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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 |

# Rückblick

- 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

### Paul Schwerdtner (TU Berlin)

Donnerstag, 23. Juli 2020

**Structure Preserving H-infinity Approximation**

We present a new method to perform model order reduction for linear port-Hamiltonian systems. The method is based on directly optimizing the reduced order model's parameters. For that we first explain how we fully parametrize a linear port-Hamiltonian system such that all structural constraints are automatically satisfied. After that, we give an insight into the optimization problem we set up to minimize distance between the transfer function of the given model and of our reduced order approximation with respect to the H-infinity norm. Finally, we highlight the effectiveness of our method by comparing it to other structure preserving model order reduction schemes on a port-Hamiltonian benchmark system.

### Philipp Schulze (TU Berlin)

Donnerstag, 16. Juli 2020

**Structure-Preserving Model Reduction for Advection-Dominated Systems **

Classical model reduction methods are usually based on projecting the full-order model (FOM) onto a suitable low-dimensional linear subspace. However, when it comes to systems whose dynamics is dominated by the advection of sharp fronts, such as shocks, linear subspace methods are often not capable of yielding sufficiently accurate low-dimensional surrogate models. In recent years, more and more attention has been paid to nonlinear model reduction methods which appear to be more suitable for these advection-dominated systems.

In this talk, we consider a recently proposed nonlinear model reduction scheme which employs dynamically transformed ansatz functions in order to follow the advection. The special focus of this talk is on preserving a port-Hamiltonian structure during the nonlinear projection. Furthermore, we present a first attempt to retain the energy balance after time discretization, by using ideas of the discrete gradient method. The proposed method is illustrated by means of a numerical example.

### Art Joschua Robert Pelling (TU Berlin)

Donnerstag, 16. Juli 2020

**Inverse Filter Design for Room Acoustics**

When dealing with electro-acoustical systems in rooms, it is often desirable to compensate for distortion introduced by surface reflections. Commonly, the room is modeled as a linear time-invariant system and a pre-filter is employed in order to equalise the room transfer function.

In this talk. we will start by introducing the room equalisation problem and giving a brief overview of inverse filter design methods prevalently used today. We will then propose a new method which utilises a linear quadratic controller (LCQ) based on a reduced order model, which is generated by randomly sampled room impulse responses with the so-called Eigenspace Realisation Algorithm (ERA). Finally, we will take a look at some numerical examples of how the RandomSVD can greatly improve the performance of the reduced order modeling step with ERA.

### Daniel Bankmann (TU Berlin)

Donnerstag, 09. Juli 2020

**Sensitivity computation of boundary value problems coming from optimal control of strangeness-free 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, although theory for adjoint equations with boundary values exists. We close this gap, by formulating the correct boundary conditions based on the standard adjoint equations for strangeness-free differential-algebraic equations. Also, we present a forward approach by directly differentiating the boundary value problem.

We finally analyze, how different requirements on the smoothness of the coefficients affect the possible methods that can be used.

### Marine Froidevaux (TU Berlin)

Donnerstag, 09. Juli 2020

**Counting eigenvalues of nonlinear eigenvalues problems: a contour integral approach**

As many other applications in physics, the numerical simulation of so-called photonic crystals requires to solve large-scale nonlinear eigenvalue problems (EVPs). In contrast to linear EVPs for which a vast collection of efficient numerical methods exists, nonlinear EVPs remain much more complex to solve.

The usual strategy to solve a nonlinear EVP is to first linearize it in some sense - typically either with a Newton method or “manually" e.g. through a Companion form. Another approach consists in integrating the resolvent over a contour enclosing the eigenvalues of interest. In this talk we will consider the contour integral method described in [Bey12] and discuss its use to estimate the number of eigenvalues within a given domain of the complex plane.

If time permits, we will discuss the use of model order reduction to fasten up the computation of the contour integral.

This is joint work with Luka Grubisic (U Zagreb), Philipp Jorkowski (TU Berlin) and Kersten Schmidt (TU Darmstadt).

[Bey12] W.-J. Beyn, An integral method for solving nonlinear eigenvalue problems, Linear Algebra and its Applications, 2012.

### Dorothea Hinsen (TU Berlin)

Freitag, 03. Juli 2020

**A port-Hamiltonian approach for modelling power networks including the telegraph equations**

In recent years, energy transition and the increasing electricity demand have led to a growing interest in modelling power networks, which have to withstand unexpected contingencies as voltage or transient instabilities. One approach to modelling power networks are port-Hamiltonian systems. The power networks we are dealing with consist of generators, loads and transmission lines.

In this talk, we discuss an approach of a power network model, where we model the network as a graph. In this graph model, each edge stands for a component. We can describe each component and, therefore, each edge as a port-Hamiltonian system. The port-Hamiltonian system describing a transmission line edge is based on the telegraph equation. By using the graph and the port-Hamiltonian system structure, we can combine each component into a global port-Hamiltonian system for partial differential algebraic equations representing a complete power network. After space discretization, we have differential algebraic equation of index 1.

