Control (CDC)}, doi = {10.1109/CDC45484.2021.9682841}, issn = {2576-2370}, pages = {6895-6900}, title = {Parametric model reduction via rational interpolation along parameters}, year = 2021 }
5. F. Black, P. Schulze, and B. Unger, “Decomposition of flow data via gradient-based transport optimization,” ArXiv e-print 2107.03481, 2021, [Online]. Available: https://arxiv.org/abs/2107.03481
  - Abstract
  - BibTeX
  Abstract
  We study an optimization problem related to the approximation of given data by a linear combination of transformed modes. In the simplest case, the optimization problem reduces to a minimization problem well-studied in the context of proper orthogonal decomposition. Allowing transformed modes in the approximation renders this approach particularly useful to compress data with transported quantities, which are prevalent in many flow applications. We prove the existence of a solution to the infinite-dimensional optimization problem. Towards a numerical implementation, we compute the gradient of the cost functional and derive a suitable discretization in time and space. We demonstrate the theoretical findings with three challenging numerical examples.
  BibTeX
  @article{BlaSU21c, abstract = {We study an optimization problem related to the approximation of given data by a linear combination of transformed modes. In the simplest case, the optimization problem reduces to a minimization problem well-studied in the context of proper orthogonal decomposition. Allowing transformed modes in the approximation renders this approach particularly useful to compress data with transported quantities, which are prevalent in many flow applications. We prove the existence of a solution to the infinite-dimensional optimization problem. Towards a numerical implementation, we compute the gradient of the cost functional and derive a suitable discretization in time and space. We demonstrate the theoretical findings with three challenging numerical examples.}, author = {Black, Felix and Schulze, Philipp and Unger, Benjamin}, journal = {ArXiv e-print 2107.03481}, pubdate = {2021-07-07}, title = {Decomposition of flow data via gradient-based transport optimization}, url = {https://arxiv.org/abs/2107.03481}, year = 2021 }
6. I. V. Gosea, S. Gugercin, and B. Unger, “Parametric model reduction via rational interpolation along parameters,” ArXiv e-print 2104.01016, 2021, [Online]. Available: https://arxiv.org/abs/2104.01016
  - Abstract
  - BibTeX
  Abstract
  We present a novel projection-based model reduction framework for parametric linear time-invariant systems that allows interpolating the transfer function at a given frequency point along parameter-dependent curves as opposed to the standard approach where transfer function interpolation is achieved for a discrete set of parameter and frequency samples. We accomplish this goal by using parameter-dependent projection spaces. Our main result shows that for holomorphic system matrices, the corresponding interpolatory projection spaces are also holomorphic. The coefficients of the power series representation of the projection spaces can be computed iteratively using standard methods. We illustrate the analysis on three numerical examples.
  BibTeX
  @article{GosGU21, abstract = {We present a novel projection-based model reduction framework for parametric linear time-invariant systems that allows interpolating the transfer function at a given frequency point along parameter-dependent curves as opposed to the standard approach where transfer function interpolation is achieved for a discrete set of parameter and frequency samples. We accomplish this goal by using parameter-dependent projection spaces. Our main result shows that for holomorphic system matrices, the corresponding interpolatory projection spaces are also holomorphic. The coefficients of the power series representation of the projection spaces can be computed iteratively using standard methods. We illustrate the analysis on three numerical examples.}, author = {{Gosea}, Ion Victor and {Gugercin}, Serkan and {Unger}, Benjamin}, journal = {ArXiv e-print 2104.01016}, pubdate = {2021-04-05}, title = {Parametric model reduction via rational interpolation along parameters}, url = {https://arxiv.org/abs/2104.01016}, year = 2021 }
7. J. Heiland and B. Unger, “Identification of linear time-invariant systems with Dynamic Mode Decomposition,” ArXiv e-print 2109.06765, 2021, [Online]. Available: https://arxiv.org/abs/2109.06765
  - Abstract
  - BibTeX
  Abstract
  Dynamic mode decomposition (DMD) is a popular data-driven framework to extract linear dynamics from complex high-dimensional systems. In this work, we study the system identification properties of DMD. We first show that DMD is invariant under linear transformations in the image of the data matrix. If, in addition, the data is constructed from a linear time-invariant system, then we prove that DMD can recover the original dynamics under mild conditions. If the linear dynamics are discretized with a Runge-Kutta method, then we further classify the error of the DMD approximation and detail that for one-stage Runge-Kutta methods even the continuous dynamics can be recovered with DMD. A numerical example illustrates the theoretical findings.
  BibTeX
  @article{HeiU21, abstract = {Dynamic mode decomposition (DMD) is a popular data-driven framework to extract linear dynamics from complex high-dimensional systems. In this work, we study the system identification properties of DMD. We first show that DMD is invariant under linear transformations in the image of the data matrix. If, in addition, the data is constructed from a linear time-invariant system, then we prove that DMD can recover the original dynamics under mild conditions. If the linear dynamics are discretized with a Runge-Kutta method, then we further classify the error of the DMD approximation and detail that for one-stage Runge-Kutta methods even the continuous dynamics can be recovered with DMD. A numerical example illustrates the theoretical findings.}, author = {Heiland, Jan and Unger, Benjamin}, journal = {ArXiv e-print 2109.06765}, pubdate = {2021-09-15}, title = {Identification of linear time-invariant systems with {Dynamic Mode Decomposition}}, url = {https://arxiv.org/abs/2109.06765}, year = 2021 }
