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## Absolventen-Seminar • Numerische Mathematik

Verantwortliche Dozenten: |
Prof. Dr. Christian Mehl [1], Prof. Dr. Volker Mehrmann
[2] |
---|---|

Koordination: | Benjamin
Unger, Dr. Matthias Voigt [3] |

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 18.10. | 10:15 Uhr | MA
376 | Vorbesprechung | |

Do
25.10. | 10:15 Uhr | MA 376 | Alexander Grimm
[4] | Empirical least-squares fitting of
parametrized dynamical systems [abstract] |

Do 01.11. | 10:15 Uhr | MA 376 | [5]no seminar | |

Do
08.11. | 10:15 Uhr | MA 376 | Andrii
Dmytryshyn [6] | Geometry of matrix
polynomial spaces [abstract] |

Christian Mehl [7] | Generic
low-rank perturbations of matrix pencils with symmetry structures
[abstract] | |||

Do
15.11. | 10:15 Uhr | MA 376 | no
seminar | |

Do 22.11. | 10:15 Uhr | MA
376 | Riccardo Morandin [8] | A new formulation for port-Hamiltonian
differential-algebraic equations [abstract] |

Olga Markova | Length realizability question for pairs of quasi-commuting
matrices [abstract] | |||

Do 29.11. | 10:15 Uhr | MA 376 | no
seminar | |

Do 06.12. | 10:15 Uhr | MA 376 | Arbi
Moses Badlyan | - canceled - |

Volker Mehrmann [9] | Computation of Stability Radii for Large-Scale Dissipative
Hamiltonian Systems [abstract] | |||

Do 13.12. | 10:15 Uhr | MA 376 | Daniel
Bankmann | Adjoint sensitivity equations in
optimal control of differential-algebraic equations
[abstract] |

Paul Schwerdtner
[10] | Robust Control of Delay Systems
[abstract] | |||

Do
20.12. | 10:15 Uhr | MA 376 | Arbi Moses
Badlyan | Generalized Port-Hamiltonian
Systems - Turbulent Pipe Flow [abstract] |

Do
10.01. | 10:15 Uhr | MA 376 | Pia Lutum | Numerical Computation of the Real Structured Stability
Radius [abstract] |

Philipp Schulze [11] | - canceled
- | |||

Do
17.01. | 10:15 Uhr | MA
376 | Manuel Radons [12] | Semi-automatically optimized calibration of internal
combustion engines [abstract] |

Dorothea Hinsen | - canceled - |
|||

Do 24.01. | 10:15 Uhr | MA
376 | Rico Berner [13] | Multi-cluster structures in networks of adaptively coupled
oscillators [abstract] |

Vander Freitas | Symmetric
circular formations with unitary speed particles
[abstract] | |||

Do
31.01. | 10:15 Uhr | MA 376 | Marine
Froidevaux | Model order reduction
applied to photonic crystals [abstract] |

Tim Mitchell [14] | H-infinity controller design with stability guarantees for
large-scale systems [abstract] | |||

Do 07.02. | 10:15 Uhr | MA 376 | Matthias Voigt
[15] | A Subspace Framework for H-infinity-Norm
Minimization [abstract] |

Christoph Zimmer | Time
discretization schemes for hyperbolic PDAEs by Epsilon-expansion
[abstract] | |||

Do
14.02. | 10:15 Uhr | MA
376 | Benjamin Unger | Port-Hamiltonian Dynamic Mode Decomposition
[abstract] |

Ines Ahrens
[16] | Finding Structure with Randomness –
Randomized Singular Value Decomposition and other Decompositions
[abstract] |

# Rückblick

- Absolventen Seminar SS 18 [17]
- Absolventen Seminar WS 17/18 [18]
- Absolventen Seminar SS 17 [19]
- Absolventen Seminar WS 16/17 [20]
- Absolventen Seminar SS 16 [21]
- Absolventen Seminar WS 15/16 [22]
- Absolventen Seminar SS 15 [23]
- Absolventen Seminar WS 14/15 [24]
- Absolventen Seminar SS 14 [25]
- Absolventen Seminar WS 13/14 [26]
- Absolventen Seminar SS 13 [27]
- Absolventen Seminar WS 12/13 [28]
- Absolventen Seminar SS 12 [29]
- Absolventen Seminar WS 11/12 [30]

