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## Absolventen-Seminar • Numerische Mathematik

Verantwortliche Dozenten: | Prof. Dr. Christian Mehl, Prof. Dr. Volker Mehrmann |
---|---|

Koordination: | 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 20.10. | 10:15 Uhr | MA 376 | Vorbesprechung | |

Do 27.10. | 10:15 Uhr | MA 376 | Robert Altmann | Splitting Methods for Constrained Diffusion-Reaction Systems [abstract] |

Christian Schröder | Quadratization for 2nd order model reduction [abstract] | |||

Do 03.11. | 10:15 Uhr | MA 376 | no seminar | |

Do 10.11. | 10:15 Uhr | MA 376 | Serkan Gugercin | On Some Recent Advances in Model Reduction [abstract] |

Do 17.11. | 10:15 Uhr | MA 376 | no seminar | |

Do 24.11. | 10:15 Uhr | MA 376 | Volker Mehrmann | Computation of Lyapunov Spectra for DAEs Using Half-Explicit Methods [abstract] |

Benjamin Unger | On well-posedness of implicit delay differential equations [abstract] | |||

Do 01.12. | 10:15 Uhr | MA 376 | Jeroen Stolwijk | On Adaptive Error Control Strategies, Rounding, and Iteration Errors for Gas Flow in Networks [abstract] |

Michelle Stahl | Numerical Computation of the Hydraulic Parts of a Waste Water Pump [abstract] | |||

Do 08.12. | 10:15 Uhr | MA 376 | Philipp Schulze | Structured Data-Driven Modeling [abstract] |

Christian Mehl | Generic rank-one perturbations of quaternionic matrices [abstract] | |||

Do 15.12. | 10:15 Uhr | MA 376 | Daniel Bankmann | Bilevel optimal control of parameter dependent differential-algebraic equations [abstract] |

Christian Schröder | Is there anything Balanced Truncation can't do? [abstract] | |||

Do 22.12. | 10:15 Uhr | MA 376 | no seminar | |

Do 05.01. | 10:15 Uhr | MA 376 | Sofia Bikopoulou | Recovery Techniques for solving Large Linear Systems with Fault Tolerance [abstract] |

Matthew Salewski | Using Lie groups to model transport phenomena [abstract] | |||

Do 12.01. | 10:15 Uhr | MA 376 | Lena Scholz | Structural Analysis for DAEs arising in the modeling of Electrical Circuits [abstract] |

Patrizia Nüßing | Model order reduction with adaptive POD [abstract] | |||

Do 19.01. | 10:15 Uhr | MA 376 | Melina Simichanidou | Discontinuous Galerkin Methods for Reactive Euler Equations [abstract] |

Arbi Moses Badlyan | Port-Hamiltonian formulation of a semi-discrete approximate model of inviscid Burgers' [abstract] | |||

Do 26.01. | 10:15 Uhr | MA 376 | Marine Froidevaux | Photonic Crystals: Applications, Modeling, and Analysis [abstract] |

Hendrik Schleifer | Discretization of the port-Hamiltonian system for the Navier Stokes equations [abstract] | |||

Do 02.02. | 10:15 Uhr | MA 376 | Andres Gonzalez Zumba | New Techniques of Dynamic Modelling and Stability Analysis for Network Systems Applied to Energy Systems with Uncertainties [abstract] |

Benjamin Unger | Hankel Singular Values vs. Kolmogorov n-widths [abstract] | |||

Do 09.02. | 10:15 Uhr | MA 376 | Christoph Zimmer | On the smoothing property of PDEs with retarded delayed terms [abstract] |

Nico Dethard | Calculating Eigenvalues of a Dissipative Port-Hamiltonian System via a Homotopy approach [abstract] | |||

Do 16.02. | 10:15 Uhr | MA 376 | Geraldine Lina Frantz | A randomized method for solving large hierarchical equation systems [abstract] |

Matthias Voigt | A Greedy Subspace Method for Computing the H-infinity-Norm [abstract] |

### Robert Altmann (TU Berlin)

Donnerstag, 27. Oktober 2016

**Splitting Methods for Constrained Diffusion-Reaction Systems**

We consider nonlinear diffusion-reaction systems which have an additional constraint such as having a prescribed integral mean. With the help of Lie and Strang splitting we would like to treat the nonlinearity separately. This means that the time integration is reduced to the solution of a linear constrained system and a nonlinear ODE.

