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## Absolventen-Seminar • Numerische Mathematik

Dieses Semester wird das Seminar online auf Zoom stattfinden.

Verantwortliche Dozenten: | Prof. Dr. Tobias Breiten, , Prof. Dr. Christian MehlProf. Dr. Volker Mehrmann |
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Koordination: | Ines Ahrens |

Termine: | Do 10:00-12:00 |

Inhalt: | Vorträge von Bachelor- und Masterstudenten, Doktoranden, Postdocs und manchmal auch Gästen zu aktuellen Forschungsthemen |

### Amon Lahr (TU Berlin)

Donnerstag, 10. Dezember 2020

**Reduced-order design of suboptimal H∞ controllers using rational Krylov subspaces**

In the field of robust control, H∞ control provides an established framework to design control laws guaranteeing stability and performance over a range of perturbations of the nominal system model. The underlying mathematical problem is usually separated into finding the (sub)optimal attenuation (γ-iteration), and designing a stabilizing controller for which the H∞ norm of the closed-loop transfer function is not greater than γ. For large-scale systems, especially the γ\gamma-iteration proves to be computationally demanding as it requires the exact solution of two algebraic Riccati equations (ARE) in every step of the bisection method. Furthermore, the dimension of the obtained control law needs to be reduced for most practical applications.

In this talk, we introduce some of the challenges related to reduced-order design of H∞ controllers. Furthermore, an accelerated implementation of the γ-iteration is presented, which is based on low-rank approximations of the ARE solutions using rational Krylov subspaces. Therein, a reduced-order controller is constructed and verified at each bisection step using a large-scale H∞ norm computation method and the calculation of a few eigenvalues of the closed-loop matrix. The results are discussed by means of numerical examples arising from control of partial differential equations.

### Daniel Bankmann (TU Berlin)

Donnerstag, 10. Dezember 2020

**Multilevel Optimization Problems with Linear Differential-Algebraic Equations**

I'm going to practice my defense talk for my dissertation. The talk is supposed to last no more than 30 minutes. Feedback is very welcome.

We discuss different multilevel optimization problems in the context of linear differential-algebraic equations. On the one hand, we address multilevel optimal control problems, where sensitivity information of the necessary conditions of the optimal control problem can be used to compute solutions of the upper level problem. When the upper level is given by a nonlinear least-squares problem, we present a step size estimator. On the other hand, we show how the analytic center of the passivity LMI can be used as a good starting point in the computation of the passivity radius.

### Martin Isoz (UCT Praque)

Donnerstag, 26. November 2020

**Simulations of fully-resolved particle-laden flows: fundamentals and challenges for model order reduction**

Particle-laden flows are present in numerous aspects of day-to-day life ranging from technical applications such as fluidisation or filtration to medicinal problems, e.g. behavior of clots in blood vessels. Nevertheless, computational fluid dynamics (CFD) simulations containing freely moving and irreguralry shaped bodies are still a challenging topic. More so, if the bodies are large enough to affect the fluid flow and distributed densely enough to come in contact both with each other and with the computational domain boundaries. In this talk, we present a finite volume-based CFD solver for modeling flow-induced movement of interacting irregular bodies. The modeling approach uses a hybrid fictitious domain-immersed boundary method (HFDIB) for inclusion of the solids into the computational domain. The bodies movement and contacts are solved via the discrete element method (DEM). Unfortunately, the coupled HFDIB-DEM model structure causes significant limitations with respect to applications of standard projection-based methods of model order reduction (MOR). While we focus mostly on the HFDIB-DEM solver development, the talk is concluded by the challenges the HFDIB-DEM approach poses for MOR.

### Volker Mehrmann (TU Berlin)

Donnerstag, 19. November 2020

**Structured backward errors for eigenvalues associated with port-Hamiltonian descriptor systems **

When computing the eigenstructure of matrix pencils associated with the passivity analysis of perturbed port-Hamiltonian descriptor systems using a structured generalized eigenvalue method, one should make sure that the computed spectrum satises the symmetries that corresponds to this structure and the underlying physical system. We perform a backward error analysis and show that for matrix pencils associated with port-Hamiltonian descriptor systems and a given computed eigenstructure with the correct symmetry structure, there always exists a nearby port-Hamiltonian descriptor system with exactly that eigenstructure. We also derive bounds for how near this system is and show that the stability radius of the system plays a role in that bound.

V. Mehrmann and P. Van Dooren, Structured backward errors for eigenvalues of linear port-Hamiltonian descriptor systems, To appear in SIAM Journal Matrix Analysis and Applications, 2020. See arxiv.org/abs/2005.04744.

### Tim Moser (TU München)

Donnerstag, 05. November 2020

**A Riemannian Framework for Ecient H2-Optimal Model Reduction of Port-Hamiltonian Systems**

The port-Hamiltonian systems paradigm provides a powerful framework for the network modeling of multi-physics systems. By exploiting inherent system characteristics such as passivity, the modeling in port-Hamiltonian form also facilitates the subsequent controller design. Therefore it is advantageous to preserve the port-Hamiltonian structure in the model reduction process for which different approaches have been proposed (see e.g. [1], [2]).

In [1], a modified version of the iterative rational Krylov algorithm (IRKA-PH) was proposed for the H2-optimal model reduction of port-Hamilonian systems. Since IRKA-PH is based on Petrov-Galerkin projections, certain degrees of freedom must be given up in order to preserve the port-Hamiltonian structure. This inevitably leads to the fact that it is generally not possible to satisfy all necessary H2- optimality conditions in this projective framework.

We address this issue and propose a novel Riemannian framework for the H2-optimal reduction of port-Hamiltonian systems. We incorporate geometric constraints using the Riemannian problem formulation of [3] and exploit the computationally efficient pole-residue formulation of the H2-error proposed in [4]. By this means, preservation of the port-Hamiltonian structure and H2-optimality upon convergence are guaranteed and the framework is also accessible for the reduction of large-scale systems.

References

[1] S. Gugercin, R. V. Polyuga, C. Beattie, and A. van der Schaft, "Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems," Automatica, vol. 48, no. 9, pp. 1963-1974, 2012.

[2] R. V. Polyuga and A. J. van der Schaft, "Effort- and flow-constraint reduction methods for structure preserving model reduction of port-Hamiltonian systems," Systems & Control Letters, vol. 61, no. 3, pp. 412-421, 2012.

[3] K. Sato, "Riemannian optimal model reduction of linear port-Hamiltonian systems," Automatica, vol. 93, pp. 428-434, 2018.

[4] L. Meier and D. Luenberger, "Approximation of linear constant systems," IEEE Transactions on Automatic Control, vol. 12, no. 5, pp. 585-588, 1967.

# Rückblick

- Absolventen Seminar SS 20
- Absolventen Seminar WS 19/20
- Absolventen Seminar SS 19
- Absolventen Seminar WS 18/19
- Absolventen Seminar SS 18
- Absolventen Seminar WS 17/18
- Absolventen Seminar SS 17
- Absolventen Seminar WS 16/17
- Absolventen Seminar SS 16
- Absolventen Seminar WS 15/16
- Absolventen Seminar SS 15
- Absolventen Seminar WS 14/15
- Absolventen Seminar SS 14
- Absolventen Seminar WS 13/14
- Absolventen Seminar SS 13
- Absolventen Seminar WS 12/13
- Absolventen Seminar SS 12
- Absolventen Seminar WS 11/12