### Tobias Breiten (TU Berlin)

Freitag, 03. Juli 2020

**The Mortensen observer, minimum energy estimation and value function approximations **

Estimating the state of a nonlinear perturbed dynamical system based on (output) measurements is a well-known control theoretic problem. While in the linear case, an optimal observer is given by the famous Kalman(-Bucy) filter, in the nonlinear case, constructing observers is significantly more complex and many approaches such as extended or unscented Kalman filters exist. The Mortensen observer relies on the concept of minimum energy estimation and a value function framework which is determined by a Hamilton-Jacobi-Bellman equation. This talk provides an introduction to these concepts as well as a neural network based approximation technique for nonlinear observer design.

### Viktoria Pauline Schwarzott (TU Berlin)

Donnerstag, 25. Juni 2020

**Dissipation inequality conserving discretization of pHDAE**

Considering Port-Hamiltonian-Systems the power balance equation holds along any solution and the dissipation inequality is fulfilled. While discretizing with Gauß-Legendre collocation methods, it is already known that these properties maintain in case of a quadratic Hamiltonian function [1]. Taking the dissipation inequality conservation as a goal, we consider other methods to discretize, by focusing on the Lobatto IIIA methods.

[1] Morandin and Mehrmann; Structure-preserving discretization for port-Hamiltonian descriptor systems, arXiv:1903.10451

### Ruili Zhang (Beijing Jiaotong University and TU Berlin)

Donnerstag, 25. Juni 2020

**Symplectic simulation for the gyrocenter dynamics of charged particles**

Gyrocenter dynamics of charged particles plays a fundamental and important role in plasma physics, which requires accuracy and conservation in a long-time simulation. Variational symplectic algorithms and canonicalized symplectic algorithms have been developed for gyrocenter dynamics. However, variational symplectic methods are always unstable, and canonicalized symplectic methods need coordinates transformation case by case, which is usually difficult to find.

In the following, we start from the degenerate Lagrangian of the gyrocenter dynamics of charged particles, and give a Hamiltonian system with constraints. The system can be written as in a port-Hamiltonian differential-algebraic equation (pHDAE). The flow on the manifold generated by the system is symplectic. So for the special form of pHDAE, we can apply the symplectic PRK methods. The implementation of the methods is described, and some numerical tests are reported.

### Marie Krause (TU Berlin)

Donnerstag, 18. Juni 2020

**Numerical methods for computing the distance to singularity, instability and higher index for port-Hamiltonian systems**

Dissipative port-Hamiltonian systems arise in many applications that describe the flow of energy. In many real world problems those systems are facing disturbances due to inaccurate data or bad modeling.

In the following we will consider the consequences of such disturbances for the system properties of stability, regularity and the index of the system. As a first case we assume the system to be stable, regular and of index one and compute the distance to the nearest unstable, singular or higher index system. Because of the special structure of dissipative port-Hamiltonian systems, it is possible to consider the distance problem in the more general context of square matrix polynomials with symmetric positive semi-definite and one skew-symmetric matrix. As a second case we consider a unstable, singular or higher index system and comupute the distance to the nearest stable, regular or index one system.

### Rebecca Beddig (TU Berlin)

Donnerstag, 18. Juni 2020

**H2 ⊗ L∞-Optimal Model Order Reduction**

In this talk, we discuss H2 ⊗ L∞-optimal model order reduction for parametric linear time-invariant systems. The H2 ⊗ L∞-error is defined as the maximum H2-error within the feasible parameter domain. First, we discuss the computation of the H2 ⊗ L∞-error with numerical optimization, Chebyshev interpolation and pointwise evaluation. We find an initial reduced-order model (ROM) with projection-based methods. The H2 ⊗ L∞-error is then minimized by optimizing over the matrix elements of the ROM. To obtain a uniformly asymptotically stable ROM, we furthermore introduce a stability check and a stability constraint. Then, we show numerical results to illustrate the method.

### Christoph Zimmer (TU Berlin)

Donnerstag, 11. Juni 2020

**Exponential Integrators for Semi-Linear PDAEs - Higher Order Integrators and Approximation of Lagrange Multipliers**

In my last talk we applied successfully exponential integrators to semi-linear partial differential equation with constraints (PDAEs). Examples for such semi-linear PDAEs are the incompressible Navier-Stokes equations or PDEs with dynamical boundary conditions.

In this talk, we construct explicit exponential integrators of convergence order up to four for finite-dimensional differential-algebraic equations (DAEs). We show how the Lagrange multiplier, e.g. the pressure in the Navier-Stokes equations, can be approximated. In the second part we discuss the additional problems which can occur for PDAEs.

### Riccardo Morandin (TU Berlin)

Donnerstag, 11. Juni 2020

**Infinite-dimensional port-Hamiltonian systems**

We present a novel formulation for infinite-dimensional (dissipative) port-Hamiltonian descriptor systems (pHDAEs). Our goal is to unify and generalize multiple different concepts of infinite-dimensional pH systems already known in the literature.

