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## Absolventen-Seminar • Numerische Mathematik

Verantwortliche Dozenten: | Prof. Dr. Christian Mehl, Prof. Dr. Volker Mehrmann |
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Koordination: | Ines Ahrens |

Termine: | Do 10:00-12:00 in MA 376 |

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

Datum | Zeit | Raum | Vortragende(r) | Titel |
---|---|---|---|---|

Do 11.04. | 10:15 Uhr | MA 376 | no seminar | |

Do 18.04. | 10:15 Uhr | MA 376 | Vorbesprechung | |

Benjamin Unger | Feedback regularization of DAEs via delays [abstract] | |||

Do 25.04. | 10:15 Uhr | MA 376 | Volker Mehrmann | Numerical analysis of finite element systems modeling elastic stents [abstract] |

Do 02.05. | 10:15 Uhr | MA 376 | Christoph Zimmer | On the solvability of port-Hamiltonian partial differential equation with linear constraints [abstract] |

Paul Van Dooren | Role modeling using a low rank similarity matrix [abstract] | |||

Do 09.05. | 10:15 Uhr | MA 376 | Michal Wojtylak | The gap distance between linear pencils [abstract] |

Paul Schwerdtner | Structure Preserving or Realization Independent H-infinity Approximation [abstract] | |||

Do 16.05. | 10:15 Uhr | MA 376 | Ninoslav Truhar | On an Eigenvector-Dependent Nonlinear Eigenvalue Problem from Perspective of Relative Perturbation Theory [abstract] |

Dorothea Hinsen | A modeling of a power network with the telegraph equations [abstract] | |||

Do 23.05. | 10:15 Uhr | MA 376 | Jennifer Przybilla | Model Reduction of Differential-Algebraic Systems by Parameter-Dependent Balanced Truncation [abstract] |

Riccardo Morandin | Energy-based hierarchical modeling of power networks [abstract] | |||

Do 30.05. | 10:15 Uhr | MA 376 | no seminar | |

Do 06.06. | 10:15 Uhr | MA 376 | Pia Marie Lutum | Numerical Computation of the Real Structured Stability Radius [abstract] |

Michelle Stahl | Analysis of optimization methods for nonlinear functions to improve the geometry of injectors used for excavator ships [abstract] | |||

Do 13.06. | 10:15 Uhr | MA 376 | Serhiy Yanchuk | On absolute stability of delay differential equations with one discrete delay [abstract] |

Do 20.06. | 10:15 Uhr | MA 376 | Philipp Schulze | |

Philipp Krah | ||||

Do 27.06. | 10:15 Uhr | MA 376 | Marine Froidevaux | |

Do 04.07. | 10:15 Uhr | MA 376 | Ines Ahrens | |

Felix Black | ||||

Do 11.07. | 10:15 Uhr | MA 376 | Daniel Bankmann | |

# Rückblick

- 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

### Serhiy Yanchuk (TU Berlin)

Donnerstag, 13. Juni 2019

**On absolute stability of delay differential equations with one discrete delay**

In this talk I discuss the properties of the eigenvalues of linear systems of DDEs of the form x'(t)=Ax(t) + B x(t-τ) with one discrete delay τ and constant matrices A and B. In particular, I show how the asymptotic spectrum for large delay determines the absolute stability, i.e. the stability for all positive delays.

### Pia Lutum (TU Berlin)

Donnerstag, 06. Juni 2019

**Numerical Computation of the Real Structured Stability Radius**

The real structured stability radius is a measurement of the robustness of a stable matrix under certain perturbations. In this talk we develop an algorithm to compute the real structured stability radius for large-scale linear time-invariant systems in the continuous-time case. Therefore, we first use model order reduction, especially rational interpolation. In the second step, we use a level-set method to compute the real structured stability radius for reduced systems. We improve this method by combining it with an algorithm which calculates the eigenvalues of an arising Hamiltonian matrix more precisely and faster. We illustrate the efficiency of the developed method using several numerical examples.

### Michelle Stahl (TU Berlin)

Donnerstag, 06. Juni 2019

**Analysis of optimization methods for nonlinear functions to improve the geometry of injectors used for excavator ships**

In this talk we will deal with numerical optimization methods for nonlinear functions to improve the geometry of injectors applied in dredging.

The reason is a working ship that is used as an hydraulic excavator. It uses 36 equal nozzles for water injection but it is known, that using nozzles of different sizes lead to better results.

