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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 |

### Marine Froidevaux (TU Berlin)

Dienstag, 09. März 2021

**PDE eigenvalue iterations with applications in dissipative two-dimensional photonic crystals**

We consider PDE eigenvalue problems as they occur in the modeling of two-dimensional photonic crystals. In particular we discuss the modelling of the electric permittivity in the case of dissipative materials and discuss how to deal with the occurring nonlinearities in the eigenvalue. Further, we extend the inverse power method to the case of infinite-dimensional and non-self-adjoint operators.

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

### Christoph Zimmer (TU Berlin)

Dienstag, 09. März 2021

**Temporal Discretization of Constrained Partial Differential Equations**

I am going to practice my defense talk for my dissertation. Feedback is very welcome.

Examples of constrained partial differential equations (PDEs) appear in all kinds of physical fields such as fluid dynamics, thermodynamics, electrodynamics, mechanics, chemical kinetics, as well as in multi-physical applications where different physical domains are coupled. In this talk, we analyze the application of time integration schemes to constrained PDEs. Among other things, time integration schemes can be used to prove the existence of solutions or to derive temporal error bounds which are independent of the mesh width of spatial discretization. These bounds are vital for spatially discretized systems, since the temporal convergence order of finite systems is maybe higher as of infinite-dimensional ones but fades into these, if the spatial mesh gets finer. In this talk, we derive spatial mesh-independent convergence orders for Runge-Kutta methods applied to constrained PDEs with time-independent constants. The results are supported by numerical examples. In the other half of this talk, we use the implicit Euler method to prove the existence of solutions of constrained PDEs with constraints with time-dependent coefficients.

### Ines Ahrens (TU Berlin)

Donnerstag, 25. Februar 2021

**A simple success check for delay differential-algebraic equations**

Solutions of differential-algebraic equations (DAEs) may depend on derivatives of some of its equations. Structural analysis, like the Sigma Method, can determine how often each equation needs to be differentiated. Unfortunately, this number is not always correct, such that a post-processing step is required to validate the result. Such a post-processing step, the so-called success check, is provided in the Sigma Method.

In this talk, I will show a generalization of this success check which can validate any given number of differentiations. This result leads to a first success check for delay differential-algebraic equations (DDAE), where solutions may depend on derivatives and future evaluations of some equations.

### Fabian Common (TU Berlin)

Donnerstag, 25. Februar 2021

**Optimal Control for linear port-Hamiltonian descriptor systems**

In control theory we are interested in properties of control systems. These include regularity and consistency, which state, if a system has a solution and if this solution is unique. Other properties are stability, stabilizability, controllability, observability, reconstructability and passivity.

In this thesis we have a closer look on control systems with a special structure, called linear port-Hamiltonian descriptor systems. We will see how we can use the port-Hamiltonian structure to our advantage to gain some of these properties.

After we obtain a system with the required properties by feedback, we are interested in finding a cost-optimal control. Here we will see, that we can apply the same theory for output feedback as we use for state feedback. We will also have a look, if the pH-structure gives us advantages regarding Karush-Kuhn-Tucker systems and the associated Riccati-equations.

### Philipp Schulze (TU Berlin)

Donnerstag, 18. Februar 2021

**Application of the Variable Projection Method in Nonlinear Model Reduction **

Classical model reduction methods are usually based on identifying suitable low-dimensional linear subspaces and subsequent (Petrov-)Galerkin projection of the full-order model (FOM). However, for systems whose dynamics feature the transport of sharp fronts, such as shocks, the FOM solution can usually not be well-approximated by a low-dimensional linear subspace. In such cases, the solution may still approximately evolve on a low-dimensional nonlinear manifold which motivates the usage of nonlinear model reduction methods. A special class of nonlinear model reduction methods is based on approximating the solution by a linear combination of transformed basis functions where the coordinate transformation aims to mimic the behavior of the transport within the system.

In this talk we consider the following problem: Given snapshot data of the FOM solution we seek for an optimal low-dimensional approximation based on a linear combination of transformed basis functions. The corresponding minimization problem is a so-called separable nonlinear least-squares problem, since a subset of the optimization parameters occurs linearly. Exploiting this special structure, we show how the variable projection may be used to replace the original optimization problem by another one with a reduced number of parameters. By means of numerical examples, we illustrate that this application of the variable projection method may lead to a significant speedup when computing an optimal approximation of the snapshot data.

