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## Absolventen-Seminar • Numerische Mathematik

Verantwortliche Dozenten: | Prof. Dr. Christian Mehl, Prof. Dr. Volker Mehrmann |
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Koordination: | Benjamin Unger, Dr. Matthias Voigt |

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 19.10. | 10:15 Uhr | MA 376 | Vorbesprechung | |

Do 26.10. | 10:15 Uhr | MA 376 | Ines Ahrens | A generalization of the Pantelides algorithm for DAEs with delay [abstract] |

Murat Manguoglu | Parallel Solution of Sparse Underdetermined Linear Least Squares Problems [abstract] | |||

Do 02.11. | 10:15 Uhr | MA 376 | no seminar | |

Do 09.11. | 10:15 Uhr | MA 376 | Philibert Pinkert | Correction of different errors in vehicular probe data to generate high definition maps [abstract] |

Benjamin Unger | Model reduction for linear systems with low-rank switching [abstract] | |||

Do 16.11. | 10:15 Uhr | MA 376 | Riccardo Morandin | Model Hierarchy and Synchronicity Analysis of Power Network Models in Port-Hamiltonian Form [abstract] |

Felix Black | Model order reduction for transport phenomena illustrated with the 1D advection equation [abstract] | |||

Do 23.11. | 10:15 Uhr | MA 376 | Marko Hajba | Numerical study of the approximation quality of 1D PDE network approximation of the 3D model of the endovascular stent [abstract] |

Matko Ljulj | Mesh-reinforced shells [abstract] | |||

Do 30.11. | 10:15 Uhr | MA 376 | ||

Do 07.12. | 10:15 Uhr | MA 376 | Murat Manguoglu | Efficient Preconditioners for Solving Sparse Linear Systems in Quadratic Eigenvalue Problems [abstract] |

Christian Mehl | Linear algebra properties of dissipative Hamiltonian descriptor systems [abstract] | |||

Do 14.12. | 10:15 Uhr | MA 376 | Matthew Salewski | - canceled - |

Punit Sharma | Port-Hamiltonian systems and various distances for control systems [abstract] | |||

Do 21.12. | 10:15 Uhr | MA 376 | Philipp Schulze | Optimal Shifted Mode Decomposition for Advection-Dominated Systems [abstract] |

Jeroen Stolwijk | Model and Discretization Error Adaptivity within Stationary Gas Transport Optimization [abstract] | |||

Do 11.01. | 10:15 Uhr | MA 376 | Carlo Cassina | The break squeal problem: computing nearest stable matrix pair to the linearization of the quadratic eigenvalue problem to avoid the brake squeal. [abstract] |

Marine Froidevaux | Estimators for discretization and algebraic errors in simulation of photonic crystals [abstract] | |||

Do 18.01. | 10:15 Uhr | MA 376 | Arbi Moses Badlyan | Metriplectic Systems [abstract] |

David Noben | A successive linear programming approach for the Adaptive Model and Discretization Control Algorithm [abstract] | |||

Do 25.01. | 10:15 Uhr | MA 376 | David Kohn | - canceled - |

Christoph Zimmer | On an Operator-GENERIC Formulation for Mixtures of Reactive Fluids [abstract] | |||

Do 01.02. | 10:15 Uhr | MA 376 | Daniel Bankmann | On error estimation of a nonlinear least squares type bilevel optimal control problem [abstract] |

Sofia Bikopoulou | An Algorithm-Based Fault Tolerance Approach for Solving Large Scale Linear Systems [abstract] | |||

Do 08.02. | 10:15 Uhr | MA 376 | Sarah Hauschild | Model Order Reduction for port-Hamiltonian Differential-Algebraic Equations [abstract] |

Andres Gonzales Zumba | - canceled - | |||

Di 15.02. | 10:15 Uhr | MA 376 | Hannes Gernandt | On the gap distance between matrix pencils [abstract] |

Volker Mehrmann | - canceled - |

# Rückblick

- 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

### Ines Ahrens (TU Berlin)

Donnerstag, 26. Oktober 2017

** A generalization of the Pantelides algorithm for DAEs with delay**

The strangeness index for DAEs is based on the derivative array, which consists of the system itself plus its time derivatives. Index reduction is performed by selecting certain important equations from the derivative array. In a large-scale setting with high index, this might become computationally infeasible. However, if it is known a priori which equations of the original systems need to be differentiated, then the computational cost can be reduced. One way to determine these equations is by means of the Pantelides algorithm.