### Alexander Grimm (Universität Suttgart)

Donnerstag, 25. Oktober 2018

**Empirical
least-squares fitting of parametrized dynamical systems**

Given a set of response observations for a parametrized dynamical system, we seek a parametrized dynamical model that will yield uniformly small response error over a range of parameter values yet has low order. Frequently, access to internal system dynamics or equivalently, to realizations of the original system is either not possible or not practical; only response observations over a range of parameter settings might be known. Respecting these typical operational constraints, we propose a two phase approach that first encodes the response data into a high fidelity intermediate model of modest order, followed then by a compression stage that serves to eliminate redundancy in the intermediate model. For the first phase, we extend non-parametric least-squares fitting approaches so as to accommodate parameterized systems. This results in a (discrete) least-squares problem formulated with respect to both frequency and parameter that identifies "local" system response features. The second phase uses an H2-optimal model reduction strategy accommodating the specialized parametric structure of the intermediate model obtained in the first phase. The final compressed model inherits the parametric dependence of the intermediate model and maintains the high fidelity of the intermediate model, while generally having dramatically smaller system order. We provide a variety of numerical examples demonstrating our approach.

### Andrii Dmytryshyn (Umeå University, Sweden)

Donnerstag, 08. November 2018

**Geometry of matrix polynomial
spaces**

We study how small perturbations of matrix polynomials may change their elementary divisors and minimal indices by constructing the closure hierarchy graphs (stratifications) of orbits and bundles of matrix polynomial Fiedler linearizations. We show that the stratification graphs do not depend on the choice of Fiedler linearization which means that all the spaces of the matrix polynomial Fiedler linearizations have the same geometry (topology). We also develop the theory for structure preserving stratification of skew-symmetric matrix polynomials. The results are illustrated by examples using the software tool Stratigraphy.

This is a joint work with Stefan Johansson (Umeå University), Bo Kågström (Umeå University), and Paul Van Dooren (Université catholique de Louvain).

### Christian Mehl (TU Berlin)

Donnerstag, 08. November 2018

**Generic low-rank
perturbations of matrix pencils with symmetry structures**

In this talk, we discuss the effect that generic structure-preserving low-rank perturbations have on the Jordan structure of a given matrix pencil with symmetry structures. The main result relies on a specific parameterizations of the sets of structured matrix pencils with a given rank.

### Riccardo Morandin (TU Berlin)

Donnerstag, 22. November 2018

**A new formulation for port-Hamiltonian
differential-algebraic equations**

Coordinate-based port-Hamiltonian system formulations in the literature are usually either limited to represent ODEs, or to be associated with square systems and quadratic Hamiltonian functions. In this talk, I will present a new formulation for port-Hamiltonian differential-algebraic equations (pHDAEs), that does not have the previous restrictions.

Two equivalent formulations are also provided, that are useful for
simplifying the notation, proving some additional results and reaching
a better understanding of the new definition. It is proven that this
class of pHDAEs is closed under a large class of variable
transformations and under linear system interconnection.

A geometric interpretation (Dirac structure) is also associated to
the new formulation, and will be useful to extend it to a more
abstract context. Collocation-based time integration is analyzed, and
with additional assumptions a discrete port-Hamiltonian structure is
achieved.

Furthermore, the definition is extended to be applied to
infinite-dimensional systems (e.g. PDEs), while preserving most of the
previous results.

Finally, a few examples of systems that can be represented with this new formulation are shown.

### Olga Markova (Lomonosov State University, Moscow, Russia)

Donnerstag, 22. November 2018

**Length realizability question
for pairs of quasi-commuting matrices**

By the length of a finite system of generators for a finite-dimensional algebra over an arbitrary field we mean the least positive integer k such that the products of length not exceeding k span this algebra (as a vector space). The maximum length for the systems of generators of an algebra is referred to as the length of the algebra. The length evaluation can be a difficult problem, for example, the length of the full matrix algebra is unknown (Paz’s Problem, 1984).