8. R. Altmann, V. Mehrmann, and B. Unger, “Port-Hamiltonian formulations of poroelastic network models,” Mathematical and Computer Modelling of Dynamical Systems, vol. 27, Art. no. 1, 2021, doi: 10.1080/13873954.2021.1975137.
  - Abstract
  - BibTeX
  Abstract
  We investigate an energy-based formulation of the two-field poroelasticity model and the related multiple-network model as they appear in the geosciences or medical applications. We propose a port-Hamiltonian formulation of the system equations, which is beneficial for preserving important system properties after discretization or model-order reduction. For this, we include the commonly omitted second-order term and consider the corresponding first-order formulation. The port-Hamiltonian formulation of the quasi-static case is then obtained by taking the formal limit. Further, we interpret the poroelastic equations as an interconnection of a network of submodels with internal energies, adding a control-theoretic understanding of the poroelastic equations.
  BibTeX
  @article{AltMU21c, abstract = {We investigate an energy-based formulation of the two-field poroelasticity model and the related multiple-network model as they appear in the geosciences or medical applications. We propose a port-Hamiltonian formulation of the system equations, which is beneficial for preserving important system properties after discretization or model-order reduction. For this, we include the commonly omitted second-order term and consider the corresponding first-order formulation. The port-Hamiltonian formulation of the quasi-static case is then obtained by taking the formal limit. Further, we interpret the poroelastic equations as an interconnection of a network of submodels with internal energies, adding a control-theoretic understanding of the poroelastic equations.}, author = {Altmann, Robert and Mehrmann, Volker and Unger, Benjamin}, doi = {10.1080/13873954.2021.1975137}, fjournal = {Mathematical and Computer Modelling of Dynamical Systems}, journal = {Mathematical and Computer Modelling of Dynamical Systems}, number = 1, pages = {429--452}, title = {Port-Hamiltonian formulations of poroelastic network models}, volume = 27, year = 2021 }
9. F. Black, P. Schulze, and B. Unger, “Model order reduction with dynamically transformed modes for the wave equation,” in Proceedings in Applied Mathematics and Mechanics 2020, Wiley, 2021, p. e202000321. doi: 10.1002/pamm.202000321.
  - Abstract
  - BibTeX
  Abstract
  In this contribution, we apply a recently introduced nonlinear model reduction framework based on dynamically transformed modes to the linear wave equation with periodic boundary conditions. We demonstrate that under reasonable assumptions, the reduced‐order model can be evaluated efficiently. Consequently, we obtain that the state variables of the reduced‐order model are constant or linear functions with respect to time.
  BibTeX
  @inproceedings{BlaSU21a, abstract = {In this contribution, we apply a recently introduced nonlinear model reduction framework based on dynamically transformed modes to the linear wave equation with periodic boundary conditions. We demonstrate that under reasonable assumptions, the reduced‐order model can be evaluated efficiently. Consequently, we obtain that the state variables of the reduced‐order model are constant or linear functions with respect to time.}, author = {Black, Felix and Schulze, Philipp and Unger, Benjamin}, booktitle = {Proceedings in Applied Mathematics and Mechanics 2020}, doi = {10.1002/pamm.202000321}, number = 1, pages = {e202000321}, publisher = {Wiley}, title = {Model order reduction with dynamically transformed modes for the wave equation}, url = {https://doi.org/10.1002%2Fpamm.202000321}, volume = 20, year = 2021 }
10. F. Black, P. Schulze, and B. Unger, “Efficient Wildland Fire Simulation via Nonlinear Model Order Reduction,” ArXiv e-print 2106.11381, 2021, [Online]. Available: https://arxiv.org/abs/2106.11381
  - Abstract
  - BibTeX
  Abstract
  We propose a new hyper-reduction method for a recently introduced nonlinear model reduction framework based on dynamically transformed basis functions and especially well-suited for advection-dominated systems. Furthermore, we discuss applying this new method to a wildland fire model whose dynamics feature traveling combustion waves and local ignition and is thus challenging for classical model reduction schemes based on linear subspaces. The new hyper-reduction framework allows us to construct parameter-dependent reduced-order models (ROMs) with efficient offline/online decomposition. The numerical experiments demonstrate that the ROMs obtained by the novel method outperform those obtained by a classical approach using the proper orthogonal decomposition and the discrete empirical interpolation method in terms of run time and accuracy.
  BibTeX
  @article{BlaSU21b, abstract = {We propose a new hyper-reduction method for a recently introduced nonlinear model reduction framework based on dynamically transformed basis functions and especially well-suited for advection-dominated systems. Furthermore, we discuss applying this new method to a wildland fire model whose dynamics feature traveling combustion waves and local ignition and is thus challenging for classical model reduction schemes based on linear subspaces. The new hyper-reduction framework allows us to construct parameter-dependent reduced-order models (ROMs) with efficient offline/online decomposition. The numerical experiments demonstrate that the ROMs obtained by the novel method outperform those obtained by a classical approach using the proper orthogonal decomposition and the discrete empirical interpolation method in terms of run time and accuracy.}, author = {Black, Felix and Schulze, Philipp and Unger, Benjamin}, journal = {ArXiv e-print 2106.11381}, pubdate = {2021-06-23}, title = {Efficient Wildland Fire Simulation via Nonlinear Model Order Reduction}, url = {https://arxiv.org/abs/2106.11381}, year = 2021 }
11. S. Trenn and B. Unger, “Unimodular transformations for DAE initial trajectory problems,” in Proceedings in Applied Mathematics and Mechanics 2020, Wiley, 2021, p. e202000322. doi: 10.1002/pamm.202000322.
  - Abstract
  - BibTeX
  Abstract