However, Strang splitting suffers from order reduction which limits its efficiency. This is caused by the fact that the nonlinear subsystem produces inconsistent initial values for the constrained subsystem. In this talk we show that the incorporation of an additional correction term resolves this problem without increasing the computational costs.

### Christian Schröder (TU Berlin)

Donnerstag, 27. Oktober 2016

**Quadratization for 2nd order model reduction **

Quadratization is the problem of finding a quadratic matrix polynomial of size n by n that has the same eigenvalues as a given matrix of size 2n by 2n. This is possible under certain, not very restricting conditions, and the computations are surprisingly simple.

Second order model reduction of linear time invariant (LTI) systems is the problem of finding a small second order dynamical LTI system of size r whose outputs are close to those of a given large second order dynamical LTI system of size n, where r<<n. A standard approach is to reformulate the second order system as a first order system using companion linearization. Then any sort of first order model reduction technique can be applied. The resulting small-scale first order system is then brought to second order in one way or another. In this step the individual approaches differ. We propose to use the above mentioned quadratization method. This allows to transfer some error bounds and optimality statements of the employed first order technique.

### Serkan Gugercin (Virginia Tech)

Donnerstag, 10. November 2016

**On Some Recent Advances in Model Reduction**

In this talk, we will discuss two recent results on model reduction of linear dynamical systems:

In the first part of the talk, we will focus on model reduction problem for systems with inhomogeneous initial conditions. Building upon the observation that the full system response is decomposable as a superposition of the response map for an unforced system having nontrivial initial conditions and the response map for a forced system having null initial conditions, we will present a new approach that involves reducing these component responses independently and then combining the reduced responses into an aggregate reduced system response. This part of the talk is a joint work with C.A. Beattie and V. Mehrmann.

In the second part of the talk, we will present a computationally effective approach for producing high-quality H-infinity approximations to large scale linear dynamical systems by combining ideas originating in interpolatory H2-optimal model reduction with complex Chebyshev approximation. The approach is able to avoid computationally demanding H-infinity norm calculations that are normally required to monitor progress within each optimization cycle through the use of "data-driven" rational approximations that are built upon previously computed function samples. Numerical examples will illustrate that the proposed approach produces high fidelity reduced models having consistently better H-infinity performance than models produced via balanced truncation; and often are as good as (and occasionally better than) models produced using optimal Hankel norm approximation as well. This part of the talk is a joint work with C.A. Beattie and A. Castagnotto.

### Volker Mehrmann (TU Berlin)

Donnerstag, 24. November 2016

**Computation of Lyapunov Spectra for DAEs Using Half-Explicit Methods **

The computation of Lyapuniv Spectra for DAEs is highly expensive, since very high dimensional DAEs have to solved for very long time periods. To make this process more efficient, half-explicit methods based on popular explicit methods like one-leg methods, linear multistep methods, and Runge-Kutta methods are proposed and analyzed. Compared with well-known implicit methods for DAEs, these half-explicit methods demonstrate their efficiency particularly for a special class of semi-linear matrix-valued DAEs which arise in the numerical computation of spectral intervals for DAEs. Numerical experiments illustrate the theoretical results. This talk is based on joint work with Vu Hoang Linh (Hanoi).