For improving the geometries as a multidimensional problem I will introduce two optimization methods that work without use of derivatives. The first one is a simplex algorithm, called “Nelder-Mead method” or “Downhill- Simplex method”. Then I will show the idea and the application of the evolution strategy by Ingo Rechenberg, a genetic algorithm that mimics the biological evolution.

### Jennifer Przybilla (TU Berlin)

Donnerstag, 23. Mai 2019

**Model Reduction of Differential-Algebraic Systems by Parameter-Dependent Balanced Truncation**

In this talk we will deal with the application of balanced truncation for parameter-dependent differential-algebraic systems. For this, we have to solve parameter-dependent Lyapunov equations, which we do with the help of the reduced-basis method. In order to use this method, we first have to deal with the algebraic parts of the system and make the system strictly dissipative in order to apply error estimators in the reduced basis method.

### Riccardo Morandin (TU Berlin)

Donnerstag, 23. Mai 2019

**Energy-based hierarchical modeling of power networks**

Recent developments in the energy market require new mathematical models and suitable algorithms for the efficient usage of the existing networks. Starting from these real-world problems, it is necessary to concentrate on methods for large-scale energy networks and, in particular, address optimization and stability analysis, model predictive control, model-order reduction, uncertainty quantification, and related topics on energy networks. The abstract setting allows for consideration of applications arising from both, gas and power networks.

Our framework of choice is the one of port-Hamitlonian descriptor systems, or pHDAEs. These are energy-based equations with a special structure, that guarantees several beneficial properties, e.g. inherent stability and passivity, physical interpretation of variables, simple interconnection, structure-preserving model reduction, robust numerical integration and simplification of feedback stabilization.

In this talk we present a model hierarchy for some components of a power grid, specifically synchronous generators and transmission lines, where all equations presented are pHDAEs. Furthermore, we show how these components can be easily interconnected to form a larger network, while preserving the port-Hamiltonian structure. To do so, we apply Kirchhoff's laws and exploit the common structure of the electrical components, in a way that can be easily extended to other devices and more complex models.

### Dorothea Hinsen (TU Berlin)

Donnerstag, 16. Mai 2019

**A modeling of a power network with the telegraph equations **

In recent years energy transition and the increasing electricity demand have led to growing interest in modeling power networks, which withstand unexpected occurrences as voltage or transient instabilities.

One way to approach modeling power networks is with port-Hamiltonian systems.

The power networks we are dealing with consist of generators, loads and transmission lines.

In this talk we discuss an approach to a power network model, where the generators and the loads are described with port-Hamiltonian equations.

However, the transmission lines are depicted with the telegraph equations. This leads us to a PDAE model, which we will discuss.

### Ninoslav Truhar (Josip Juraj Strossmayer University of Osijek)

Donnerstag, 16. Mai 2019

**On an Eigenvector-Dependent Nonlinear Eigenvalue Problem from Perspective of Relative Perturbation Theory **

The talk contains two part. I the first part we consider the eigenvector-dependent nonlinear eigenvalue problem (NEPv) H(V) V = V Λ, where H(V) ∈ C^{n,n} is an Hermitian matrix-valued function of V ∈ C^{n,k} with orthonormal columns, i.e., V^{H} V = I_{k}, k < n (usually k ≪ n). We present the conditions on existence and uniqueness for the solvability of NEPv using the well known results of the relative perturbation theory. Our results are motiveted by the results on NEPv presented in Y. Cai, L.-H. Zhang, Z. Bai, and R.-C. Li, *On an eigenvector-dependent nonlinear eigenvalue problem*, SIAM J. Matrix Anal. Appl. 2018, where among the other results one can find conditions for existence and uniqueness for the solvability of an NEPv. These results are based on well-known standard perturbation theory for Hermitian matrices. The differences between so called standard perturbation theory approach, and our new (relative perturbation theory) approach have been illustrated in several numerical examples.

In the second part we present an upper and a lower bound for the the Frobenius norm of the matrix sin(Θ), of the sines of canonical angles between unperturbed and perturbed eigenspaces of a regular generalized Hermitian eigenvalue problem A x = λ B x where A and B are Hermitian n × n matrices, under a feasible non Hermitian perturbation. As one application of the obtained bounds we present the corresponding upper and the lower bounds for eigenspaces of a matrix pair (A,B) obtained by a linearization of regular quadratic eigenvalue problem ( λ^{2} M + λ D + K ) u = 0, where M is positive definite and D and K are semidefinite. We also apply obtained upper and lower bounds to the important problem which considers the influence of adding a damping on mechanical systems. The new results show that for certain additional damping the upper bound can be too pessimistic, but the lower bound can reflect a behaviour of considered eigenspaces properly.