  We consider linear time‐invariant differential‐algebraic equations (DAEs). For high‐index DAEs, it is often the first step to perform an index reduction, which can be realized with a unimodular matrix. In this contribution, we illustrate the effect of unimodular transformations on initial trajectory problems associated with DAEs.
  BibTeX
  @inproceedings{TreU21a, abstract = {We consider linear time‐invariant differential‐algebraic equations (DAEs). For high‐index DAEs, it is often the first step to perform an index reduction, which can be realized with a unimodular matrix. In this contribution, we illustrate the effect of unimodular transformations on initial trajectory problems associated with DAEs.}, author = {Trenn, Stephan and Unger, Benjamin}, booktitle = {Proceedings in Applied Mathematics and Mechanics 2020}, doi = {10.1002/pamm.202000322}, number = 1, pages = {e202000322}, publisher = {Wiley}, title = {Unimodular transformations for {DAE} initial trajectory problems}, url = {https://doi.org/10.1002/pamm.202000322}, volume = 20, year = 2021 }
12. R. Altmann, R. Maier, and B. Unger, “A semi-explicit integration scheme for weakly-coupled poroelasticity with nonlinear permeability,” in Proceedings in Applied Mathematics and Mechanics 2020, Wiley, 2021, p. e202000061. doi: 10.1002/pamm.202000061.
  - Abstract
  - BibTeX
  Abstract
  Within the time integration of linear elliptic‐parabolic problems such as poroelasticity, large systems have to be solved in every time step. With a semi‐explicit approach, these systems decouple, leading to a remarkable speed‐up. This paper indicates that this idea can also be applied to nonlinear problems such as poroelasticity with nonlinear permeability. To see this, we consider a related toy problem.
  BibTeX
  @inproceedings{AltMU21a, abstract = {Within the time integration of linear elliptic‐parabolic problems such as poroelasticity, large systems have to be solved in every time step. With a semi‐explicit approach, these systems decouple, leading to a remarkable speed‐up. This paper indicates that this idea can also be applied to nonlinear problems such as poroelasticity with nonlinear permeability. To see this, we consider a related toy problem.}, author = {Altmann, Robert and Maier, Roland and Unger, Benjamin}, booktitle = {Proceedings in Applied Mathematics and Mechanics 2020}, doi = {10.1002/pamm.202000061}, number = 1, pages = {e202000061}, publisher = {Wiley}, title = {A semi-explicit integration scheme for weakly-coupled poroelasticity with nonlinear permeability}, url = {https://doi.org/10.1002/pamm.202000061}, volume = 20, year = 2021 }
13. T. Breiten and B. Unger, “Passivity preserving model reduction via spectral factorization,” ArXiv e-print 2103.13194, 2021, [Online]. Available: https://arxiv.org/abs/2103.13194
  - Abstract
  - BibTeX
  Abstract
  We present a novel model-order reduction (MOR) method for linear time-invariant systems that preserves passivity and is thus suited for structure-preserving MOR for port-Hamiltonian (pH) systems. Our algorithm exploits the well-known spectral factorization of the Popov function by a solution of the Kalman-Yakubovich-Popov (KYP) inequality. It performs MOR directly on the spectral factor inheriting the original system's sparsity enabling MOR in a large-scale context. Our analysis reveals that the spectral factorization corresponding to the minimal solution of an associated algebraic Riccati equation is preferable from a model reduction perspective and benefits pH-preserving MOR methods such as a modified version of the iterative rational Krylov algorithm (IRKA). Numerical examples demonstrate that our approach can produce high-fidelity reduced-order models close to (unstructured) $H_2$-optimal reduced-order models.
  BibTeX
  @article{BreU21, abstract = {We present a novel model-order reduction (MOR) method for linear time-invariant systems that preserves passivity and is thus suited for structure-preserving MOR for port-Hamiltonian (pH) systems. Our algorithm exploits the well-known spectral factorization of the Popov function by a solution of the Kalman-Yakubovich-Popov (KYP) inequality. It performs MOR directly on the spectral factor inheriting the original system's sparsity enabling MOR in a large-scale context. Our analysis reveals that the spectral factorization corresponding to the minimal solution of an associated algebraic Riccati equation is preferable from a model reduction perspective and benefits pH-preserving MOR methods such as a modified version of the iterative rational Krylov algorithm (IRKA). Numerical examples demonstrate that our approach can produce high-fidelity reduced-order models close to (unstructured) $\mathcal{H}_2$-optimal reduced-order models.}, author = {Breiten, Tobias and Unger, Benjamin}, journal = {ArXiv e-print 2103.13194}, pubdate = {2021-03-25}, title = {Passivity preserving model reduction via spectral factorization}, url = {https://arxiv.org/abs/2103.13194}, year = 2021 }
14. J. Nicodemus, J. Kneifl, J. Fehr, and B. Unger, “Physics-informed Neural Networks-based Model Predictive Control for Multi-link Manipulators,” ArXiv e-print 2109.10793, 2021, [Online]. Available: https://arxiv.org/abs/2109.10793
  - Abstract
  - BibTeX
  Abstract
  We discuss nonlinear model predictive control (NMPC) for multi-body dynamics via physics-informed machine learning methods. Physics-informed neural networks (PINNs) are a promising tool to approximate (partial) differential equations. PINNs are not suited for control tasks in their original form since they are not designed to handle variable control actions or variable initial values. We thus present the idea of enhancing PINNs by adding control actions and initial conditions as additional network inputs. The high-dimensional input space is subsequently reduced via a sampling strategy and a zero-hold assumption. This strategy enables the controller design based on a PINN as an approximation of the underlying system dynamics. The additional benefit is that the sensitivities are easily computed via automatic differentiation, thus leading to efficient gradient-based algorithms. Finally, we present our results using our PINN-based MPC to solve a tracking problem for a complex mechanical system, a multi-link manipulator.
  BibTeX