### Benjamin Unger (TU Berlin)

Donnerstag, 24. November 2016

**On well-posedness of implicit delay differential equations**

It is a standard task in control theory to investigate stability of dynamical systems. If the stationary point under investigation is not stable, then one might try to stabilize it via feedback control. A feedback requires the observation (measurement) of the state and based on this data, the computation of the actual feedback. Both actions require a certain amount of time (which is often neglected in literature) yielding the class of time-delayed systems. In this talk we investigate implicit delay differential equations of the form F(t,x(t),\dot{x}(t),x(t-\tau)) = 0 with \tau>0. Such systems present several challenges, requiring index-reduction, consistent initialization, non-smooth crossings at integer multiples of the delay time \tau, and a-causal behavior. We demonstrate these challenges via examples and offer a regularization strategy for a certain class of such systems. In addition to the theoretical properties, the regularized system is also suitable for numerical integration. As application we discuss multibody systems subject to delayed feedback.

### Jeroen Stolwijk

Donnerstag, 01. Dezember 2016

**On Adaptive Error Control Strategies, Rounding, and Iteration Errors for Gas Flow in Networks**

Natural gas plays a crucial role in the world's energy supply. Moreover, the amount of gas that is consumed increases every year. This calls for an efficient mathematical modeling, simulation, and optimization of the gas flow through the pipeline network. In order to make real-time decisions, e.g., opening or closing a valve, the simulations should be performed both quickly and reliably.

This presentation consists of two parts. The first part concerns itself with three adaptive error control strategies. Having a pipeline network with the total error being larger than the tolerance, these strategies determine for which pipeline a refinement should be made in space, time or model. The strategies are applied on a three level model hierarchy that is based on the one-dimensional isothermal Euler equations. Computational cost functionals for these three models are determined, model and discretization error estimators for a general functional of interest are known from the literature. It is demonstrated that our newly developed strategies outperform the one that is currently implemented in the gas simulation software ANACONDA.

In the second part of the presentation we derive a measure for the error in the solution of a nonlinear system of equations due to rounding and the preliminary stopping of the Newton iteration. Therefore, we discuss a paper by T.J. Ypma (1983) in which a condition is derived for which the computed sequence of the Newton method converges to the solution of the nonlinear system despite rounding errors. The derived error measure is applied to a nonlinear system resulting from a discretization of the Euler equations in semilinear form. We find that the rounding and iteration error in the solution can be neglected compared to the model, discretization, and data uncertainty error.

The first part is joint work with P. Domschke, A. Dua, and J. Lang. The second part is joint work with V. Mehrmann.

### Michelle Stahl (TU Berlin)

Donnerstag, 01. Dezember 2016

**Numerical Computation of the Hydraulic Parts of a Waste Water Pump **

Introduction of my Bachelor's thesis, the numerical computation of the hydraulic parts of a waste water pump in ANSYS, a computational Fluid Dynamic Software. For that I will describe the mathematical equations used to model fluid flow in ANSYS CFX, which is based on the unsteady Navier-Stokes equations in their conservative form. Focusing on turbulence models I will explain the k-epsilon model, which is commonly used in CFX.

### Christian Mehl (TU Berlin)

Donnerstag, 08. Dezember 2016

**Generic rank-one perturbations of quaternionic matrices **

In recent years, generic rank-one perturbations of real and complex matrices have been studied intensively and by now the theory of both general and structure-preserving perturbations is well understood for both unstructured matrices as well as many classes of structured matrices.

In this talk, we investigate the effect of generic rank-one perturbations on matrices over the quaternions, where, surprisingly, we observe a different behavior than in the case of real or complex matrices.

### Philipp Schulze (TU Berlin)

Donnerstag, 08. Dezember 2016

**Structured Data-Driven Modeling **

In applications where only little is known about the system of interest, data-driven modeling provides an attractive alternative to physical modeling since it is based on measured or simulated data and does not require a deeper understanding of the underlying process. The Loewner framework is a data-driven modeling approach which constructs a generalized realization (descriptor system) based on frequency measurements of the transfer function of an input/ouput system. This interpolation-based method has been shown to deliver good approximations in many cases, however lacks accuracy in some applications, especially when dealing with systems with irrational transfer functions (e. g. time-delay systems).

