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## Absolventen-Seminar • Numerische Mathematik

Verantwortliche Dozenten: | Prof. Dr. Christian Mehl, Prof. Dr. Volker Mehrmann |
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Koordination: | Ann-Kristin Baum, Benjamin Unger |

Termine: | Do 10:00-12:00 in MA 376 |

Inhalt: | Vorträge von Diplomanden, Doktoranden, Postdocs und manchmal auch Gästen zu aktuellen Forschungsthemen |

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

Do 16.10. | 10:15 Uhr | MA 376 | Vorbesprechung | |

Namita Behera | Fiedler Linearizations for LTI State-Space Systems and for Rational Eigenvalue Problems | |||

Ann-Kristin Baum | A flow-on-manifold formulation ofdifferential-algebraic equations.Application to positive systems. | |||

Do 23.10. | 10:15 Uhr | MA 376 | Benjamin Unger | Analysis of the Structure of the POD Basis |

Pratibhamony Das | A priori and a posteriori uniform error estimates based on moving meshes for singularly perturbed problems | |||

Do 30.10. | 10:15 Uhr | MA 376 | - kein Seminar - | |

Do 06.11. | 10:15 Uhr | MA 376 | Christian Schröder | A special topic in Krylov subspaces |

| ||||

Do 13.11. | 10:15 Uhr | MA 376 | Cornelia Gamst | Maxwell's eigenvalue problem for photonic crystals |

Phi Ha | Analysis and Numerical solutions of Delay-DAEs | |||

Do 20.11. | 10:15 Uhr | MA 376 | - kein Seminar - | |

Do 27.11. | 10:15 Uhr | MA 376 | - kein Seminar - | |

Do 04.12. | 10:15 Uhr | MA 376 | Michael Overton | Investigation of Crouzeix's Conjecture via Optimization |

Volker Mehrmann | When is a linear control system equivalent to a port-Hamiltonian system | |||

Do 11.12. | 10:15 Uhr | MA 376 | Michał Wojtylak | On deformations of classical Jacobi matrices |

Do 18.12. | 10:15 Uhr | MA 376 | Christoph Conrads | The Off-Diagonal Block Method |

Helia Niroomand Rad | Time-Harmonic Scattering Problem Considering The Pole Condition | |||

Do 08.01. | 10:15 Uhr | MA 376 | - kein Seminar - | |

Do 15.01. | 10:15 Uhr | MA 376 | Judith Simon | Structure Preserving Discretization Methods for Self-Adjoint Differential-Algebraic Equations |

Matthias Voigt | The Kalman-Yakubovich-Popov Inequality for Differential-Algebraic Equations | |||

Do 22.01. | 10:15 Uhr | MA 376 | Jeroen Stolwijk | Error Analysis for the Euler Equations in Purely Algebraic Form |

Do 29.01. | 10:15 Uhr | MA 376 | Philipp Schulze | Structure-Preserving Model Reduction |

Robert Altmann | Optimal Control for Problems with Servo Constraints | |||

Do 05.02. | 10:15 Uhr | MA 376 | Leo Batzke | Rank-k Perturbations of Hamiltonian Matrices |

Sarosh Quraishi | Parametric study of a brake squeal using machine learning | |||

Do 12.02. | 10:15 Uhr | MA 376 | Ute Kandler | Computation of extreme eigenvalues of a large-scale symmetric eigenvalue problem |

Deepika Gill | Multi-Mode Control of Tall Building using Distributed Multiple Tuned Mass Dampers |

### Namita Behera (TU Berlin)

Donnerstag, 16. Oktober 2014

**Fiedler Linearizations for LTI State-Space Systems and forRational Eigenvalue Problems**

In this seminar, we introduce a family of linearizations, which we refer to as

Fiedler linearizations, of the Rosenbrock system matrix of an LTI system in state space form for computation of transmission and invariant zeros of the system. We define linearizations for the transfer function of the LTI system and show that under appropriate assumptions a Fiedler linearization of the system matrix is also a linearization of the transfer function. Thus, given a rational eigenvalue problem, we reformulate the problem of computing eigenvalues of a rational matrix function to that of computation of transmission zeros of an LTI state space system. Hence we compute the eigenvalues by solving a generalized eigenvalue problem for Fiedler pencil of the system matrix.