  @article{NicKFU21, abstract = {We discuss nonlinear model predictive control (NMPC) for multi-body dynamics via physics-informed machine learning methods. Physics-informed neural networks (PINNs) are a promising tool to approximate (partial) differential equations. PINNs are not suited for control tasks in their original form since they are not designed to handle variable control actions or variable initial values. We thus present the idea of enhancing PINNs by adding control actions and initial conditions as additional network inputs. The high-dimensional input space is subsequently reduced via a sampling strategy and a zero-hold assumption. This strategy enables the controller design based on a PINN as an approximation of the underlying system dynamics. The additional benefit is that the sensitivities are easily computed via automatic differentiation, thus leading to efficient gradient-based algorithms. Finally, we present our results using our PINN-based MPC to solve a tracking problem for a complex mechanical system, a multi-link manipulator.}, author = {Nicodemus, Jonas and Kneifl, Jonas and Fehr, Jörg and Unger, Benjamin}, journal = {ArXiv e-print 2109.10793}, pubdate = {2021-09-23}, title = {Physics-informed Neural Networks-based Model Predictive Control for Multi-link Manipulators}, url = {https://arxiv.org/abs/2109.10793}, year = 2021 }
2020
1. I. Ahrens and B. Unger, “The Pantelides algorithm for delay differential-algebraic equations,” Transactions of Mathematics and Its Application, vol. 4, Art. no. 1, 2020, doi: 10.1093/imatrm/tnaa003.
  - Abstract
  - BibTeX
  Abstract
  We present a graph-theoretical approach that can detect which equations of a delay differential-algebraic equation (DDAE) need to be differentiated or shifted to construct a solution of the DDAE. Our approach exploits the observation that differentiation and shifting are very similar from a structural point of view, which allows us to generalize the Pantelides algorithm for differential-algebraic equations to the DDAE setting. The primary tool for the extension is the introduction of equivalence classes in the graph of the DDAE, which also allows us to derive a necessary and sufficient criterion for the termination of the new algorithm.
  BibTeX
  @article{AhrU20, abstract = {We present a graph-theoretical approach that can detect which equations of a delay differential-algebraic equation (DDAE) need to be differentiated or shifted to construct a solution of the DDAE. Our approach exploits the observation that differentiation and shifting are very similar from a structural point of view, which allows us to generalize the Pantelides algorithm for differential-algebraic equations to the DDAE setting. The primary tool for the extension is the introduction of equivalence classes in the graph of the DDAE, which also allows us to derive a necessary and sufficient criterion for the termination of the new algorithm.}, author = {{Ahrens}, Ines and {Unger}, Benjamin}, doi = {10.1093/imatrm/tnaa003}, file = {/homes/numerik/unger/WWW/Publications/AhrU20.pdf}, journal = {Transactions of Mathematics and Its Application}, number = 1, pages = {1--36}, preprint = {https://arxiv.org/abs/1908.01514}, project = {SFB910,DAE:Delay}, pubdate = {2020-09-30}, title = {The {P}antelides algorithm for delay differential-algebraic equations}, volume = 4, year = 2020 }
2. R. Altmann, R. Maier, and B. Unger, “Semi-explicit discretization schemes for weakly coupled elliptic-parabolic problems,” Mathematics of Computation, p. 1, Oct. 2020, doi: 10.1090/mcom/3608.
  - BibTeX
  BibTeX
  @article{Altmann_2020, author = {Altmann, Robert and Maier, Roland and Unger, Benjamin}, doi = {10.1090/mcom/3608}, journal = {Mathematics of Computation}, month = {10}, pages = 1, publisher = {American Mathematical Society ({AMS})}, title = {Semi-explicit discretization schemes for weakly coupled elliptic-parabolic problems}, url = {https://doi.org/10.1090%2Fmcom%2F3608}, year = 2020 }
3. R. Altmann, V. Mehrmann, and B. Unger, “Port-Hamiltonian formulations of poroelastic network models,” ArXiv e-print 2012.01949, Dec. 2020, [Online]. Available: https://arxiv.org/abs/2012.01949
  - Abstract
  - BibTeX
  Abstract
  We investigate an energy-based formulation of the two-field poroelasticity model and the related multiple-network model as they appear in the geosciences or medical applications. We propose a port-Hamiltonian formulation of the system equations, which is beneficial for preserving important system properties after discretization or model-order reduction. For this, we include the commonly omitted second-order term and consider the corresponding first-order formulation. The port-Hamiltonian formulation of the quasi-static case is then obtained by taking the formal limit. Further, we interpret the poroelastic equations as an interconnection of a network of submodels with internal energies, adding a control-theoretic understanding of the poroelastic equations.
  BibTeX
  @article{AltMU20c, abstract = {We investigate an energy-based formulation of the two-field poroelasticity model and the related multiple-network model as they appear in the geosciences or medical applications. We propose a port-Hamiltonian formulation of the system equations, which is beneficial for preserving important system properties after discretization or model-order reduction. For this, we include the commonly omitted second-order term and consider the corresponding first-order formulation. The port-Hamiltonian formulation of the quasi-static case is then obtained by taking the formal limit. Further, we interpret the poroelastic equations as an interconnection of a network of submodels with internal energies, adding a control-theoretic understanding of the poroelastic equations.}, author = {Altmann, Robert and Mehrmann, Volker and Unger, Benjamin}, journal = {ArXiv e-print 2012.01949}, month = {12}, pubdate = {2020-12-04}, title = {Port-Hamiltonian formulations of poroelastic network models}, url = {https://arxiv.org/abs/2012.01949}, year = 2020 }
4. F. Black, P. Schulze, and B. Unger, “Projection-based model reduction with dynamically transformed modes,” ESAIM: Mathematical Modelling and Numerical Analysis, vol. 54, Art. no. 6, Oct. 2020, doi: 10.1051/m2an/2020046.
  - BibTeX
  BibTeX