To reach our goal, we investigate the concise structure that identifies finite-dimensional pH systems, the difficulties that emerge with partial-differential equations, and what are the minimal passages necessary to include all these systems in one formulation, without compromising its generality. The classical concepts of Dirac structure, symmetry, skew-symmetry and positive definiteness are modified and extended to be applied to the infinite-dimensional case, without ambiguity or imprecision.

The resulting general pHDAE is in the form of an input-output control system, and includes the boundary conditions as algebraic or output equations. The obtained structure allows straightforward change of variables, model reduction and energy-preserving interconnection. The example of gas networks is presented as a possible application.

### Philipp Krah (TU Berlin)

Donnerstag, 04. Juni 2020

**A scalable non-linear model order reduction approach for complex moving fronts in combustion systems.**

Model order reduction (MOR) aims at describing large and numerical complex systems by much smaller ones. Unfortunately, model reduction fails for transport dominated systems with sharp fronts, like propagating flames, moving shocks or traveling acoustic waves. In this talk, I will present some of my ongoing work on model order reduction (MOR) of transport dominated systems with complex moving fronts in combustion systems. Our non-linear MOR approach parametrizes the flow field with the help of a front shape function and a levelset function. The levelset function is used to generate a local coordinate, which parametrizes the distance to the front. The freedom of choice of the levelset function far away from the zero level is used to obtain a low dimensional description of the full order model in a constrained optimization problem. Here, we utilize low rank and smoothness constrains to obtain a reduced system. The optimization problem is solved with the help of the alternating direction method of multipliers (ADMM). This enables us to split the optimization problem into smaller pieces, each of which are then easier to handle and computationally efficient for large scale problems.

My talk will be structured as follows: First, I explain the basic challenge for MOR when being applied to advective transport on a 1D and 2D example of a sharp front moving with constant speed. I will then introduce the constrained optimization problem and the ADMM algorithm and show some numerical results for a 2D traveling flame front.

Keywords: Nonlinear model order reduction, proper orthogonal decomposition, combustion, advection systems

### Onkar Jadhav (TU Berlin)

Donnerstag, 28. Mai 2020

**Hierarchical modeling to establish a model order reduction framework for financial risk analysis.**

We propose a model order reduction framework for the financial risk analysis based on a proper orthogonal decomposition (POD) method. The study involves the computations of high dimensional parametric convection-diffusion reaction partial differential equations (PDEs).

The model hierarchy simplifies the process of obtaining a reduced-order model and states as follows. The discretization of a PDE generates a full order model (FOM). Furthermore, the POD approach relies on the method of snapshots in which the FOM is solved for only a certain number of training parameters to obtain a reduced-order basis. Finally, this reduced-order basis is used to construct a reduced-order model. The training parameters are chosen using the classical/adaptive greedy sampling approach. In this work, we also analyze different errors associated with the numerical methods, namely modeling error, discretization error, parameter sampling error, and model order reduction error.**Keywords:** Model order reduction, financial risk analysis, model hierarchy, greedy sampling, proper orthogonal decomposition.

### Michał Wojtylak (Jagiellonian University, Krakow)

Donnerstag, 14. Mai 2020

**Deformed numerical range, dilations, spectral constants.**

For a matrix T we introduce a a deformed numerical range W_r(T), r is a parameter between 0 and 2. It is a convex set, containing the spectrum, depending continuously on r.

Other properties of W_r(T) will be shown as well. We will review the dilation theory and discuss the spectral constants for these sets. We will also show the relation of W_r(T) to other variations on the numerical range.

Joint work with Patryk Pagacz and Paweł Pietrzycki.

### Benjamin Unger (TU Berlin)

Donnerstag, 07. Mai 2020

**Delay differential-algebraic equations in earthquake engineering**

In earthquake engineering, many models feature complicated dynamics 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. To remedy this problem, one 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 rendering the complete dynamical system as a (nonlinear) delay differential-algebraic equation (DDAE). In this talk, I present the challenges that come with such an approach and discuss the solvability of the resulting DDAE.

### Serhiy Yanchuk (TU Berlin)

Donnerstag, 30. April 2020

**Modeling active optical networks**

We develop a nonlinear formalism for active optical networks. The propagation along active links is treated via suitable rate equations, which require the inclusion of an auxiliary variable: the population inversion. Altogether, the resulting mathematical model can be viewed as an abstract network, its nodes corresponding to the optical fields in the physical links. The dynamical equations are differential delay-algebraic equations. The stationary states of a generic setup with a single active medium are discussed, showing that the role of the passive components can be combined into a single transfer function that takes into account the corresponding resonances.

### Volker Mehrmann (TU Berlin)

Donnerstag, 23. April 2020

**Model reduction for linear port-Hamiltonian descriptor systems**

We show how to combine model order reduction tecniques for differential-algebraic equations with port-Hamiltonian structure preservation. For this, we extend three classes of model reduction techniques (reduction of the Dirac structure, moment matching, and tangential interpolation) to handle linear port-Hamiltonian differential-algebraic equations. There are several challenges that have to be addressed. These include the preservation of constraints, the preservation of the structure, and the proof of error estimates. The performance of the methods is illustrated for benchmark examples originating from semi-discretized flow problems, acoustic fields in gas networks, and mechanical multibody systems.