In this talk, we present an extension of the Loewner framework which allows to construct models with more general transfer functions parametrized by some coefficient functions. The specific choice of coefficient functions imposes the system structure, which covers, for instance, second-order systems, integro-differential equations and systems with time-delayed state variables. The presented approach not only allows to impose a model structure but in many cases also yields realizations matching more interpolation data than the Loewner framework while maintaining the number of state variables. We demonstrate the performance of the new method by means of numerical examples and compare the results to those of the classical Loewner framework.

This is joint work with Benjamin Unger, Christopher Beattie, and Serkan Gugercin.

### Daniel Bankmann (TU Berlin)

Donnerstag, 15. Dezember 2016

**Bilevel optimal control of parameter dependent differential-agebraic equations**

Differential-algebraic optimal control problems can be obtained from modeling the dynamical behavior of various applications, such as multibody systems and electrical circuits. However, sometimes the system's description is depending on certain parameters which might have to be fixed before building the real-world plant. To obtain these parameters in an 'optimal' way it is natural to consider multiple levels of optimizations.

In this talk I will present the topic of my PhD. We will discuss the problem setting of bilevel optimal control and some of the difficulties that may even arise in the linear case; for instance non-smoothness with respect to the parameters of the solution and the transformation matrices leading to a strangeness-free formulation.

### Christian Schröder (TU Berlin)

Donnerstag, 15. Dezember 2016

**Is there anything Balanced Truncation can't do?**

Balanced truncation (BT) is a model order reduction method that is famous for its elegant H_\infty error bound that holds for zero initial values. We will look at two variants of BT, one for nonzero initial values, and one aimed at H_2 error bounds. In both cases the error bounds are almost as elegant as the original.

### Sofia Bikopoulou (TU Berlin)

Donnerstag, 05. Januar 2017

**Recovery Techniques for solving Large Linear Systems with Fault Tolerance**

Modern society relies on highly complex computational systems; from 1.000 cores to 1.000.000 cores are used to achieve massive parallelism in scientific or industrial software applications. As current trends denote, it is essential to provide methods that avoid system failure and deliver correct service even in the presence of faults. Consequently, effective fault tolerance solutions for such High Performance Computing (HPC) platforms are more than necessary.

One way of achieving fault tolerance is by using recovery techniques based on checkpointing (CKPT) and rollback protocols. The state of the entire application is periodically saved and can be retrieved in case the original application crashes. The application is then restarted or recovered from the last checkpoint created and continued from that point on, thereby minimizing the time lost due to the failure.

A wide variety of checkpointing and rollback recovery techniques is used in different applications. In this talk I will present a recovery method for solving large linear systems with fault tolerance in HPC platforms.

### Matthew Salewski (TU Berlin)

Donnerstag, 05. Januar 2017

**Using Lie groups to model transport phenomena **

Transport phenomena pose a problem for conventional modelling techniques. Here I discuss the use of Lie groups to enhance reduced-basis methods.

First, I will review the use of Lie groups in modelling of advection-dominated problems. In such cases, the Lie group implies a time-dependent basis. Fortunately, the time dependence is decoupled from the dynamics and can be moved to the offline postprocessing stage.

Next, I introduce a method which builds on these concepts to construct a model in which some of the basis vectors are time-dependent; these particular basis vectors can describe the transport phenomena and their evolution can largely be determined by the Lie group. However, the time-dependence destroys the online-offline decomposition which would otherwise contribute to the model's efficiency. I will discuss some strategies to address this issue.

### Lena Scholz (TU Berlin)

Donnerstag, 12. Januar 2017

**Structural Analysis for DAEs arising in the modeling of Electrical Circuits **

We consider the Signature Method for the structural analysis of differential-algebraic equations (DAEs) that arise in the modeling and simulation of electrical circuits. Different formulations of the set of model equations are considered. For some formulations we show that the structural approach may fail for certain circuit topologies, while other formulations are better suited for a structural analysis. The results are illustrated by a number of examples.

### Patrizia Nüßing (TU Berlin)

Donnerstag, 12. Januar 2017

**Model order reduction with adaptive POD**

In this talk I will introduce the topic of my master thesis. It deals with the model order reduction applied to the heat equation. For parabolic PDEs, with distributed input and output, Volker Mehrmann, Jan Heiland and Michael Schmidt developped a framework which constructs a low-order Input/Output map by a direct discretization of the original I/O map and the subsequent use of the Tucker decomposition. My task now is to extend their approach by an adaptive POD, using a combination of deal II and matlab in order to reduce the computing time.