  @article{Black_2020, author = {Black, Felix and Schulze, Philipp and Unger, Benjamin}, doi = {10.1051/m2an/2020046}, journal = {{ESAIM}: Mathematical Modelling and Numerical Analysis}, month = {10}, number = 6, pages = {2011--2043}, publisher = {{EDP} Sciences}, title = {Projection-based model reduction with dynamically transformed modes}, url = {https://doi.org/10.1051%2Fm2an%2F2020046}, volume = 54, year = 2020 }
5. B. Unger, “Delay differential-algebraic equations in real-time dynamic substructuring,” ArXiv e-print 2003.10195, Mar. 2020, [Online]. Available: https://arxiv.org/abs/2003.10195
  - Abstract
  - BibTeX
  Abstract
  Hybrid numerical-experimental testing is a standard approach for complex dynamical structures that are, on the one hand, not easy to model due to complexity and parameter uncertainty and, on the other hand, too expensive for full-scale experiments. The main idea is to subdivide the structure in a part that can be accurately simulated with numerical methods and an experimental component. The numerical simulation and the experiment are coupled in real-time by a so-called transfer system, which induces a time-delay into the system. In this paper, we study the solvability of the resulting hybrid numerical-experimental system, which is typically described by a set of nonlinear delay differential-algebraic equations, and extend existing results from the literature to this case.
  BibTeX
  @article{Ung20, abstract = {Hybrid numerical-experimental testing is a standard approach for complex dynamical structures that are, on the one hand, not easy to model due to complexity and parameter uncertainty and, on the other hand, too expensive for full-scale experiments. The main idea is to subdivide the structure in a part that can be accurately simulated with numerical methods and an experimental component. The numerical simulation and the experiment are coupled in real-time by a so-called transfer system, which induces a time-delay into the system. In this paper, we study the solvability of the resulting hybrid numerical-experimental system, which is typically described by a set of nonlinear delay differential-algebraic equations, and extend existing results from the literature to this case.}, author = {{Unger}, B.}, file = {/homes/numerik/unger/WWW/Publications/Ung20-ppt.pdf}, journal = {ArXiv e-print 2003.10195}, month = {03}, project = {SFB910,DAE,DAE:Delay}, pubdate = {2020-03-24}, title = {Delay differential-algebraic equations in real-time dynamic substructuring}, url = {https://arxiv.org/abs/2003.10195}, year = 2020 }
2019
1. S. Trenn and B. Unger, “Delay regularity of differential-algebraic equations,” in 2019 IEEE 58th Conference on Decision and Control (CDC), Dec. 2019, pp. 989–994. doi: 10.1109/CDC40024.2019.9030146.
  - Abstract
  - BibTeX
  Abstract
  We study linear time-invariant delay differential-algebraic equations (DDAEs). Such equations can arise if a feedback controller is applied to a descriptor system and the controller requires some time to measure the state and to compute the feedback resulting in the time-delay. We present an existence and uniqueness result for DDAEs within the space of piecewise-smooth distributions and an algorithm to determine whether a DDAE is delay-regular.
  BibTeX
  @inproceedings{TreU19, abstract = {We study linear time-invariant delay differential-algebraic equations (DDAEs). Such equations can arise if a feedback controller is applied to a descriptor system and the controller requires some time to measure the state and to compute the feedback resulting in the time-delay. We present an existence and uniqueness result for DDAEs within the space of piecewise-smooth distributions and an algorithm to determine whether a DDAE is delay-regular.}, author = {{Trenn}, Stephan and {Unger}, Benjamin}, booktitle = {2019 IEEE 58th Conference on Decision and Control (CDC)}, doi = {10.1109/CDC40024.2019.9030146}, issn = {2576-2370}, month = {12}, pages = {989-994}, title = {Delay regularity of differential-algebraic equations}, url = {https://ieeexplore.ieee.org/document/9030146/}, year = 2019 }
2. E. Fosong, P. Schulze, and B. Unger, “From Time-Domain Data to Low-Dimensional Structured Models,” ArXiv e-print 1902.05112, Feb. 2019, [Online]. Available: https://arxiv.org/abs/1902.05112
  - Abstract
  - BibTeX
  Abstract
  We present a framework for constructing a structured realization of a linear time-invariant dynamical system solely from a discrete sampling of an input and output trajectory of the system. We estimate the transfer function of the original model at selected frequencies using a modification of the empirical transfer function estimation that was recently presented in Peherstorfer, Gugercin, Willcox, SIAM J.~Sci.~Comput., 39(5):2152--2178, 2017. Our realization interpolates the transfer function estimates and can be seen as a generalization of the Loewner framework to structured systems. We demonstrate the presented framework by means of a delay example.
  BibTeX
  @article{FosSU19-ppt, abstract = {We present a framework for constructing a structured realization of a linear time-invariant dynamical system solely from a discrete sampling of an input and output trajectory of the system. We estimate the transfer function of the original model at selected frequencies using a modification of the empirical transfer function estimation that was recently presented in [Peherstorfer, Gugercin, Willcox, SIAM J.~Sci.~Comput., 39(5):2152--2178, 2017]. Our realization interpolates the transfer function estimates and can be seen as a generalization of the Loewner framework to structured systems. We demonstrate the presented framework by means of a delay example.}, archiveprefix = {arXiv}, author = {{Fosong}, Elliot and {Schulze}, Philipp and {Unger}, Benjamin}, eprint = {1902.05112}, file = {/homes/numerik/unger/WWW/Publications/FosSU19-ppt.pdf}, journal = {ArXiv e-print 1902.05112}, month = {02}, project = {SFB910, MOR}, pubdate = {2019-02-13}, title = {From Time-Domain Data to Low-Dimensional Structured Models}, url = {https://arxiv.org/abs/1902.05112}, year = 2019 }
3. B. Unger and S. Gugercin, “Kolmogorov n-widths for linear dynamical systems,” Advances in Computational Mathematics, vol. 45, Art. no. 5, Dec. 2019, doi: 10.1007/s10444-019-09701-0.
  - Abstract
  - BibTeX
  Abstract