### Ann-Kristin Baum (TU Berlin)

Donnerstag, 16. Oktober 2014

**A flow-on-manifold formulation ofdifferential-algebraic equations.Application to positive systems.**

Differential-algebraic equations (DAEs) are coupled systems of differential and algebraic equations that model dynamical processes constrained by auxiliary algebraic conditions, like e.g. connected joints in multibody systems, connections or loops in networks or balance equations and conservation laws in advection-diffusion equations. Considering applications in economy, social sciences, biology or chemistry, the analyzed values typically represent nonnegative quantities like the amount of goods, individuals or the density of a chemical or biological species. This leads to positive systems, i.e., systems for which every solution that starts with a componentwise nonnegative initial value remains nonnegative for its lifetime.

In this talk, we derive a systematic description of positive systems that allows to validate if a given model possesses the property of positivity. We generalize the notion of a flow from ordinary differential equations to DAEs and give an explicit solution formula using a projection approach. We characterize flow invariant sets for DAEs and derive a uniform framework of flow invariant sets for implicit and explicit differential equations. Considering the nonnegative orthant, in particular, we thus give a uniform description of positive systems, constrained or unconstrained.

### Benjamin Unger (TU Berlin)

Donnerstag, 23. Oktober 2014

**Analysis of the Structure of the POD Basis **

Over the past two decades model reduction of very high dimensional systems, arising from physical measurements or generated by partial differential equations, has become increasingly important to the CFD and optimal control community. In this talk, we outline the general idea of reduced order modeling (ROM) and focus on Proper Orthogonal Decomposition (POD) as a particular technique. A analysis of the structure of the POD basis is performed and conclusions concerning the efficiency of the method are drawn. Moreover, the nonlinear ROM technique Discrete Empirical Interpolation Method (DEIM) is analyzed with respect to different discretisation techniques.

### Pratibhamony Das (TU Berlin)

Donnerstag, 23. Oktober 2014

**A priori and a posteriori uniform error estimates based on moving meshes for singularly perturbed problems **

In this talk, I will give a small introduction of singularly perturbation and moving mesh methods. Then, a nonlinear parametrized problem will be considered to show the graphical differences between a priori and a posteriori generated meshes. Thereafter, a posteriori errror estimate based on equidistribution principle will be considered for system of reaction diffusion problems. If time permits, a two parametric parabolic convection diffusion problem will be considered for convergence analysis on moving meshes.

### Christian Schröder (TU Berlin)

Donnerstag, 06. November 2014

**A special topic in Krylov subspaces **

I will talk about a recent project dealing with numerical linear algebra, Krylov subspaces and Arnoldi's recurrence.

### Cornelia Gamst (TU Berlin)

Donnerstag, 13. November 2014

**Maxwell's eigenvalue problem for photonic crystals **

Photonic crystals are periodic optical nanostructures that affect the motion of photons through their geometry. They can be modeled as infintely periodic geometric structures in order to calculate their characteristic resonant wavelengths for the electromagnetic fields. The model can be reduced to the eigenvalue problem of Maxwell's equations with a form depending on the parameters of the material and geometry. In this talk I will give a description of the mathematical model for photonic crystals and illustrate some results on the existence of solutions for different examples of material and geometric parameters. Afterwards, I will consider the finite element method for solving the resulting eigenvalue problem and present some results on error estimation for simple cases of material and geometric parameters and indicate directions for further work regarding more complicated cases of parameters.

### Phi Ha (TU Berlin)

Donnerstag, 13. November 2014

**Analysis and Numerical solutions of Delay-DAEs **

Differential-algebraic equations (DAEs) have an important role in modeling practical systems, wherever the system needs to satisfy some algebraic

constraints due to conservation laws or surface conditions. On the other hand, time-delays occur naturally in various dynamical systems, both physically, when the transfer phenomena (energy, signal, material) is not instantaneous, and artificially, when a time-delay is used in the controller. The combination of differential-algebraic equations and time-delays leads to a new mathematical object: "delay differential-algebraic equations (Delay-DAEs)", which is a source of many complex behavior.