### Riccardo Morandin (TU Berlin)

Donnerstag, 18. Februar 2021

**Port-Hamiltonian modeling and discretization of isothermal gas networks**

In this talk we introduce a modeling paradigm for gas networks, based on port-Hamiltonian descriptor systems (pHDAE). I will present three different PDE models to represent pipes containing isothermal gas, with an increasing degree of accuracy. These models are then replaced with finite-dimensional pHDAEs, through space-discretization, or manipulation of the explicit solution. Simplified pHDAE models for valves and compressors are also introduced.

It is then shown how to interconnect the network while preserving mass, energy, and the port-Hamiltonian structure, introducing pipe junctions, sources and sinks. The three gas pipe models are allowed to appear at the same time as different edges of the network. The resulting system is a semi-explicit nonlinear differential-algebraic equation of index 2.

I shortly present an algorithm that extracts necessary and sufficient conditions for the non-singularity of the DAE from the structure of the graph associated to the gas network. As a by-product of this algorithm, one can achieve index reduction, while at the same time preserving the pHDAE structure of the system. One can then apply structure-preserving time-discretization methods, for example the implicit midpoint rule, or an associated partitioned Runge-Kutta scheme.

A partial numerical implementation of these results is shown.

If time allows for it, some details missing because of time constraints can be discussed at the end of the talk.

### Simon Michael Bäse (TU Berlin)

Donnerstag, 04. Februar 2021

**Time-limited balanced truncation model order reduction for descriptor systems**

Balanced truncation is a well-known model order reduction technique for large-scale systems. In recent years, time-limited balanced truncation, which restricts the system Gramians to finite time intervals, has been investigated for different system types. We extend the ideas to linear time-invariant continuous-time descriptor systems using the framework of projected generalized Lyapunov equations. The formulation of the resulting Lyapunov equations is challenging since the right-hand sides are unknown a priori. We propose Krylov subspace methods for the efficient computation of the right-hand sides for different system structures. Since the right-hand sides may become indefinite, we use an LDLT factorization based ADI iteration to solve the Lyapunov equations and obtain the system balancing transformation. Comparing the time-limited to the classical approach in numerical experiments, we observe a steeper decay of the Hankel singular values. This behavior renders useful, especially when employing low-rank approximation techniques. Further, we show that time-limited balanced truncation can deliver reduced-order models that are more accurate in the prescribed time domain.

### Onkar Jadhav (TU Berlin)

Donnerstag, 28. Januar 2021

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

A parametric model order reduction (MOR) approach for simulating the high dimensional models arising in financial risk analysis is proposed.

We implement the proper orthogonal decomposition (POD) approach to generate small model approximations for the high dimensional parametric convection-diffusion reaction partial differential equations (PDE). The proposed technique uses an adaptive greedy sampling approach based on surrogate modeling to efficiently locate the most relevant training parameters, thus generating the optimal reduced basis. The best suitable reduced model is procured such that the total error is less than the user-defined tolerance. The three major errors considered are the discretization error associated with the full model obtained by discretizing the PDE, the model order reduction error, and the parameter sampling error.

The developed technique is analyzed, implemented, and tested on industrial data of different financial instruments under two prominent models (one-factor and two-factor Hull-White models).

The results illustrate that the reduced model provides a significant speedup with excellent accuracy over a full model approach, demonstrating its potential applications in the historical or Monte Carlo value at risk calculations.

### Miriam Goldack (TU Berlin)

Donnerstag, 28. Januar 2021

**A simple shifted Proper Orthogonal Decomposition suitable for large scale problems**

In 2018 Reiss et. al introduced the shifted Proper Orthogonal Decomposition (POD) to speed up the convergence when decomposing fields of transport dominated systems with the help of the POD. The method builds on the idea that a single traveling wave or moving localized structure can be perfectly described by its wave profile and a time-dependent shift. Therefore, the shifted POD decomposes transport fields by shifting the data field in a so called co-moving frame, in which the wave is stationary and can be described by a few spatial basis functions determined with the help of the POD. In the presence of multiple transports, the decomposition procedure becomes more complex and costly, because high dimensional non-linear optimization problems have to be solved.