If the DAE features in addition a delay term then taking derivatives might not be sufficient and instead, the derivative array must additionally be shifted in time thus increasing the computational complexity even further. In this talk I will explain how one can modify the Pantelides algorithm such that it determines the equations which need to be shifted and/or differentiated. This is joint work with Benjamin Unger.

### Murat Manguoglu (Middle East Technical University)

Donnerstag, 26. Oktober 2017

**Parallel Solution of Sparse Underdetermined Linear Least Squares Problems**

Sparse underdetermined systems of equations in which the minimum norm solution needs to be computed arise in many applications, such as geophysics, signal processing, and computational finance. In this talk, we introduce a parallel algorithm for obtaining the minimum 2-norm solution of sparse underdetermined system of equations. The proposed algorithm assumes a generalized banded form where the coefficient matrix has a column overlapped block structure in which the blocks are sparse. The blocks are handled independently by any existing solver and a smaller reduced system is formed and needs to be solved before obtaining the minimum norm solution of the original system in parallel. We implement the proposed algorithm by using the message passing paradigm. We show the parallel scalability of the proposed algorithm and compare it against an existing state-of-the-art solver on both shared and distributed memory platforms. This is a joint work with F. Sukru Torun (Bilkent University) and Cevdet Aykanat (Bilkent University).

### Philibert Pinkert (TU Berlin)

Donnerstag, 09. November 2017

**Correction of different errors in vehicular probe data to generate high definition maps**

High definition maps play an important role in enabling highly automated driving as well as other applications. Since the standard method to generate these maps is rather expensive, we want to create a highway map using the abundantly available sensor data of ordinary vehicles, which are driving on the road for another purpose than generating a map. The drawback about this data is its low precision and other serious errors that need to be corrected first.

One of the errors is a temporal offset between the different sensors. I will show how I correct this offset by minimizing an error function. Another error causes the traces on curvy parts of the road to be farther away from their actual position than on straight parts. I will present my progress on finding a model for this error assuming it is a side effect of the Kalman Filter in the GPS module and fitting it to the data. Finally, I will talk about my ideas on how to generate a map using the corrected sensor data.

### Benjamin Unger (TU Berlin)

Donnerstag, 09. November 2017

**Model reduction for linear systems with low-rank switching**

In this talk we present a new strategy for model reduction of linear switched systems. The main idea is to derive an abstract model without switching - called the envelope system - that is able to reproduce the behavior of the switched system if a suitable feedback is supplied. That advantage of this formalism is that one can use standard MOR techniques. Moreover, the envelope system has a physical interpretation that describes the energy that comes with switching between different models. If additionally the subsystems of the switched system are port-Hamiltonian systems, that this structure can be preserved in the reduced model. This talk describes joint work with Philipp Schulze (TU Berlin).

### Riccardo Morandin (TU Berlin)

Donnerstag, 16. November 2017

**Model Hierarchy and Synchronicity Analysis of Power Network Models in Port-Hamiltonian form **

In view of highly decentralized and diversified power generation concepts, in particular with renewable energies such as wind and solar power, the analysis and control of the stability and the synchronization of power networks is an important task that requires different levels of modeling detail for different tasks. A frequently used qualitative approach relies on simplified nonlinear network models like the Kuramoto model.

In this talk we present the first draft of a model hierarchy for power networks. In particular, we present a new energy-based formulation of the Kuramoto model as port-Hamiltonian system of differential-algebraic equations. This leads to a very robust representation of the system with respect to disturbances, and it encodes the underlying physics, such as the dissipation inequality or the deviation from synchronicity, directly in the structure of the equations.

This talk describes joint work with Volker Mehrmann (TU Berlin), Simona Olmi (TU Berlin) and Eckehard Schöll (TU Berlin).

### Felix Black (TU Berlin)

Donnerstag, 16. November 2017

**Model Order Reduction for transport phenomena illustrated with the 1D advection equation**

In many physical applications such as thermodynamics and fluid dynamics, transport phenomena are observed. Transport phenomena usually describe the advection of mass, energy, or other physical quantities without diffusion. Unfortunately, they also present a challenge for classical model order reduction (MOR) techniques, in cases where the solution exhibits high gradients, e.g., a shock.

The goal of my master thesis is to analyse and compare MOR techniques that are tailored to advection-dominated problem models by means of the 1D wave equation and the 1D Burgers' equation. In this talk, we will instead use the 1D advection equation, for simplicity, and investigate some of the MOR techniques which will be considered later in the thesis.