In this talk we discuss the length evaluation problem for quasi-commuting pairs of matrices (we say that matrices A and B quasi-commute if the products AB and BA are linearly dependent). We provide sharp upper and lower bounds for the length of such pairs and show how the interval between these extremal values is divided into intervals of realizable values for the length and ``gaps'', i.e. non-realizable values.

The talk is based on a joint work with Alexander Guterman and Volker Mehrmann.

### Arbi Moses Badlyan (TU Berlin)

Donnerstag, 06. Dezember 2018

**Generalized Port-Hamiltonian
Systems – Turbulent Pipe Flow **

In this talk I present the notion of a generalized port-Hamiltonian system based on modified, explicitly state dependent Stokes-Dirac structures.

I will demonstrate the mathematical framework by taking the example of a turbulent pipe flow. The mathematical model of this pipe flow is given as system of coupled partial-differential equations complemented by closure relations and contains a non-local heat-conduction term. The presented results are part of a joint work with Prof. Volker Mehrmann (TU Berlin).

### Volker Mehrmann (TU Berlin)

Donnerstag, 06. Dezember 2018

**Computation of Stability
Radii for Large-Scale Dissipative Hamiltonian Systems**

A linear time-invariant dissipative Hamiltonian (DH) system is always Lyapunov stable and under weak further conditions even asymptotically stable. In various applications there is uncertainty on the system matrices, and it is desirable to know whether the system remains asymptotically stable uniformly against all possible uncertainties within a given perturbation set. Such robust stability considerations motivate the concept of stability radius for DH systems, i.e., what is the maximal perturbation permissible to the coefficients, while preserving the asymptotic stability. We consider two stability radii, the unstructured one where the coefficients are subject to unstructured perturbation, and the structured one where the perturbations preserve the DH structure.

We propose new algorithms to compute these stability radii for large scale problems by tailoring subspace frameworks that are interpolatory and guaranteed to converge at a super-linear rate in theory. At every iteration, they first solve a reduced problem and then expand the subspaces in order to attain certain Hermite interpolation properties between the full and reduced problems. The reduced problems are solved by means of the adaptations of existing level-set algorithms for H-infinity-norm computation in the unstructured case, while, for the structured radii, we benefit from algorithms that approximate the objective eigenvalue function with a piece-wise quadratic global underestimator. The performance of the new approaches is illustrated with several examples including a system that arises from a finite-element modeling of an industrial disk brake.

Joint work with N. Aliyev, V. Mehrmann, and E. Mengi

### Daniel Bankmann (TU Berlin)

Donnerstag, 13. Dezember 2018

**Adjoint sensitivity equations in optimal
control of differential-algebraic equations **

Optimal control problems for differential-algebraic equations appear in a variety of applications coming from electrical or mechanical engineering.

Sometimes, the system's descriptions are also depending on parameters and one is not only interested in computing the solution of such a problem. In addition, we are interested in computing how the solution changes for small changes in the parameters, i. e. the sensitivities of the optimal solution with respect to the parameters.

Solutions of the optimal control problem can be characterized by so-called necessary conditions. These constitute a boundary value problem for differential-algebraic equations.

In the current literature, only sensitivity analysis for initial value problems of differential-algebraic equations or boundary value problems for ordinary differential equations has been carried out.

We close this gap, by formulating the correct boundary conditions based on the standard adjoint equations for differential-algebraic equations of index 1. This includes jump conditions on certain variables which pose new challenges on solvability and existence of solutions.

In this talk, we present an adjoint sensitivity boundary value problem for index 1 differential-algebraic equations with boundary values. Based on a flow formulation, we analyze properties like existence and uniqueness and apply these results to the necessary conditions coming from the optimal control problem.

### Paul Schwerdtner (TU Berlin)

Donnerstag, 13. Dezember 2018

**Robust Control of Delay
Systems**

We review the shortcomings of an algorithm that computes the H-infinity norm of possibly irrational and large scale transfer functions using a subspace projection based approach. After that, we show how these can be overcome by employing rational interpolation in the Loewner matrix framework and compare our new approach to state-of-the-art methods in various numerical examples.

After that, we utilize the new computation method for H-infinity norms in an optimization loop to compute robust fixed order controllers for time delay systems. Specifically, we alter the controller parameters of a given fixed order controller to optimize the H-infinity norm of the resulting closed loop transfer function.

