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## Absolventen-Seminar • Numerische Mathematik

Dieses Semester wird das Seminar online auf Zoom stattfinden.

Verantwortliche Dozenten: |
Prof. Dr. Tobias Breiten [1], | Prof. Dr. Christian Mehl
[2], Prof. Dr. Volker Mehrmann
[3]
---|---|

Koordination: | Ines Ahrens
[4] |

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

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

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

Do 23.04. | 10:15 Uhr | Vorbesprechung | |

Volker Mehrmann [5] | Model
reduction for linear port-Hamiltonian descriptor systems | ||

Do
30.04. | 10:15 Uhr | Serhiy Yanchuk [6] | Modeling
active optical networks |

Do
07.05. | 10:15 Uhr | Benjamin Unger | Delay
differential-algebraic equations in earthquake
engineering |

Do 14.05. | 10:15 Uhr | Michal
Wojtylak | Deformed numerical range, dilations,
spectral constants. |

Do
28.05. | 10:15 Uhr | Onkar Jadhav | Hierarchical
modeling to establish a model order reduction framework for financial
risk analysis. |

Do
04.06. | 10:15 Uhr | Philipp Krah [7] | A scalable
non-linear model order reduction approach for complex moving fronts in
combustion systems. |

Do
11.06. | 10:15 Uhr | Christoph Zimmer | Exponential
Integrators for Semi-Linear PDAEs - Higher Order Integrators and
Approximation of Lagrange Multipliers |

Riccardo Morandin [8] | Infinite-dimensional port-Hamiltonian
systems | ||

Do
18.06. | 10:15 Uhr | Marie Krause | Numerical methods
for computing the distance to singularity, instability and higher
index for port-Hamiltonian systems |

Rebecca Beddig | H2 ⊗
L∞-Optimal Model Order Reduction | ||

Do 25.06. | 10:15 Uhr | Viktoria Pauline
Schwarzott | Dissipation inequality conserving
discretization of pHDAE |

Ruili
Zhang | Symplectic simulation for the
gyrocenter dynamics of charged particles | ||

Do 02.07. | 10:15 Uhr | Dorothea Hinsen | A
port-Hamiltonian approach for modelling power networks including the
telegraph equations |

Tobias
Breiten | The Mortensen observer, minimum
energy estimation and value function approximations | ||

Do 09.07. | 10:15 Uhr | Daniel
Bankmann | Sensitivity computation of boundary
value problems coming from optimal control of strangeness-free
differential-algebraic equations |

Marine
Froidevaux | Counting eigenvalues of nonlinear
eigenvalues problems: a contour integral approach | ||

Do 16.07. | 10:15 Uhr | Philipp Schulze
[9] | Structure-Preserving Model Reduction for
Advection-Dominated Systems |

Art Joshua
Robert Pelling | Inverse Filter Design for Room
Acoustics | ||

Do
23.07. | 10:15 Uhr | Paul Schwerdtner [10] | Structure
Preserving H-infinity Approximation |

# Rückblick

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

### 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.

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