The methods under investigation are

(1) a modified version of Proper Orthogonal Decomposition (POD) that is able to preserve port-Hamiltonian structure,

(2) a symmetry reduction approach, and

(3) an extended finite element method (XFEM).

The effectiveness of the methods will be demonstrated via numerical examples.

### Marko Hajba (University of Zagreb, Croatia)

Donnerstag, 23. November 2017

**Numerical study of the approximation quality of 1D PDE network approximation of the 3D model of the endovascular stent**

We present a prototype model of an endovacular stent in COMSOL Multiphysics. The 3D model is discretized with tetrahedral elements. The 3D object has then been sampled on all intersection points of struts. Based on this information a graph has been constructed where two nodes adjacent to the same strut areconnected by an edge. 1D PDE network model of Canic and Tambaca has been built on this structured and the convergence of the 3D model when reducing the thickness has been numerically studied. We present results of numerical convergence experiments. The experiments show that a 1D PDE network model of 5e3 DOFs agrees with the 3D model of order 2e6 DOFs with as little as 5% error.

### Matko Ljulj (University of Zagreb, Croatia)

Donnerstag, 23. November 2017

**Mesh-reinforced shells**

We formulate a new free-boundary type mathematical model describing the interaction between a shell (described by a two-dimensional (2D) Naghdi type shell model) and mesh-like structure consisting of thin rods (described by a 1D network model of curved rods). We apply this model in interaction between vascular walls treated with vascular devices called stents, by developing a solver within Freefem++ and apply it to four commercially available coronary stents as the interact with the vascular wall.

This is a joint work with S. Čanić, M. Galović and J. Tambača.

### Murat Manguoglu (Middle East Technical University)

Donnerstag, 07. Dezember 2017

**Efficient Preconditioners for Solving Sparse Linear Systems in Quadratic Eigenvalue Problems**

In the course of minimizing disk brake squeal the solution of a special quadratic eigenvalue problems (QEP) is required. Solving such problems usually involve linearization of the QEP, resulting in a two times bigger but linear eigenvalue problem with the same eigenvalues. After linearization, a classical shift-and-invert Arnoldi or other eigensolvers can be used to find the eigenvalues which requires the solution of shifted and complex linear systems. In this talk, we will present a novel multilevel preconditioning scheme for solving such linear systems based on their block structures. As part of the proposed scheme we also present a new robust iterative scheme for solving linear systems that are symmetric and indefinite which is also a general purpose iterative scheme. This is a joint work with Volker Mehrmann.

### Christian Mehl (TU Berlin)

Donnerstag, 07. Dezember 2017

**Linear algebra properties of dissipative Hamiltonian descriptor systems**

The properties of regular and singular matrix pencils arising from dissipative Hamiltonian descriptor systems are investigated. In particular, we will see that under mild assumptions such pencils have the following properties: all eigenvalues are in the closed left half plane, the nonzero finite eigenvalues on the imaginary axis are semisimple, the index is at most two, and there are restrictions for the possible left and right minimal indices.

### Punit Sharma (University of Mons, Belgium)

Donnerstag, 14. Dezember 2017

**Port-Hamiltonian systems and various distances for control systems**

Motivated by the structure of LTI port-Hamiltonian systems, we define the DH matrix: a matrix A ∈ F^{nxn}, where F ∈ {R, C} is said to be a DH matrix if A = (J − R)Q for some matrices J, R, Q ∈ F^{nxn} such that J is skew-Hermitian, R is Hermitian positive semidefinite and Q is Hermitian positive definite.

In the first half of the talk, I will briefly discuss the various structured distances to instability for LTI port-Hamiltonian systems using DH matrices. In the second half of the talk, I will talk about the distance to stability for general LTI control systems, i.e., the minimal perturbation under which an unstable system becomes stable. This is the converse problem of the distance to instability problem. We will show that a system is stable if and only if its state matrix is a DH matrix. We propose new algorithms to solve this problem.

These ideas can be generalized to get good approximate solutions to some other nearness problems for control systems like, distance to stability for descriptor systems, distance to positive realness, and minimizing the norm of static feedback.

### Philipp Schulze (TU Berlin)

Donnerstag, 21. Dezember 2017

**Optimal Shifted Mode Decomposition for Advection-Dominated Systems**

Model order reduction (MOR) schemes have become a useful tool in applications where a model needs to be evaluated many times with different input settings, for instance, in optimization and control. Most MOR techniques for nonlinear systems are based on snapshots of the full-order solution to identify a small number of modes, a linear combination of which describes the main features of the solution sufficiently well. However, for systems whose dynamics are dominated by advection, the solution often cannot be approximated sufficiently well by a linear combination of just a few modes.