In this talk, I will present a new shifted POD method, which is a simplification of the formulation used in [1]. In contrast to the former formulation, the new method allows to minimize the singular-value-based objective function without Newton-type methods and is therefore well suited for large-scale problems. I will provide results for 1D and 2D datasets. The latter stem from simulations of incompressible Navier-Stokes equations, with moving object boundaries, such as moving wings of an insect or two moving cylinders at Reynolds number 200.

[1] Reiss, Julius. "Optimization-based modal decomposition for systems with multiple transports." arXiv preprint arXiv:2002.11789 (2020).

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

Donnerstag, 21. Januar 2021

**Eigenvalues problem of G-Hamiltonian matrix**

We find that the linearization of the Drude-Lorentz mode has G-Hamiltonian structure, and its eigenvalue problem is actually the eigenvalue problem of G-Hamiltonian matrix. The eigenvalues of a G-Hamiltonian matrix are symmetric with respect to the imaginary axis. Its eigenvalues move out of the imaginary only when two different kinds of eigenvalues collide on the imaginary axis. To give the structure-preserving numerical computation for the eigenvalues of a G-Hamiltonian matrix, we view the G-Hamiltonian matrix as a generation of a Hamiltonian matrix and G-skew-Hamiltonian matrix as a generation of a skew-Hamiltonian matrix by replacing iJ with a Hermitian matrix G. We generate the idea in the ref [1] and give some relative lemmas about the G-Hamiltonian matrix.

[1] P. Benner, R. Byers, V. Mehrmann, and H. Xu, Numerical computation of deflating subspaces of skew-Hamiltonian/Hamiltonian pencils. SIAM Journal on Matrix Analysis and Applications, 24(1), (2002)

### Tobias Breiten (TU Berlin)

Donnerstag, 07. Januar 2021

**Error bounds for port-Hamiltonian model and controllerreduction based on system balancing**

Linear quadratic Gaussian (LQG) control design for port-Hamiltonian systems is studied. A recently proposed method from the literature is reviewed and modified such that the resulting controllers have a port-Hamiltonian (pH) realization. Based on this new modification, a reduced-order controller is obtained by truncation of a balanced system. The approach is shown to be closely related to classical LQG balanced truncation and shares a similar a priori error bound with respect to the gap metric. With regard to this error bound, a theoretically optimal pH-representation is derived. Consequences for pH-preserving balanced truncation model reduction are discussed and shown to yield two different classical $\mathcal{H}_\infty$-error bounds.

### Felix Black (TU Berlin)

Donnerstag, 07. Januar 2021

**Model reduction with dynamically transformed modes: offline stage and path minimization**

The key goal of model order reduction is to determine high fidelity approximations of solutions of large-scale dynamical systems to reduce computational effort. Many classical model order reduction methods are formulated in a projection framework; the solution to the original system is approximated within a suitable low-dimensional subspace. The particular way how the subspace is determined is one of the distinct features of the different model reduction methods. Commonly, the subspace is determined by solving a minimization problem for the basis vectors that form the subspace, and the full order solution is approximated via a linear combination of the fixed basis vectors with time-dependent coefficients. If the dynamical system exhibits advective transport, however, classical methods often fail to produce low-dimensional models that result in a high fidelity approximation. One strategy to remedy this problem is the shifted proper orthogonal decomposition (shifted POD, see [1]), or, more generally, a projection-based ansatz with dynamically transformed modes (see [2]), which extends the classical approximation ansatz by introducing transformation operators associated with the basis vectors. Those transformation operators are parametrized by paths in suitable vector spaces, allowing the (now non-stationary) subspace to cope with the advection. However, while in the classical approach, it is sufficient to solve a minimization problem that depends only on the basis vectors, the approach with dynamically transformed modes requires to solve a minimization problem that depends on the basis vectors, as well as the time-dependent coefficients and also the path variables that parametrize the transformations. In this talk, we discuss the resulting minimization problem for the determination of suitable basis vectors, coefficients, and paths, and aim to prove that, under certain assumptions, there exist solutions.

References:

[1] J. Reiss, P. Schulze, J. Sesterhenn, V. Mehrmann, The Shifted Proper Orthogonal Decomposition: A Mode Decomposition for Multiple Transport Phenomena, SIAM J. Sci. Comput. 40 (2018), no. 3, A1322 - A1344.

[2] F. Black, P. Schulze, B. Unger, Projection-based model reduction with dynamically transformed modes, ESAIM: Math. Model. Numer. Anal. 54 (2020), no. 6, 2011 - 2043.