In this talk, we address the computational problem for numerical solutions to general linear Delay-DAEs.

First, we discuss the characteristic properties, which have not been mentioned in prior studies of numerical solutions to Delay-DAEs.

Then, we propose an algorithm, which extends the classical (Bellman) method of steps, to determine the solution of general linear Delay-DAEs.

Second, we examine the functionality and the efficiency of our algorithm by some illustrative examples

### Michael Overton (Courant Institute, NYU)

Donnerstag, 04. Dezember 2014

**Investigation of Crouzeix's Conjecture via Optimization**

We investigate a challenging problem in the theory of non-normal matrices

called Crouzeix's conjecture, which we will explain in some detail.

We present experimental results using nonsmooth optimization and CHEBFUN,

a very useful tool for computing with functions in MATLAB.

### Volker Mehrmann (TU Berlin)

Donnerstag, 04. Dezember 2014

When is a linear control system equivalent to a port-Hamiltonian system Joint work with Christopher Beattie and Hongguo Xu The synthesis of system models describing physical phenomena often follows a system-theoretic network paradigm that formalizes the interconnection of naturally specified subsystems. When the models of dynamic subsystem components arise from variational principles, the aggregate system model typically has structural features that characterize it as a port-Hamiltonian system. We discuss the structure of linear port-Hamiltonian systems and then answer the question when there exists a state space transformation that turns a general linear input-output system into a port-Hamiltonian system.

### Michał Wojtylak (Jagiellonian University, Cracow)

Donnerstag, 11. Dezember 2014

**On deformations of classical Jacobi matrices **

Let A be an infinite tridiagonal Hermitian matrix. We will try to reveal the spectrum of the product HA, where H=diag(-1,1,1,...). Our main interest will lie in locating the (unique) nonpositive eigenvalue of HA, i.e. the unique eigenvalue with the eigenvector x with x'Hx less or equal zero. It appears that this eigenvalue can be computes as a limit of eigenvalues of nonpositive type of the finite sections of the matrix HA. The character of the convergence will be discussed in detail. The research is motivated by a problem in signal analysis: detecting damped oscillations in a highly noisy signal. Joint work with Maxim Derevyagin (University of Mississippi).

### Christoph Conrads (TU Berlin)

Donnerstag, 18. Dezember 2014

**The Off-Diagonal Block Method**

The Off-Diagonal Block Method We present a new method for finding eigenspaces of large, sparse, symmetric positive semidefinite matrices. The method is well suited for computer implementation and can be used, f. e., to compute eigenpairs anywhere in the spectrum of the matrix or to calculate approximations to all matrix eigenvalues. We will explain how the method works, compare it to existing methods, and present numerical examples.

Joint work with Volker Mehrmann.

The slides can be found at

ftp://ftp.math.tu-berlin.de/pub/numerik/conrads/Conrads-ODBM-20141218.pdf

### Helia Niroomand Rad (TU Berlin)

Donnerstag, 18. Dezember 2014

**Time-Harmonic Scattering Problem Considering The Pole Condition**

Scattering problem in the field of wave propagation describes a large variety of problems in acoustic and electro-magnetics. In such problems, in order to have a proper formulation for the propagation in an unbounded domain, one may consider the radiating condition. In addition, to deal with the problem numerically within a bounded computational domain, one should impose an artificial boundary with a suitable boundary condition illustrating the radiating condition, which is canonically described by the pole condition. In this talk, we mainly show that one can apply the Laplace technique leading to the pole condition, and it is followed by some existence and uniqueness results.