In this talk, we present the shifted proper orthogonal decomposition (sPOD) which extends the classical approach by shifting the modes in space in order to follow the advection. We propose a new algorithm which computes a sPOD based on the problem of minimizing the residual of the approximation. The numerical examples reveal that even strongly advection-dominated phenomena can be described well with small numbers of shifted modes.

### Jeroen Stolwijk (TU Berlin)

Donnerstag, 21. Dezember 2017

**Model and Discretization Error Adaptivity within Stationary Gas Transport Optimization**

In this talk the minimization of operation costs for natural gas transport networks is studied. Based on a recently developed model hierarchy ranging from detailed models of instationary partial differential equations with temperature dependence to highly simplified algebraic equations, modeling and discretization error estimates are presented to control the overall error in an optimization method for stationary and isothermal gas flows. The error control is realized by switching to more detailed models or finer discretizations if necessary to guarantee that in the end a prescribed model and discretization error tolerance is satisfied. The adaptively controlled optimization method converges and the new approach is illustrated with numerical examples.

This is joint work with Volker Mehrmann and Martin Schmidt.

### Carlo Cassina (TU Berlin)

Donnerstag, 11. Januar 2018

**The brake squeal problem: computing nearest stable matrix pair to the linearization of the quadratic eigenvalue problem to avoid the brake squeal.**

The brake squeal problem is a noise pollution problem that affects people's everyday life nowadays. It is possible to define the macroscopic equations of motion of the brake disk from finite element modeling. The problem can be transformed into a Quadratic Eigenvalue Problem dependent from one parameter. The QEP can be reduced with model order reduction and studied for all values of the parameter. Proper orthogonal decomposition permits to reduce the order of the QEP in a definetely more successful way than classical modal truncation. Finally it is possible to define the nearest stable matrix pair to the original pair coming from the linearization using fast gradient method (FGM). The nearest matrix pair represents now a new brake that avoids squeal.

### Marine Froidevaux (TU Berlin)

Donnerstag, 11. Januar 2018

**Estimators for discretization and algebraic errors in simulations of photonic crystals.**

Adaptive finite element methods (AFEM) are the methods of choice for discretizing a number of PDE eigenvalue problems. In this framework, the grid refinement procedure is based on bounds for the discretization errors that are derived from the PDE. The resulting discretized system is usually solved with the help of iterative algebraic solvers, which admit a certain error tolerance. The so-called algebraic error introduced by the iterative solvers is typically not taken into account in the grid refinement procedure.

In this talk, we consider a Maxwell eigenvalue problem arising from the simulation of light waves in photonic crystals. We derive error bounds for the discretization errors, as well as for the algebraic errors, and discuss how balancing both types of errors can decrease the overall cost of the AFEM.

### Arbi Moses Badlyan (TU Berlin)

Donnerstag, 18. Januar 2018

**Metriplectic Systems**

Metriplectic systems are state space formulations that have become famous under the acronym GENERIC (General Equation for the Non-Equilibrium Reversible Irreversible Coupling). GENERIC is supplemented by complementary non-interacting conditions and as unifying framework combines a Hamiltonian with a special kind of gradient system such that the invariants of the Hamiltonian flow are preserved.

In this talk I present the mathematical framework of GENERIC/metriplectic systems and by means of examples of physical phenomena (e.g. heat conduction) modeled as metriplectic systems lay the ground for the upcoming talk of my collegue Christoph Zimmer (TU Berlin) who will present the results of our joint work.

### David Noben (TU Berlin)

Donnerstag, 18. Januar 2018

**A successive linear programming approach for the Adaptive Model and Discretization Control Algorithm**

In the stationary gas transport optimization, pressure change is described by the Euler equation for compressible fluids. To save computational costs, the usual approach is to use simplified models of this equation. This can have a large impact on the solutions accuracy and computational speed. The aim of the Adaptive Model and Discretization Control by Mehrmann et al. (2017) is to keep computational effort acceptable while using also higher level model descriptions.

The talk focuses on the constrained nonlinear programs (NLP) which have to be solved during this algorithm. A technique called, successive linear programming’ is introduced. Here we compute the solution to an NLP by solving a sequence of linear programs. Further we want to illustrate the work of the algorithm on a gas network instance and compare its result to different approaches.