### Florian Stelzer (TU Berlin)

Donnerstag, 17. Dezember 2020

**Deep Learning with a Single Neuron: Folding a Deep Neural Network in Time using Feedback-Modulated Delay Loops**

Deep neural networks are among the most widely applied machine learning tools showing outstanding performance in a broad range of tasks. We present a method for folding a deep neural network of arbitrary size into a single neuron with multiple time-delayed feedback loops. This single-neuron deep neural network comprises only a single nonlinearity and appropriately adjusted modulations of the feedback signals. The network states emerge in time as a temporal unfolding of the neuron's dynamics. By adjusting the feedback-modulation within the loops, we adapt the network's connection weights. These connection weights are determined via a modified back-propagation algorithm that we designed for such types of networks. Our approach fully recovers standard Deep Neural Networks (DNN), encompasses sparse DNNs, and extends the DNN concept toward dynamical systems implementations. The new method, which we call Folded-in-time DNN (Fit-DNN), exhibits promising performance in a set of benchmark tasks.

F. Stelzer, A. Röhm, R. Vicente, I. Fischer and S. Yanchuk, Deep Learning with a Single Neuron: Folding a Deep Neural Network in Time using Feedback-Modulated Delay Loops. See arxiv.org/abs/2011.10115.

### Malte Krümel (TU Berlin)

Donnerstag, 17. Dezember 2020

**Index-Aware Model Reduction for Optimization of Gas Networks **

Optimization problems of gas networks became increasingly important in the age of energy transition. Solving them numerically poses many challenges and requires model order reduction (MOR) of non-linear differential-algebraic-equations (DAE). The underlying model consists of 1D Euler equations for flow modelling, Kirchhoff’s law and further boundary conditions that describe the dynamics and relations between elemtents of a directed graph. Applying reasonable simplifications and spatial discretization leads to a DAE system. The system is re-formulated to obtain an Input-Output system that has traceability index-2.

We will first look at an often used approach of reducing the index to an ODE system. For such systems common large-scale MOR can be applied. Because of computational issues this approach has some disadvantages. Thus, we will explore the approach of Index-Aware MOR. Here, the system is decoupled into differential, index-1 and index-2 equations via projections. The decoupled system can now be reduced by making use of the properties of each type of equations. The talk will conclude with the outlook to incorporating the non-linear part of the model into the model reduction process and testing the reduced model in optimal control.

### Amon Lahr (TU Berlin)

Donnerstag, 10. Dezember 2020

**Reduced-order design of suboptimal H∞ controllers using rational Krylov subspaces**

In the field of robust control, H∞ control provides an established framework to design control laws guaranteeing stability and performance over a range of perturbations of the nominal system model. The underlying mathematical problem is usually separated into finding the (sub)optimal attenuation (γ-iteration), and designing a stabilizing controller for which the H∞ norm of the closed-loop transfer function is not greater than γ. For large-scale systems, especially the γ\gamma-iteration proves to be computationally demanding as it requires the exact solution of two algebraic Riccati equations (ARE) in every step of the bisection method. Furthermore, the dimension of the obtained control law needs to be reduced for most practical applications.

In this talk, we introduce some of the challenges related to reduced-order design of H∞ controllers. Furthermore, an accelerated implementation of the γ-iteration is presented, which is based on low-rank approximations of the ARE solutions using rational Krylov subspaces. Therein, a reduced-order controller is constructed and verified at each bisection step using a large-scale H∞ norm computation method and the calculation of a few eigenvalues of the closed-loop matrix. The results are discussed by means of numerical examples arising from control of partial differential equations.

### Daniel Bankmann (TU Berlin)

Donnerstag, 10. Dezember 2020

**Multilevel Optimization Problems with Linear Differential-Algebraic Equations**

I'm going to practice my defense talk for my dissertation. The talk is supposed to last no more than 30 minutes. Feedback is very welcome.

We discuss different multilevel optimization problems in the context of linear differential-algebraic equations. On the one hand, we address multilevel optimal control problems, where sensitivity information of the necessary conditions of the optimal control problem can be used to compute solutions of the upper level problem. When the upper level is given by a nonlinear least-squares problem, we present a step size estimator. On the other hand, we show how the analytic center of the passivity LMI can be used as a good starting point in the computation of the passivity radius.