### Judith Simon (TU Berlin)

Donnerstag, 15. Januar 2015

Structure Preserving Discretization Methods for Self-Adjoint Differential-Algebraic Equations

We consider a fundamental problem in control theory of minimizing a cost functional, subject to constraints that are modeled by a differential-algebraic equation with time-variant coefficient matrices. This problem is called the linear quadratic optimal control problem. Developing numerical methods for finding an optimal solution is an important and difficult task in relation with differential-algebraic equations. In general, there are two different approaches, which still form a big field in research. Firstly, one can apply common optimization techniques to the problem and discretize the resulting system afterwards. On the other hand, it is also possible to discretize the initial problem first and then apply optimization techniques. Here, we focus on the former approach.

It has been shown that the necessary optimality conditions of the discrete-time optimal control problem underly certain properties,

i.e., the optimality system is a self-conjugate operator associated with self-adjoint triples of coefficient matrices. We analyze the

continuous-time setting and are partly able to show the same result if we apply particular discretization methods to the continuous-time optimality system. Finally, we examine the results with the help of an example from the multibody mechanics.

### Matthias Voigt (TU Berlin)

Donnerstag, 15. Januar 2015

**The Kalman-Yakubovich-Popov Inequality for Differential-Algebraic Equations **

Abstract: The Kalman-Yakubovich-Popov lemma is one of the most famous results in systems and control theory. Loosely speaking, it states

equivalent conditions for the positive semi-definiteness of a so-called Popov function on the imaginary axis in terms of the solvability of a certain linear matrix inequality, namely the Kalman-Yakubovich-Popov (KYP) inequality. In applications, this lemma plays an important role in assessing feasibility of linear-quadratic optimal control problems or characterizing dissipativity of linear control systems.

In the literature, there exist manifold attempts to generalize this lemma to differential-algebraic equations. However, most of these approaches make certain restrictive assumptions such as a bounded index or impulse controllability. In this talk we show how to drop these restrictions by considering the KYP inequality on the system space, i.e., the subspace in which the solution trajectories of the system evolve. Moreover, we present results on the solution structure of this inequality. In particular, we consider rank-minimizing, stabilizing, and extremal solutions. These results can then be interpreted as a generalization of the algebraic Riccati equation to a very general class of differential-algebraic control systems.

### Jeroen Stolwijk (TU Berlin)

Donnerstag, 22. Januar 2015

**Error Analysis for the Euler Equations in Purely Algebraic Form **

Natural gas plays a crucial role in the energy supply of Europe and the world. It is sufficiently and readily available, is traded, and is storable. After oil, natural gas is the second most used energy supplier in Germany. The high and probably increasing demand for natural gas calls for a mathematical modeling, simulation, and optimisation of the gas transport through the existing pipeline network.

The gas flow through a pipeline can be accurately modeled by the one-dimensional Euler equations. However, their numerical solution requires much computational effort, such that several simplifications are often performed. The Euler equations in purely algebraic form are analysed in this presentation. More specifically, an error analysis is performed for both the mass flux and the temperature (the pressure was analysed in the previous talk). We ask ourselves: Are rounding and measurement errors amplified in the solution? This question is answered both theoretically and statistically.

Finally, we will investigate whether the model can be further simplified by assuming that the temperature is constant. Which condition(s) should be satisfied such that this assumption can be made safely?

### Philipp Schulze (TU Berlin)

Donnerstag, 29. Januar 2015

**Structure-Preserving Model Reduction **

In the past decades, model order reduction has gained increasing attention in all fields where numerical simulations are performed. Especially, in large-scale optimisation and control applications, reduced order models are needed to allow reasonable computation times and storage requirements while still maintaining an acceptable level of accuracy. When applying standard model reduction techniques, important properties of the original model (e.g. stability, passivity) are in general not included in the reduced model. Since these properties are often related to certain structures, this issue motivates for the field of structure-preserving model reduction. In this talk, we consider the Loewner framework, proposed by Mayo and Antoulas (2004), which is a technique for generalised realisation but may also be applied for the purpose of model reduction. The task is to extent this approach in order to preserve the so-called /port-Hamiltonian/ (pH) structure which very often arises by nature when modelling physical systems. Some first results regarding linear time-invariant pH systems are presented.