Furthermore, we show how the realization independence of the H-infinity norm computation method can be exploited to facilitate imposing performance requirements on the closed loop transfer function.

Finally, we describe future work that is concerned with an efficient check for asymptotic stability of delay systems, since that is currently the bottleneck in our optimization based approach.

### Arbi Moses Badlyan (TU Berlin)

Donnerstag, 20. Dezember 2018

**Generalized Port-Hamiltonian
Systems - Turbulent Pipe Flow**

In this talk I will present the notion of a generalized port-Hamiltonian system based on explicitly state dependent Stokes-Dirac structures.

I will demonstrate the mathematical framework by taking the example of a turbulent pipe flow. The mathematical model of this pipe flow is given as system of coupled partial-differential equations complemented by closure relations and contains a non-local transport term. The presented results are part of a joint work with Prof. Christopher Beattie (Virginia Tech) and Prof. Volker Mehrmann (TU Berlin).

### Pia Lutum (TU Berlin)

Donnerstag, 10. Januar 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 real perturbations. In this talk we briefly discuss an iterative algorithm to compute the stability radius. Based on this algorithm, we further discuss an iterative method to compute the real structured stability radius for large-scale matrices by projections.

### Manuel Radons (TU Berlin)

Donnerstag, 17. Januar 2019

**Semi-automatically optimized calibration
of internal combustion engines**

Modern combustion engines incorporate a number of actuators and sensors that can be used to control and optimize the performance and emissions. We describe a semi-automatic method to simultaneously measure and calibrate the actuator settings and the resulting behavior of the engine. The method includes an adaptive process for refining the measurements, a data cleaning step, and an optimization procedure. The optimization works in a discretized space and incorporates the conditions to describe the dependence between the actuators and the engine behavior as well as emission bounds. We demonstrate our method on practical examples.

### Rico Berner (TU Berlin)

Donnerstag, 24. Januar 2019

**Multi-cluster structures in
networks of adaptively coupled oscillators**

Dynamical systems on networks with adaptive couplings appear naturally in real-world systems such as power grid networks, social networks as well as neuronal networks. We investigate collective behaviour in a paradigmatic network of adaptively coupled phase oscillators. The coupling topology of the network changes slowly depending on the dynamics of the oscillators. We show that such a system gives rise to numerous complex dynamics, including relative equilibria and hierarchical multi-cluster states. An analytic treatment for equilibria and multi-cluster solutions as well as the existence of continuous families of these states is presented and parameter regimes of high multi-stability are found. In addition, we give an interpretation for equilibria as functional units which are building blocks in multi-cluster structures. Our results contribute to the understanding of mechanisms for pattern formation in adaptive networks, such as the emergence of multi-layer structure in neural systems.

### Vander Freitas (National Institute for Space Research INPE, Brazil)

Donnerstag, 24. Januar 2019

**Symmetric circular formations with unitary
speed particles**

Several living beings such as birds, fish, flies and bees perform collective motion when foraging, migrating or escaping from predators. They are able to flock to a certain direction, rotate around a common center, or swarm within an area. Biologists try to understand the local rules that lead single individuals to macroscopic group behaviors. Inspired on that, it is possible to develop models for mobile vehicles, whose applications include data collection, surveillance, monitoring, etc. Motivated by such problems, we study a model of particles with phase-coupled oscillators dynamics, focusing on the special case in which they rotate around a common center and are able to group into clusters. Those formations arise from an optimization procedure of certain potentials. We then investigate the parameter space, the impact of adding and removing particles, and how to exchange from one configuration to another.

### Marine Froidevaux (TU Berlin)

Donnerstag, 31. Januar 2019

**Model order reduction applied to photonic
crystals**

Photonic crystals are composite materials having a periodic structure which affects the propagation of light. Their optical characteristics depend on the composing materials and on the geometry of their periodic structure. We are interested in analyzing the optical characteristics of multiple photonic crystal types by computing their so-called band diagram, i.e. the graph of the eigenfrequencies versus the wavevector.

In order to build these band diagrams for some given parameter values, multiple large-scale nonlinear eigenvalue problems derived from the Maxwell equations need to be solved. In this talk, we will show how a reduced model accounting for several parameters can be built using the reduced-basis method and present some numerical results.