  Kolmogorov n-widths and Hankel singular values are two commonly used concepts in model reduction. Here, we show that for the special case of linear time-invariant (LTI) dynamical systems, these two concepts are directly connected. More specifically, the greedy search applied to the Hankel operator of an LTI system resembles the minimizing subspace for the Kolmogorov n-width and the Kolmogorov n-width of an LTI system equals its (n +þinspace1)st Hankel singular value once the subspaces are appropriately defined. We also establish a lower bound for the Kolmorogov n-width for parametric LTI systems and illustrate that the method of active subspaces can be viewed as the dual concept to the minimizing subspace for the Kolmogorov n-width.
  BibTeX
  @article{Unger2019, abstract = {Kolmogorov n-widths and Hankel singular values are two commonly used concepts in model reduction. Here, we show that for the special case of linear time-invariant (LTI) dynamical systems, these two concepts are directly connected. More specifically, the greedy search applied to the Hankel operator of an LTI system resembles the minimizing subspace for the Kolmogorov n-width and the Kolmogorov n-width of an LTI system equals its (n +{\thinspace}1)st Hankel singular value once the subspaces are appropriately defined. We also establish a lower bound for the Kolmorogov n-width for parametric LTI systems and illustrate that the method of active subspaces can be viewed as the dual concept to the minimizing subspace for the Kolmogorov n-width.}, author = {Unger, Benjamin and Gugercin, Serkan}, day = 01, doi = {10.1007/s10444-019-09701-0}, issn = {1572-9044}, journal = {Advances in Computational Mathematics}, month = {12}, number = 5, pages = {2273-2286}, title = {Kolmogorov n-widths for linear dynamical systems}, url = {https://doi.org/10.1007/s10444-019-09701-0}, volume = 45, year = 2019 }
4. D. Kern, M. Bartelt, and B. Unger, “How to write an article for GAMMAS and a longer title,” GAMM Archive for Students, vol. 1, Art. no. 1, Feb. 2019, doi: 10.14464/gammas.v1i1.417.
  - BibTeX
  BibTeX
  @article{Kern_2019, author = {Kern, Dominik and Bartelt, Matthias and Unger, Benjamin}, doi = {10.14464/gammas.v1i1.417}, journal = {{GAMM} Archive for Students}, month = {02}, number = 1, pages = {1--5}, publisher = {Technische Universitat Chemnitz}, title = {How to write an article for {GAMMAS} and a longer title}, url = {https://doi.org/10.14464%2Fgammas.v1i1.417}, volume = 1, year = 2019 }
2018
1. P. Schulze, B. Unger, Christopher. Beattie, and S. Gugercin, “Data-driven Structured Realization,” Linear Algebra and its Applications, vol. 537, pp. 250–286, 2018, doi: 10.1016/j.laa.2017.09.030.
  - Abstract
  - BibTeX
  Abstract
  We present a framework for constructing structured realizations of linear dynamical systems having transfer functions of the form $C(\sum_k=1^K h_k(s)A_k)^-1B$ where $h_1, h_2, łdots, h_K$ are prescribed functions that specify the surmised structure of the model. Our construction is data-driven in the sense that an interpolant is derived entirely from measurements of a transfer function. Our approach extends the Loewner realization framework to a more general system structure that includes second-order (and higher) systems as well as systems with internal delays. Numerical examples demonstrate the advantages of this approach.
  BibTeX
  @article{SchUBG18, abstract = {We present a framework for constructing structured realizations of linear dynamical systems having transfer functions of the form $\widetilde{C}(\sum_{k=1}^K h_k(s)\widetilde{A}_k)^{-1}\widetilde{B}$ where $h_1, h_2, \ldots, h_K$ are prescribed functions that specify the surmised structure of the model. Our construction is data-driven in the sense that an interpolant is derived entirely from measurements of a transfer function. Our approach extends the Loewner realization framework to a more general system structure that includes second-order (and higher) systems as well as systems with internal delays. Numerical examples demonstrate the advantages of this approach.}, author = {Schulze, Philipp and Unger, Benjamin and Beattie, Christopher. and Gugercin, Serkan}, doi = {10.1016/j.laa.2017.09.030}, issn = {0024-3795}, journal = {Linear Algebra and its Applications}, pages = {250-286}, preprint = {http://www3.math.tu-berlin.de/cgi-bin/IfM/show_abstract.cgi?Report-23-2016.rdf.html}, title = {Data-driven Structured Realization}, url = {http://www.sciencedirect.com/science/article/pii/S0024379517305645}, volume = 537, year = 2018 }
2. P. Schulze and B. Unger, “Model reduction for linear systems with low-rank switching.,” SIAM J. Control and Optimization, vol. 56, Art. no. 6, 2018, doi: 10.1137/18M1167887.
  - Abstract
  - BibTeX
  Abstract
  We introduce a novel model order reduction method for large-scale linear switched systems (LSS) where the coefficient matrices are affected by a low-rank switching. The key idea is to replace the LSS by a non-switched system with extended input and output vectors - called the envelope system - which is able to reproduce the dynamical behavior of the original LSS by applying a certain feedback law. The envelope system can be reduced using standard model order reduction schemes and then transformed back to an LSS. Furthermore, we present an upper bound for the output error of the reduced-order LSS and show how to preserve quadratic Lyapunov stability. The approach is tested by means of various numerical examples demonstrating the efficacy of the presented method.
  BibTeX
  @article{SchU18, abstract = {We introduce a novel model order reduction method for large-scale linear switched systems (LSS) where the coefficient matrices are affected by a low-rank switching. The key idea is to replace the LSS by a non-switched system with extended input and output vectors - called the envelope system - which is able to reproduce the dynamical behavior of the original LSS by applying a certain feedback law. The envelope system can be reduced using standard model order reduction schemes and then transformed back to an LSS. Furthermore, we present an upper bound for the output error of the reduced-order LSS and show how to preserve quadratic Lyapunov stability. The approach is tested by means of various numerical examples demonstrating the efficacy of the presented method.}, author = {Schulze, Philipp and Unger, Benjamin}, doi = {10.1137/18M1167887}, journal = {SIAM J. Control and Optimization}, number = 6, pages = {4365-4384}, preprint = {https://arxiv.org/abs/1801.09445}, title = {Model reduction for linear systems with low-rank switching.}, volume = 56, year = 2018 }