### Martin Isoz (UCT Praque)

Donnerstag, 26. November 2020

**Simulations of fully-resolved particle-laden flows: fundamentals and challenges for model order reduction**

Particle-laden flows are present in numerous aspects of day-to-day life ranging from technical applications such as fluidisation or filtration to medicinal problems, e.g. behavior of clots in blood vessels. Nevertheless, computational fluid dynamics (CFD) simulations containing freely moving and irreguralry shaped bodies are still a challenging topic. More so, if the bodies are large enough to affect the fluid flow and distributed densely enough to come in contact both with each other and with the computational domain boundaries. In this talk, we present a finite volume-based CFD solver for modeling flow-induced movement of interacting irregular bodies. The modeling approach uses a hybrid fictitious domain-immersed boundary method (HFDIB) for inclusion of the solids into the computational domain. The bodies movement and contacts are solved via the discrete element method (DEM). Unfortunately, the coupled HFDIB-DEM model structure causes significant limitations with respect to applications of standard projection-based methods of model order reduction (MOR). While we focus mostly on the HFDIB-DEM solver development, the talk is concluded by the challenges the HFDIB-DEM approach poses for MOR.

### Volker Mehrmann (TU Berlin)

Donnerstag, 19. November 2020

**Structured backward errors for eigenvalues associated with port-Hamiltonian descriptor systems **

When computing the eigenstructure of matrix pencils associated with the passivity analysis of perturbed port-Hamiltonian descriptor systems using a structured generalized eigenvalue method, one should make sure that the computed spectrum satises the symmetries that corresponds to this structure and the underlying physical system. We perform a backward error analysis and show that for matrix pencils associated with port-Hamiltonian descriptor systems and a given computed eigenstructure with the correct symmetry structure, there always exists a nearby port-Hamiltonian descriptor system with exactly that eigenstructure. We also derive bounds for how near this system is and show that the stability radius of the system plays a role in that bound.

V. Mehrmann and P. Van Dooren, Structured backward errors for eigenvalues of linear port-Hamiltonian descriptor systems, To appear in SIAM Journal Matrix Analysis and Applications, 2020. See arxiv.org/abs/2005.04744.

### Tim Moser (TU München)

Donnerstag, 05. November 2020

**A Riemannian Framework for Ecient H2-Optimal Model Reduction of Port-Hamiltonian Systems**

The port-Hamiltonian systems paradigm provides a powerful framework for the network modeling of multi-physics systems. By exploiting inherent system characteristics such as passivity, the modeling in port-Hamiltonian form also facilitates the subsequent controller design. Therefore it is advantageous to preserve the port-Hamiltonian structure in the model reduction process for which different approaches have been proposed (see e.g. [1], [2]).

In [1], a modified version of the iterative rational Krylov algorithm (IRKA-PH) was proposed for the H2-optimal model reduction of port-Hamilonian systems. Since IRKA-PH is based on Petrov-Galerkin projections, certain degrees of freedom must be given up in order to preserve the port-Hamiltonian structure. This inevitably leads to the fact that it is generally not possible to satisfy all necessary H2- optimality conditions in this projective framework.

We address this issue and propose a novel Riemannian framework for the H2-optimal reduction of port-Hamiltonian systems. We incorporate geometric constraints using the Riemannian problem formulation of [3] and exploit the computationally efficient pole-residue formulation of the H2-error proposed in [4]. By this means, preservation of the port-Hamiltonian structure and H2-optimality upon convergence are guaranteed and the framework is also accessible for the reduction of large-scale systems.

References

[1] S. Gugercin, R. V. Polyuga, C. Beattie, and A. van der Schaft, "Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems," Automatica, vol. 48, no. 9, pp. 1963-1974, 2012.

[2] R. V. Polyuga and A. J. van der Schaft, "Effort- and flow-constraint reduction methods for structure preserving model reduction of port-Hamiltonian systems," Systems & Control Letters, vol. 61, no. 3, pp. 412-421, 2012.

[3] K. Sato, "Riemannian optimal model reduction of linear port-Hamiltonian systems," Automatica, vol. 93, pp. 428-434, 2018.

[4] L. Meier and D. Luenberger, "Approximation of linear constant systems," IEEE Transactions on Automatic Control, vol. 12, no. 5, pp. 585-588, 1967.

# Rückblick

- Absolventen Seminar SS 20
- 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