### Christoph Zimmer (TU Berlin)

Donnerstag, 25. Januar 2018

**On an Operator-GENERIC Formulation for Mixtures of Reactive Fluids**

Mathematical models of reactive fluid mixtures play an important role in industrial applications like in the simulation of energy technologies as for example simulation of hydrogen fuel cells. Good descriptions of dynamical physical phenomena should not only describe the dynamics but also their structure should enforce important physical laws explicitly. Under structure-preserving discretization these laws are then also satisfied for the simulation results. In the first part of this talk, we shortly derive field equations which cover the dynamics of a mixture of compressible Newtonian fluids consisting of reactive constituents. The required closure relations are based on the phenomenological theory of classical irreversible thermodynamics. In the second part, we consider metriplectic systems also known under the acronym GENERIC (General Equations for the Non-Equilibrium Reversible Irreversible Coupling). The geometric framework behind metriplectic systems was briefly presented by Arbi Moses Badlyan in the last seminar. We introduce a GENERIC based formulation in an operator setting that encodes the weak formulation of the field equations of the fluid mixture. Amongst others, we will show that the conservation of total energy and the second law of thermodynamics for isolated systems can simply be derived from the structure of the formulation.

This is a joint work with Arbi Moses Badlyan.

### Daniel Bankmann (TU Berlin)

Donnerstag, 01. Februar 2018

**On error estimation of a nonlinear least squares type bilevel optimal control problem**

Optimal control tasks arise in a variety of applications from, e.g., mechanical or electrical engineering, where one wants to minimize a certain cost functional with respect to some input function and the resulting state trajectory. Usually, the systems describing the dynamical behavior of these applications are governed by differential equations. Furthermore, in certain applications, e.g. humanoid locomotion, the cost functional descriptions may depend on parameters.

The goal then is to find a 'good' set of parameters such that the trajectory generated by the corresponding optimal control problem is close to some reference trajectory. This is a nonlinear least squares problem and introduces an upper level of optimization. It can be solved numerically using Gauß-Newton type methods. However, one needs to recompute the solution of the inner optimal control problem at every step. Thus, one can save computation effort if one has control over the error in the inner solution.

In this talk we will revisit results on error estimation for Gauß-Newton methods. Then, we will apply these result to the structure of our problem and derive an error estimator for the inner problem.

### Sofia Bikopoulou (TU Berlin)

Donnerstag, 01. Februar 2018

**An Algorithm-Based Fault Tolerance Approach for Solving Large Scale Linear Systems**

High Performance Computing (HPC) systems are widely used for industrial and academic purposes. The high complexity of the hardware components of such large multicore structures leads to the vulnerability of the system, so that supercomputers can experience unexpected failures throughout the execution of applications.

In this talk I will present a technique for detecting and correcting soft errors when solving large scale linear systems on HPC platforms. The way of achieving fault resilience is by using Algorithm-Based Fault Tolerance with checksums, in combination with checkpointing and rollback recovery protocols. The method detects multiple soft errors and corrects up to two soft errors that occur due to incorrect numerical computations on a large-scale platform.

### Sarah Hauschild (TU Berlin)

Donnerstag, 08. Februar 2018

**Model Order Reduction for port-Hamiltonian Differential-Algebraic Equations**

Port-Hamiltonian systems are often used to model large physical systems, which lead to large Systems after discretization. Therefore, model reduction methods are needed. Furthermore, these methods have to be structure preserving to keep all the nice properties of pH-systems.

In my talk I am going to summarize some structure preserving model reduction methods for ordinary port-Hamiltonian systems, like Moment Matching and power conservation based methods. Furthermore, it will be shown how they can be applied to port-Hamiltonian DAEs and some numerical examples will be given.

### Hannes Gernandt (TU Ilmenau)

Donnerstag, 15. Februar 2018

**On the gap distance between matrix pencils**

It is well known that the solutions of linear time-invariant differential algebraic equations (DAEs) can be given in terms of the solution of an eigenvalue problem for matrix pencils. Therefore, robustness of the DAE is intimately related to the robustness of the underlying pencil. In particular, for matrix pencils which are in some sense close to a singular pencil, the numerical solution of the DAE is no longer possible.

In this talk, we use the gap metric, a classic distance for linear operators, to characterize the distance between matrix pencils. For a given regular matrix pencil, the gap distance to the closest singular pencil is bounded from above and below. Finally, we compare the gap distance with the distance to singularity introduced by Byers, He and Mehrmann.

The talk is based on a joint work with Thomas Berger, Carsten Trunk, Henrik Winkler and Michał Wojtylak.