### Melina Simichanidou (TU Berlin)

Donnerstag, 19. Januar 2017

**Discontinuous Galerkin Methods for Reactive Euler Equations**

The nonreactive Euler equations of gas dynamics are nonlinear conservation laws of mass, momentum and energy. Extending this set with equations describing chemical processes yields the complete set of reactive Euler equations. In order to solve this enriched system in this talk, we present the framework of the Discontinuous Galerkin Finite Element Method. We illustrate the behavior of the method using the so-called ZND model with the one-step chemical reaction involving two (2) species. Furthermore, the corresponding formulation for the three (3) species water reaction model will be presented.

### Arbi Moses Badlyan (TU Berlin)

Donnerstag, 19. Januar 2017

**Port-Hamiltonian formulation of a semi-discrete approximate model of inviscid Burgers'**

Modeling of complex physical phenomena can involve systems of coupled partial differential equations, which through spatial discretization result in systems of ordinary differential equations of large space-state dimension.** **Structure-preserving model reduction approaches applicable to large-scale, nonlinear port-Hamiltonian systems require the original model to be formulated as finite-dimensional port-Hamiltonian system.** **The inviscid Burgers' equation is of interest, since it is the prototype of a first order hyperbolic conservation equation that can develop shock waves.

In this talk I present some results of a yet unfinished work where a standard finite-volume scheme is used to obtain an approximate finite-dimensional port-Hamiltonian representation of the inviscid Burgers' equation.

After a brief review of the finite-volume method, I will show how the inviscid Burgers' equation is discretized using local Lax-Fridrichs numerical flux. Then I discuss the conditions under which a reformulation of the semi-discrete approximate model can be identified as a non-linear finite-dimensional input-state output Port-Hamiltonian system.

### Marine Froidevaux (TU Berlin)

Donnerstag, 26. Januar 2017

**Photonic Crystals: Applications, Modeling, and Analysis**

Photonic crystals are structures which interact with light in a special way and which have developed as a promising technology in the last few decades. Experts predict a wide range of applications, including laser guides for cancer surgery or optical computing.

In this talk, I will first give an insight into the physics behind some recent applications. We will then discuss the modeling of the physical phenomena based on the Maxwell equations of electromagnetism and address the challenges linked to the discretization of these equations, as well as their solving with techniques from numerical linear algebra.

### Hendrik Schleifer (TU Berlin)

Donnerstag, 26. Januar 2017

**Discretization of the port-Hamiltonian system for the Navier Stokes equations**

The talk will introduce my master thesis topic of how to discretize the port-Hamiltonian formulation for the Navier Stokes Equations. A short introduction to the port-Hamiltonian framework and splitting methods will be presented and the problems that arise due to discretization especially concerning the dissipative part of the system. Furthermore an example implementation for a port-Hamiltonian formulation of the one dimensional shallow water equations for an open channel flow will be shown.

### Andres Gonzalez Zumba (TU Berlin)

Donnerstag, 02. Februar 2017

**New Techniques of Dynamic Modelling and Stability Analysis for Network Systems Applied to Energy Systems with Uncertainties**

Deterministic approach has been traditionally used for modelling, simulation, analysis, control and optimization of energy systems and multi-physical systems in general. Nowadays in the engineering area, the most of dynamic stability analysis procedures are performed under this perspective. However, a deterministic viewpoint does not provide a realistic assessment of the system performance when important uncertain variables are present. In the current presentation I propose the identification, study and evaluation of new stability analysis methods in order to deal with dynamic stability analysis of complex network energy systems with variant parameters. I introduce a review covering numerical time-domain integration of the general Stochastic Differential-Algebraic Equations (SDAE) which describe the systems behaviour, process known in engineering area as *time-domain analysis*. Subsequently, I focus on methods based on *Lyapunov stability concepts* applied to the original nonlinear systems models, and *spectral analysis* for linearized version models of the systems with time-variant parameters (considering the statistical moments of the uncertain variables in both cases). These techniques will be applied to electric power systems and gas systems modelled in an energy-based and structure preserving Port-Hamiltonian form descripted by Differential-Algebraic Equations (DAE). Although the current investigation is mainly a theoretical description about the research developed so far, it is expected that these new approaches in the formulation and analysis of finite systems with lumped parameters can pose valuable contributions in order to enhance the study and evaluation of energy network systems.