### Tim Mitchell (MPI Magdeburg)

Donnerstag, 31. Januar 2019

**H-infinity controller design
with stability guarantees for large-scale systems**

We consider the problem of designing low-order controllers for large-scale linear time-invariant (LTI) dynamical systems with input and output. While the high cost of working with large-scale systems can mostly be avoided by first applying model order reduction, this can often result in controllers which fail to stabilize the closed-loop plant of the original full-order system. In this talk, we discuss two different approaches to guaranteeing stability of the original large-scale system. The first works by working directly with the large-scale data but requires H-infinity norm approximation techniques to remain tractable, while the second incorporates both full- and reduced-order model data in order to remain efficient.

### Matthias Voigt (TU Berlin)

Donnerstag, 07. Februar 2019

**A Subspace Framework for H-infinity-Norm
Minimization**

We deal with the minimization of the H-infinity-norm of the transfer function of a parameter-dependent descriptor system. A subspace framework is proposed for such minimization problems where the involved systems are of large order. The algorithm is a greedy interpolatary approach inspired by our recent work [Aliyev et al., SIAM J. Matrix Anal. Appl., 38(4):1496--1516, 2017] for the computation of the L-infinity-norm. In this work, we minimize the H-infinity-norm of a reduced-order parameter-dependent system obtained by two-sided Petrov-Galerkin projections onto certain subspaces. Then we expand the subspaces so that Hermite interpolation properties hold between the full and reduced-order system at the optimal parameter value for the reduced order system. From that we can establish a superlinear rate of convergence under some smoothness assumptions.

Joint work with Nicat Aliyev (Istanbul Zaim Sabahattin University), Peter Benner (MPI Magdeburg), and Emre Mengi (Koc University, Instanbul).

### Christoph Zimmer (TU Berlin)

Donnerstag, 07. Februar 2019

**Time discretization schemes
for hyperbolic PDAEs by Epsilon-expansion**

Transport problems like the transport of gas in a single pipe or electrical energy on a transmission line can be described by hyperbolic partial differential equations (PDEs). If one considers a network of pipes or transmission lines, the topology of the network introduces additional constraints. The resulting system of equations is then a constrained hyperbolic PDE (PDAE).

In this talk we recall how hyperbolic PDAEs with a slow and a fast moving state can be approximated by parabolic PDAEs and we extend this approach to get approximations of higher order. Furthermore, we discuss how time discretization affects the approximation.

This is joint work with Robert Altmann (Uni Augsburg).

### Benjamin Unger (TU Berlin)

Donnerstag, 14. Februar 2019

**Port-Hamiltonian Dynamic Mode
Decomposition**

Dynamic Mode Decomposition (DMD), is a popular method designed to create a discrete-time dynamical system from measurements only. In general, there is no guarantee, that the resulting dynamical system is stable or passive. Since these properties are encoded in the structure of port-Hamiltonian systems, it is desirable to modify DMD such that the resulting system has a port-Hamiltonian structure.

In this talk we introduce a modification of DMD that produces a discrete-time port-Hamiltonian realization. To this end, we split the problem into several different tasks, including a least-squares solution of the dissipation inequality and the solution of a skew-symmetric Procrustes problem. If time permits, we illustrate our theoretical findings with numerical examples.

### Ines Ahrens (TU Berlin)

Donnerstag, 14. Februar 2019

**Finding Structure with Randomness –
Randomized Singular Value Decomposition and other
Decompositions**

The singular value decomposition (SVD) of low rank matrices plays a central role in data analysis and scientific computing. Nowadays, matrices are often big and might be stored in slow memory, which leads to high computational time. The paper [1] presents randomized algorithms that compute highly accurate approximate matrix decompositions in less time than classical algorithms. It is even possible to calculate an approximate SVD, that accesses the matrix only once. In this talk I present the main ideas of [1] and illustrate its wide applicability.

[1] N. Halko, P. G. Martinsson, and J. A. Tropp. Finding Structure with Randomness: Probabilistic Algorithms for Constructing Aproximate Matrix Decompositions. SIAM Review, 53(2):217–288, 2011.

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