3. B. Unger, “Discontinuity Propagation in Delay Differential-Algebraic Equations,” The Electronic Journal of Linear Algebra, vol. 34, pp. 582–601, Feb. 2018, doi: 10.13001/1081-3810.3759.
  - BibTeX
  BibTeX
  @article{Unger_2018, author = {Unger, Benjamin}, doi = {10.13001/1081-3810.3759}, journal = {The Electronic Journal of Linear Algebra}, month = {02}, pages = {582-601}, publisher = {University of Wyoming Libraries}, title = {Discontinuity Propagation in Delay Differential-Algebraic Equations}, url = {https://doi.org/10.13001%2F1081-3810.3759}, volume = 34, year = 2018 }
2016
1. P. Schulze and B. Unger, “Data-driven interpolation of dynamical systems with delay.,” Systems Control Lett., vol. 97, pp. 125–131, 2016, doi: 10.1016/j.sysconle.2016.09.007.
  - Abstract
  - BibTeX
  Abstract
  We present a data-driven realization for systems with delay, which generalizes the Loewner framework. The realization is obtained with low computational cost directly from measured data of the transfer function. The internal delay is estimated by solving a least-square optimization over some sample data. Our approach is validated by several examples, which indicate the need for preserving the delay structure in the reduced model.
  BibTeX
  @article{SchU16, abstract = {We present a data-driven realization for systems with delay, which generalizes the Loewner framework. The realization is obtained with low computational cost directly from measured data of the transfer function. The internal delay is estimated by solving a least-square optimization over some sample data. Our approach is validated by several examples, which indicate the need for preserving the delay structure in the reduced model.}, author = {Schulze, Philipp and Unger, Benjamin}, doi = {10.1016/j.sysconle.2016.09.007}, journal = {Systems Control Lett.}, pages = {125-131}, preprint = {http://www3.math.tu-berlin.de/cgi-bin/IfM/show_abstract.cgi?Report-17-2015.rdf.html}, title = {Data-driven interpolation of dynamical systems with delay.}, volume = 97, year = 2016 }
2015
1. V. Mehrmann and B. Unger, “Index preserving polynomial representation of nonlinear differential-algebraic systems,” Institut für Mathematik, TU Berlin, Str. des 17. Juni 136, D-10623 Berlin, FRG, Preprint 2––2015, 2015. [Online]. Available: http://www3.math.tu-berlin.de/cgi-bin/IfM/show_abstract.cgi?Report-02-2015.rdf.html
  - Abstract
  - BibTeX
  Abstract
  Recently in Gu (2011), Comput. Des. Integr. Circuits Syst., 30(9):1207--1320 a procedure was presented that allows to reformulate nonlinear ordinary differential equations in a way that all the nonlinearities become polynomial on the cost of increasing the dimension of the system. We generalize this procedure (called `polynomialization') to systems of differential-algebraic equations (DAEs). In particular, we show that if the original nonlinear DAE is regular and strangeness-free (i.\,e., it has differentiation index one) then this property is preserved by the polynomial representation. For systems which are not strangeness-free, i.\,e., where the solution depends on derivatives of the coefficients and inhomogeneities, we also show that the index is preserved for arbitrary strangeness index. However, to avoid ill-conditioning in the representation one should first perform an index reduction on the nonlinear system and then construct the polynomial representations. Although the analytical properties of the polynomial reformulation are very appealing, care has to be given to the numerical integration of the reformulated system due to additional errors. We illustrate our findings with several examples.
  BibTeX
  @techreport{MehU15, abstract = {Recently in [Gu (2011), Comput. Des. Integr. Circuits Syst., 30(9):1207--1320] a procedure was presented that allows to reformulate nonlinear ordinary differential equations in a way that all the nonlinearities become polynomial on the cost of increasing the dimension of the system. We generalize this procedure (called `polynomialization') to systems of differential-algebraic equations (DAEs). In particular, we show that if the original nonlinear DAE is regular and strangeness-free (i.\,e., it has differentiation index one) then this property is preserved by the polynomial representation. For systems which are not strangeness-free, i.\,e., where the solution depends on derivatives of the coefficients and inhomogeneities, we also show that the index is preserved for arbitrary strangeness index. However, to avoid ill-conditioning in the representation one should first perform an index reduction on the nonlinear system and then construct the polynomial representations. Although the analytical properties of the polynomial reformulation are very appealing, care has to be given to the numerical integration of the reformulated system due to additional errors. We illustrate our findings with several examples.}, address = {Str. des 17. Juni 136, D-10623 Berlin, FRG}, author = {Mehrmann, Volker and Unger, Benjamin}, file = {/homes/numerik/unger/WWW/Publications/MehU15_ppt.pdf}, institution = {Institut f{\"u}r Mathematik, TU Berlin}, number = {2--2015}, project = {SFB910, DAE}, title = {Index preserving polynomial representation of nonlinear differential-algebraic systems}, type = {Preprint}, url = {http://www3.math.tu-berlin.de/cgi-bin/IfM/show_abstract.cgi?Report-02-2015.rdf.html}, year = 2015 }
2013
1. B. Unger, “Impact of Discretization Techniques on Nonlinear Model Reduction and Analysis of the Structure of the POD Basis,” Master’s Thesis, Virginia Polytechnic and State University, Blacksburg, Virginia, USA, 2013. [Online]. Available: https://vtechworks.lib.vt.edu/handle/10919/24197
  - BibTeX
  BibTeX
  @phdthesis{Ung13, address = {Blacksburg, Virginia, USA}, author = {Unger, Benjamin}, language = {English}, school = {Virginia Polytechnic and State University}, title = {Impact of Discretization Techniques on Nonlinear Model Reduction and Analysis of the Structure of the {POD} Basis}, type = {Master's Thesis}, url = {https://vtechworks.lib.vt.edu/handle/10919/24197}, year = 2013 }