### Robert Altmann (TU Berlin)

Donnerstag, 29. Januar 2015

**Optimal Control for Problems with Servo Constraints**

Consider a robot with an end effector which should follow a prescribed trajectory. The corresponding model equations lead to a DAE structure with a so-called servo-constraint. If the input and output variables do not 'fit well', we obtain a system which is of index 5. In this talk, we consider an alternative approach using an optimal control setting.

### Leo Batzke (TU Berlin)

Donnerstag, 05. Februar 2015

**Rank-k Perturbations of Hamiltonian Matrices **

Many applications give rise to Hamiltonian matrices, that

is, matrices H with H^TJ=-JH, where J is skew-symmetric and

invertible. It is our goal to determine the change of the canonical

form of H when it is subjected to a generic (`typical') perturbation

that preserves the Hamiltonian structure.

Previously, generic Hamiltonian rank-1 perturbations were investigated

by Mehl, Mehrmann, Ran, and Rodman and it was shown that they may

produce different results from non-Hamiltonian perturbations.

Even so, it is a nontrivial task to extend these results to the case

of rank-k perturbations. In this talk, we will discuss the

difficulties of going from rank-1 perturbations to ones of arbitrary

rank and then derive a generic canonical form for Hamiltonian matrices

under Hamiltonian rank-k perturbations employing a new technique.

### Sarosh Quraishi (TU Berlin)

Donnerstag, 05. Februar 2015

**Parametric study of a brake squeal using machine learning**

In this talk we review POD (proper orthogonal decomposition) based model reduction for parametric studies of brake squeal and address some shortcomings and areas that need further improvement. I also present some preliminary results on using machine learning (classification using Support Vector Machine (SVM)) for the task of classifying parameter values responsible for a brake squeal using a data-set generated from a minimal model for brake squeal [1].

Reference:

[1] Utz von Wagner, Daniel Hochlenert, and Peter Hagedorn, Minimal models for disk brake squeal. Journal of Sound and Vibration 302 (2007), 527-539.

### Ute Kandler (TU Berlin)

Donnerstag, 12. Februar 2015

**Computation of extreme eigenvalues of a large-scale symmetric eigenvalue problem **

We consider a computation which approximates a few minimal eigenvalues of a symmetric large-scale eigenvalue problem. The application we have in mind comes from strongly correlated quantum spin systems where we can assume that the eigenvectors are representable in a low-rank tensor format. We use the tensor train format (TT) for vectors and matrices in order to overcome the curse of dimensionality and to make storage and computation feasible. The eigenstates of interest are computed by the minimization of a block Rayleigh quotient which is performed in an alternating scheme for all dimensions.

### Deepika Gill (TU Berlin)

Donnerstag, 12. Februar 2015

**Multi-Mode Control of Tall Building using Distributed Multiple Tuned Mass Dampers **

In the past decades, tuned mass damper (TMD) has gained increasing attention in the control of structural vibrations. The TMD has a mass, a spring and a dashpot connected to main system. The natural frequency of the TMD is tuned to natural frequency of the main system, the vibrations of the main system causes the damper to vibrate in resonance, and as a result, the vibration energy is dissipated through the damping in the TMD. The main disadvantage of a single TMD is its sensitivity of the effectiveness to the error in the natural frequency of the structure. This sensitivity can be reduced by multiple tuned mass dampers (MTMD). The basic configurations of MTMD consist of large number of dampers whose natural frequencies are distributed around the natural frequency of the controlled mode of structure and all dampers are placed at the topmost floor. The MTMD configurations result into placement intricacies. The problem of placement is solved by use of distributed multi tuned mass dampers (d-MTMDs). The dampers of d-MTMDs are distributed in the vertical direction along the height of building.

In this talk, we will review TMD, MTMD and d-MTMDs. Moreover, we will see variation of different responses such as displacement, acceleration etc. for TMD, MTMD and d-MTMDs. We will also discuss equation of motion for TMD, MTMD and d-MTMDs and its solution using Newmark’s beta method.