### Michal Wojtylak (Jagiellonian University, Krakow)

Donnerstag, 09. Mai 2019

**The gap distance between linear pencils**

We introduce a new distance between linear pencils, which is based on their graphs ker[A,-E]. Several basic properties of this distance will be shown. Next, we will formulate the problem of distance to singularity and compare it with the original one given by Byers, He and Mehrmann in 1998. The talk is based on a joint work: Berger, Gernandt, Trunk, Winkler, MW, LAA 2019.

### Paul Schwerdtner (TU Berlin)

Donnerstag, 09. Mai 2019

**Structure Preserving or Realization Independent H-infinity Approximation**

In this talk we revisit linorm_subsp, an algorithm that is used to compute the H-infinity norm of transfer functions of large-scale dynamical systems and show how it can be made more efficient for irrational transfer functions using an inner loop outer loop strategy.

Then, we present a greedy interpolation approach to address the H-infinity model order reduction problem that is using the evaluation of the H-infinity norm computation provided by linorm_subsp.

Starting from an initial reduced order model, we compute the H-infinity norm of the error transfer function and then place an new interpolation point where this H-infinity norm is attained. In this way, we construct a sequence of reduced order models aiming at minimizing the H-infinity norm of the error transfer function.

Using interpolation for this allows to either preserve the given model structure or construct surrogate models with a structure defined by the user to approximate the given system, respectively.

### Christoph Zimmer (TU Berlin)

Donnerstag, 02. Mai 2019

**On the solvability of port-Hamiltonian partial differential equation with linear constraints **

Through its natural approach of energy as underlying quantity and its many nice properties, port-Hamiltonan (pH) systems rose as a popular modeling tool for dynamical system like electrical, mechanical, and hydrodynamical ones. Therefore, it is not surprising that the concept was extended to non-autonomous systems, to descriptor systems and to infinite dimensional systems. In this talk, we investigate non-autonomous semi-explicit pH descriptor systems. Mainly focusing on the existence of solution, we will start with finite dimensional systems. Afterwards, we will transfer the used tricks to the infinite dimensional case and prove the existence of solutions as well as their smoothness.

### Paul Van Dooren (Catholic University of Louvain)

Donnerstag, 02. Mai 2019

**Role modeling using a low rank similarity matrix**

Community detection is a popular approach used analyze large networks and obtain relevant statistical properties by finding community structures in networks. However, community detection algorithms cannot be used to find non-community structures in networks, such as cyclic graph structures which appear in food web networks. The role extraction problem represents large networks by a smaller, general graph structure, called role graphs. In this presentation, we analyze the neighborhood pattern similarity measure used to solve the role extraction problem. We show theoretically that under certain assumptions the role graphs can be recovered from a low-rank factorization of the similarity matrix due to the relationship between the dominant eigenvalues of the similarity matrix and the number of roles. We also show how it applies to the detection of overlapping communities and bipartite communities.

### Volker Mehrmann (TU Berlin)

Donnerstag, 25. April 2019

** Numerical analysis of finite element systems modeling elastic stents**

A new model description for the numerical simulation of elastic stents is proposed. Based on the new formulation an inf-sup inequality for the finite element discretization is proved and the proof of the inf-sup inequality for the continuous problem is simplified. The new formulation also leads to faster simulation times despite an increased number of variables. The techniques also simplify the analysis and numerical solution of the evolution problem describing the movement of the stent under external forces. The results are illustrated via numerical examples, see [1].

[1] L. Grubišić, M. Ljulj, V. Mehrmann, and J. Tambača, Modeling and discretization methods for the numerical simulation of elastic stents, arxiv.org/1812.10096, Preprint 01-2019, Institute of Mathematics, TU Berlin, submitted for publication, 2019.

### Benjamin Unger (TU Berlin)

Donnerstag, 18. April 2019

**Feedback regularization of DAEs via delays**

We study linear time-invariant delay differential-algebraic equations (DDAEs). Such equations arise naturally, if a feedback controller is applied to a descriptor system, since 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. As a consequence, we show that a DAE can be regularized by a feedback if and only if it can be regularized by a delayed feedback. This is joint work with Stephan Trenn (University of Groningen).