Funded Projects

Artificial intelligence meets molecular dynamics: certified machine learning for resolution transformation
Together with Dr. K. Pluhackova, Aug. 2021 – Dez. 2023
Decoupling integration schemes of higher order for poroelastic networks
Together with Dr. habil. R. Altmann, Oct. 2021 – Sept. 2024
Data-Integrated Low-Dimensional Surrogate Models with Guarantees
June 2021 – Oct. 2024

Talks

Invited Talks

Model reduction for transport phenomena via state-dependent projections
MOR Seminar (online), Universität Stuttgart, Apr. 2021.
Co-Design via port-Hamiltonian systems: advantages, challenges, and perspectives
Workshop of the GAMM activity group on dynamics and control theory (online), Feb. 2021.
Structure-Preserving Model-Order Reduction for Port-Hamiltonian Systems
Oberseminar Numerical Optimization (online), Universität Konstanz, Jan. 2021.
Kolmogorov n-widths and Hankel normapproximation
Applied Numerical Analysis Seminar (Virginia Tech), Blacksburg, USA, Feb. 2017.

Minisymposium Talks

Model reduction for mixed data and first-principle models
Mechanistic Machine Learning and Digital Twins for Computational Science, Engineering & Technology, San Diego, USA, Sept. 2021.
The Digital Twin Philosophy for Additive Manufacturing
SIAM Conference on Computational Science and Engineering, online, Mar. 2021.
Nonlinear Galerkin model reduction for transport-dominated problems
DMV Jahrestagung 2020, Chemnitz, Deutschland, Sept. 2020.
From Data to Structured Reduced-orderModels
SIAM Conference on Computational Science and Engineering, Spokane, USA, Feb. 2019.
Well-posedness and numerical solution of nonlinear delay differential-algebraic equations
International Conference on Scientific Computation and Differential Equations, Bath, UK, Sept. 2017.
Model Reduction, Transport Problems, and Structure
SIAM Conference on Computational Science and Engineering, Atlanta, USA, Feb. 2017.
Structured surrogate models of complex dynamical systems based on input/output data
7th European Congress ofMathematics (7ECM), Berlin, Germany, Juli 2016.
Structured input/output realization of dynamical systems
20th Conference of the International Linear Algebra Society, Leuven, Belgium, July 2016.

Selected other talks and poster

Passivity preserving MOR via spectral factorization
GAMM-Workshop 2021 -- Sion Digital online), Mar. 2021.
Asymptotic stability of neutral delay equations via semi-explicit time-integration
GAMM Jahrestagung (online), Kassel, Germany, Mar. 2021.
Delay regularity of differential-algebraic equations
58th IEEE Conference on Decision and Control, Nizza, France, Dec. 2019.
Model reduction for switched systems with low-rank switching (Poster)
IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 2018.
Delay Differential-Algebraic Equations: Primary Discontinuities and Regularization
Analysis andNumerical Approximation ofConstrained Systems, Sion, Switzerland, April 2018.
Kolmogorov n-widths and optimal Hankel norm approximation
GAMM Annual Meeting, Munich, Germany, March 2018
What is a well-posed delay differential-algebraic equation?
Workshop on Control of Self-OrganizingNonlinear Systems Lutherstadt Wittenberg, Germany, Aug. 2017.
Nonintrusive Realization Theory for Structured Problems
KoMSO Challenge Workshop Reduced-Order Modeling for Simulation and Optimization, Renningen, Germany, Nov. 2016.

Conference organization

Editorial Duties

16th IFACWorkshop on Time Delay Systems (TDS 2021)
Associate Editor
GAMM Archive for Students
Editor from 2018-2019, Founding Editor

Research Visits

Aug. 2019, Centre International De Rencontres Mathématiques, Lumini, France
Research in Pairsmit Dr. R. Altmann and Dr. R. Maier
June 2018, University of Groningen, Groningen, Netherlands
Invited by Prof. Dr. S. Trenn
Feb. 2017, Virginia Tech in Blacksburg, Virginia, USA
Invited by Prof. Dr. S. Gugercin

Supervised Students

Ph.D. students:

Birgit Hillebrecht, University of Stuttgart (since Nov. 2021)
Jonas Nicodemus, University of Stuttgart (since July 2021)
Abdullah Mujahid, Augsburg University and University of Stuttgart (since Nov. 2021)
Christian Pfaendner, University of Stuttgart (since Oct. 2021)

Master theses:

Abdullah Mujahid, Ruhr Universität Bochum
Master thesis (Oct. 2021): Monolithic and iterative discretizations of linear poroelasticity (together with P. Henning and R. Altmann)
Felix Black, Technische Universität Berlin
Master thesis (Jan. 2019): Model order reduction for transport phenomena (together with V. Mehrmann und P. Schulze)
Ines Ahrens, Technische Universität Berlin
Master thesis (Sept. 2018): A Structural Approach for Delay Differential-Algebraic Equations (together with V. Mehrmann)
Viola Paschke, Technische Universität Berlin
Master thesis (July 2016): Delayed Multibody Systems – Analysis and Numerical Solution (together with V. Mehrmann)

Scientific assistants, research projects, and interns:

Zubair Asif, University of Stuttgart, scientific assistant (since Nov. 2021)
Lisa Meisinger, University of Stuttgart, research project (since Oct. 2021)
Ahsan Ayaz, University of Stuttgart, scientific assistant (since June 2021)
Elliot Fosong, University of Cambridge, DAAD RISE internship (June-Sept. 2018)
Oumayma Bentamer, ENSEEIHT Toulouse, internship (Juni - Sept. 2017)
Yuliya Semibratova, University of Illinois at Urbana-Champaign, DAAD RISE internship (May - Aug. 2016)
Ines Ahrens, Technische Universität Berlin, scientific assistant (Apr. 2016 - Sept. 2018)
Viola Paschke, Technische Universität Berlin, scientific assistant (Feb. 2015 - Apr. 2016)

Teaching

University of Stuttgart (independent teaching)

Seminar: Control theory (summer term 2022)
Seminar: Novel methods in simulation science (summer term 2022)
Seminar: Advanved topics in simulation science (winter term 2021/22)

Technische Universität Berlin

Assistent Lineare Algebra I für Mathematiker (Sommersemester 2020)
Assistent Analysis II für Mathematiker (Sommersemester 2019)
Assistent Analysis I für Mathematiker (Wintersemester 2018/2019)
Assistent Analysis II für Ingenieure (Early Bird, Sommersemester 2018)
Assistent Lineare Algebra II für Mathematiker (Wintersemester 2017/18)
Assistent Lineare Algebra II für Mathematiker (Sommersemester 2017)
Assistent Lineare Algebra I+II für Mathematiker (Wintersemester 2016/17)
Tutor Analysis I für Ingenieure (Wintersemester 2015)
Organisator Absolventen-Seminar NumerischeMathematik (Februar 2015 - Juli 2018)

Karlsruhe Institute for Technology

Tutor NumerischeMathematik II (Sommersemester 2012)
Tutor NumerischeMathematik I (Wintersemester 2011/12)
Tutor VorkursMathematik (Wintersemester 2011/12)
Tutor Einstieg in die Informatik und algorithmischeMathematik mit C++ (Sommersemester 2011)
Tutor Einstieg in die Informatik und algorithmische Mathematik mit Java (Wintersemester 2010/11)
Tutor Vorkurs Mathematik (Wintersemester 2010/11)

Channels

Find me on GoogleScholar, ResearchGate, ORCID.

Team page

Benjamin Unger

Contact

Office Hours

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2023

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