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Absolventen-Seminar • Numerische Mathematik

Dieses Semester wird das Seminar online auf Zoom stattfinden.

Verantwortliche Dozenten:
Prof. Dr. Tobias BreitenProf. Dr. Christian MehlProf. Dr. Volker Mehrmann
Ines Ahrens
Do 10:00-12:00
Vorträge von Bachelor- und Masterstudenten, Doktoranden, Postdocs und manchmal auch Gästen zu aktuellen Forschungsthemen

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.

[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.

[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.

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