### Benjamin Unger (TU Berlin)

Donnerstag, 02. Februar 2017

**Hankel Singular Values vs. Kolmogorov n-widths**

The model reduction community has seen major efforts in the last two decades and numerous improvements have been made, for example concerning linear time invariant systems, bilinear systems, nonlinear and parametric model reduction, data-driven model reduction, and structure preservation. Due to its wide applicability the research is carried out by different communities using their own tools and language. For fruitful collaborations, it is utmost important to have a common ground to work on. In this talk we contribute to this common ground by relating some of the concepts from the dynamical system and the PDE community. More precisely, we derive a connection between the Hankel singular values and the Kolmogorov n-widths - both concepts used to measure the applicability of (standard) MOR methods.

### Christoph Zimmer (TU Berlin)

Donnerstag, 09. Februar 2017

**On the smoothing property of PDEs with retarded delayed terms**

In the last decades ordinary (ODEs) and partial differential equations (PDEs) with delayed terms were used to describe practical problems as in population dynamics, fluid dynamics, climate modeling, or in control problems. A special class of this equation types is the class of problems where only the state - and not the time derivatives - appears with a delay. It is well-known, that the solutions of these systems become smoother over time for the ODE case. In this talk, we investigate if the solutions for linear delay PDEs have the same smoothing property. We show a counterexample and discuss under which conditions the solution of a so-called retarded delay PDE behaves like in the ODE case.

This is a joined work with Robert Altmann.

### Nico Dethard (TU Berlin)

Donnerstag, 09. Februar 2017

**Calculating Eigenvalues of a Dissipative Port-Hamiltonian System via a Homotopy approach **

In this talk I will present my bachelor thesis topic of how to calculate eigenvalues of a dissipative port-hamiltonian system using a homotopy approach. At first glance it might seem like there are no properties of the system which would help calculating its eigenvalues. I will talk about reducing the system, show why this makes it easier to calculate eigenvalues and how we can use the continuity of eigenvalues and eigenvectors to get the ones we originally wanted. Also I will give a short introduction of all the steps which need to be taken in between.

### Geraldine Lina Frantz (TU Berlin)

Donnerstag, 18. Februar 2016

**A randomized method for solving large hierarchical equation systems**

In this talk I will introduce the topic of my bachelor thesis, in which I try to develop an algorithm to approximate the solution of hierarchical linear equation systems in a fast way using randomization. This approach is motivated by the fact, that the computational effort of the numerical solution of PDEs grows very fast in the number of basis functions that are used for the finite element method. I consider a hierarchical stiffness matrix, whose submatrices and corresponding equation systems are considered iteratively in the algorithm. I reduce the number of constraints by premultiplying the current system with a random matrix and hereby obtaining a random linear combination of the rows of the considered matrix. I will present the idea, implementation and first results of this concept and give an overview of possible enhancements of the method.

### Matthias Voigt (TU Berlin)

Donnerstag, 16. Februar 2017

**A Greedy Subspace Method for Computing the H-infinity-Norm **

We consider the computation of the H-infinity-norm for transfer functions of a general class of linear and non-linear systems. We focus on the case where the state-space dimension is very large. We propose a subspace projection method to obtain approximations of the transfer function by interpolation techniques. The H-infinity-norms are computed for the resulting reduced functions, then the subspaces are refined by means of the optimal points on the imaginary axis where the maximum singular value of the reduced function is attained. In this way we obtain much better performance compared to existing methods.