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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 | Ute Kandler | Inexact methods for the solution of large scale Hermitian eigenvalue problems [abstract] |

Philipp Krah | Towards mode-based model order reduction for flows with sharp fronts and complex topolgies [abstract] | |||

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

Do 04.07. | 10:15 Uhr | MA 376 | Ines Ahrens | The Pantelides Algorithm for DAEs with delay [abstract] |

Felix Black | Computation of reduced order models for transport phenomena via shifted proper orthogonal decomposition [abstract] | |||

Do 11.07. | 10:15 Uhr | MA 376 | Daniel Bankmann | Optimizing condition numbers in robust stabilization [abstract] |

Marine Froidevaux | Structure-preservation in non-linear eigenvalue problems arising from simulations of photonic crystals [abstract] |

# 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

### Daniel Bankmann (TU Berlin)

Donnerstag, 11. Juli 2019

**Optimizing condition numbers in robust stabilization**

Stabilization is a well-established task for linear control systems, in which one tries to push all the eigenvalues of the system matrix via a feedback to a region contained in the complex half plane while optimizing certain condition numbers as the condition number of the closed loop matrix.

The feedback gain can either be computed by pole-placement techniques or by solving an optimal control problem with the same resulting feedback.

It has been shown, that for the numerically robust computation of these feedback matrices, it is necessary to compute the solutions of the equivalent optimal control problem via structure-exploiting computations of deflating subspaces of an associated even matrix pencil. However, also in that case, the computation of the feedback may be ill-conditioned for a bad choice of weight parameters in the optimal control problem.

In this talk we present an approach for detecting when this computation is ill-conditioned. Also, we construct relations that only keep well-conditioned parameters and decrease the dimension of the parameter set. Finally, we show with some small study example, that with this smaller parameter set, the results of the overall optimization may improve in accuracy.

### Marine Froidevaux (TU Berlin)

Donnerstag, 11. Juli 2019

**Structure-preservation in non-linear eigenvalue problems arising from simulations of photonic crystals**

Band-structure diagrams corresponding to photonic crystals can be numerically approximated by combining and discretizing the famous Maxwell equations of electromagnetism. In this talk we will focus on 2D photonic crystals where the electric permittivity is approximated by the rational Drude-Lorentz model. A finite element discretization of the system leads to parametric non-linear eigenvalue problems (NLEVPs) that are typically large, sparse and possess a symmetry in the spectrum.

Possible approaches to solve the NLEVPs are the use of contour integral methods, Newton-type methods or some linearization “by hand” followed by an iterative Krylov method. We will focus on the third approach and recall how to reduce the large-scale matrix equation with the help of the reduced basis method. We will see how to preserve the structure of the problem through the linearization and model reduction steps, as well as during the solution of the reduced system. We will also see that the extension of an existing structure-preserving Krylov method to large-scale complex even problems contains complications.

### Ines Ahrens (TU Berlin)

Donnerstag, 04. Juli 2019

**The Pantelides Algorithm for DAEs with delay**

The strangeness index for DAEs is based on the derivative array, which consists of the system itself plus its time derivatives. Index reduction is performed by selecting certain important equations from the derivative array. In a large-scale setting with high index, this might become computationally infeasible. However, if it is known a priori which equations of the original system need to be differentiated, then the computational cost can be reduced. One way to determine these equations is the Pantelides algorithm.

If the DAE features in addition a delay term, taking derivatives might not be sufficient, and instead, the derivative array must be shifted in time, thus increasing the computational complexity even further. In this talk I will explain how one can modify the Pantelides algorithm such that it determines the equations which need to be shifted and/or differentiated. This is joint work with Benjamin Unger.

### Felix Black (TU Berlin)

Donnerstag, 04. Juli 2019

**Computation of reduced order models for transport phenomena via shifted proper orthogonal decomposition**

Many classical model order reduction methods are formulated in a projection framework, building the reduced order model (ROM) by projecting the full order model (FOM) onto a suitable subspace. If the FOM exhibits advective transport, classical methods often fail to produce low-dimensional models with a small approximation error. One strategy to remedy this problem is the shifted proper orthogonal decomposition (shifted POD), which extends the classical POD by introducing additional transformation operators associated with the modes. The transformation operators are parametrized by paths in a suitable parameter space, thus allowing the transformed modes to cope with the convection. In contrast to classical methods that project onto a fixed subspace, the ROM of the shifted POD is thus obtained by projecting onto a time- and/or state-dependent subspace that adapts itself to the problem. On the one hand this approach is very flexible, on the other hand it introduces additional complexity to the online stage, since in addition to the time-dependent coefficients, also the paths need to be computed from the ROM. In this talk, we present the online stage of the shifted POD and discuss how to build a ROM from which we may compute the coefficients and the paths. This is joint work with Philipp Schulze and Benjamin Unger.

### Ute Kandler (Fraunhofer Institut)

Donnerstag, 20. Juni 2019

**Inexact methods for the solution of large scale Hermitian eigenvalue problems**

This talk focuses on the solution of high dimensional Hermitian eigenproblems in situations where vector operations cannot be carried out exactly. To this end an inexact Arnoldi method with the aim to approximate extreme eigenvalues and eigenvectors is introduced. This method is particularly well-suited for large scale problems as it efficiently reduces the storage and computational requirements by constructing an orthonormal basis of the associated Krylov subspace. In particular, we investigate the influence of the inexactness of the vector operations on the convergence behavior of the inexact Arnoldi method and more general of inexact Krylov subspace methods. By using the concept of the angle of inclusion, a bound for the distance of the exact invariant subspace to an inexact Krylov subspace is analysed. To avoid the sensitivity of the bound to the gap between the desired and the remaining eigenvalues the concept of nested subspaces is used.

### Philipp Krah (TU Berlin)

Donnerstag, 20. Juni 2019

**Towards mode-based model order reduction for flows with sharp fronts and complex topolgies**

I will present a model order reduction approach based on snapshots of 1D/2D flow fields, where the flow features sharp fronts. The approach parametrizes the flow field with the help of a levelset function which is zero at the location of the front and a nonlinear function which approximates the jump. I will explain how the freedom of choice of the levelset function can help to improve mode-based model order reduction. Therefore I will consider a 1D and 2D example of synthetic data and a 2D example of a simulated burning flame.

### 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).