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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 |

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 21.04. | 10:15 Uhr | MA 376 | Vorbesprechung | |

Do 28.04. | 10:15 Uhr | MA 376 | - kein Seminar - | |

Do 05.05. | 10:15 Uhr | MA 376 | - kein Seminar - | |

Do 12.05. | 10:15 Uhr | MA 376 | Matthew Salewski | Equivariance and reduced-order modelling [abstract] |

Volker Mehrmann | Newton-Type Methods for Nonlinear Eigenvalue Problems Arising in Photonic Crystal Modeling [abstract] | |||

Do 19.05. | 10:15 Uhr | MA 376 | Christian Schröder | Convergence analysis of the simplified inexact Jacobi-Davidson-algorithm to compute eigenpairs of a Hermitian matrix [abstract] |

Lena Scholz | The Sigma-method for electrical circuit equations [abstract] | |||

Do 26.05. | 10:15 Uhr | MA 376 | Viola Paschke | Numerical solution of delayed multibody systems [abstract] |

Luka Grubišić | Mixed finite element formulations for pde problems on graphs [abstract] | |||

Do 02.06. | 10:15 Uhr | MA 376 | Stephan Rave | Model Order Reduction for Electrochemistry Simulations [abstract] |

Benjamin Unger | Efficient simulations via model reduction enhanced with operator splitting [abstract] | |||

Do 09.06. | 10:15 Uhr | MA 376 | Arbi Moses Badlyan | On the Port-Hamiltonian Structure of the Navier-Stokes Equations for Reactive Flows – A General Consideration [abstract] |

Volker Mehrmann | On computing the distance to the nearest singular pencil [abstract] | |||

Do 16.06. | 10:15 Uhr | MA 376 | Jeroen Stolwijk | Perturbation Analysis for the Isothermal Semilinear Euler Equations [abstract] |

Punit Sharma | Structure preserving perturbations to port-Hamiltonian system [abstract] | |||

Do 23.06. | 10:15 Uhr | MA 376 | Melina Simichanidou | Discontinuous Galerkin Finite Element Methods for the Reactive Euler Equations [abstract] |

Philipp Schulze | The shifted proper orthogonal decomposition [abstract] | |||

Do 30.06. | 10:15 Uhr | MA 376 | Igor Pontes Duff Pereira | H2-optimal model approximation by structured time-delay reduced order models [abstract] |

Michael Dellnitz | Set Oriented Numerical Methods for Dynamical Systems and Optimization Problems [abstract] | |||

Do 07.07. | 10:15 Uhr | MA 376 | Marine Froidevaux | Towards a spectral analysis of photonic crystals [abstract] |

Andres Gonzalez Zumba | Energy Based Modeling and Stability Analysis of Multi-Physical Systems with DAEs, Applied to Electric Power Systems [abstract] | |||

Do 14.07. | 10:15 Uhr | MA 376 | Christoph Zimmer | Model approximation with multiplicative isometric operators in H_2 <link177389#751141>[abstract] |

Sofia Bikopoulou | Fault Tolerant Methods for Solving Linear Systems in High-Performance Computing [abstract] | |||

Do 21.07. | 10:15 Uhr | MA 376 | - kein Seminar - | |

Do 28.07. | 10:15 Uhr | MA 376 | Tobias Breiten | Control strategies for the Fokker-Planck equation [abstract] |

Kevin Carlberg | Nonlinear model reduction: discrete optimality and time parallelism [abstract] | |||

### Matthew Salewski (TU Berlin)

Donnerstag, 12. Mai 2016

**Equivariance and reduced-order modelling**

The construction of reduced-order models from a dynamical system can be enhanced when one uses properties of the system, such as the equivariance of the system under the action of a Lie group. Using the equivariance allows the dynamics to be reduced to a subspace where the action of the group has been removed. This effect can be advantageous when applied to systems of transport-dominated phenomena where the transport can be neutralized, resulting dynamics which may be more accurately modelled. Here, I discuss a protocol for constructing reduced-order models using equivariance, and demonstrate this protocol with equivariant systems with transport-dominated phenomena. In addition, I will comment on systems whose equivariance is not explicitly clear and show some approaches used to deal with this when constructing a model.

### Volker Mehrmann (TU Berlin)

Donnerstag, 12. Mai 2016

**Newton-Type Methods for Nonlinear Eigenvalue Problems Arising in Photonic Crystal Modeling **

The numerical simulation of the band structure of three-dimensional dispersive metallic photonic crystals with face-centered cubic lattices leads to large-scale nonlinear eigenvalue problems, which are very challenging due to a high dimensional subspace associated with the eigenvalue zero and the fact that the desired eigenvalues (with smallest real part) cluster and close to the zero eigenvalues. For the solution of the nonlinear eigenvalue problem, a Newton-type iterative method is proposed and the nullspace-free method is applied to exclude the zero eigenvalues from the associated generalized eigenvalue problem. To find the successive eigenvalue/eigenvector pairs, we propose a new non-equivalence deflation method to transform converged eigenvalues to infinity, while all other eigenvalues remain unchanged. The deflated problem is then solved by the same Newton-type method, which is used as a hybrid method that combines with the Jacobi-Davidson and the nonlinear Arnoldi methods to compute the clustering eigenvalues. Numerical results illustrate that the proposed method is robust even for the case of computing many clustering eigenvalues in very large problems. This talk describes joint work with T.-M. Huang and W.-W. Lin.

### Christian Schröder (TU Berlin)

Donnerstag, 19. November 2015

**Convergence analysis of the simplified inexact Jacobi-Davidson-algorithm to compute eigenpairs of a Hermitian matrix**

The Jacobi-Davidson method is a variant of Newton's method to compute eigenpairs of a matrix. In the talk I will present a convergence analysis of a simplified version of this algorithm for the case of a Hermitian matrix. In exact arithmetic the method converges cubically. I will consider the case of inexact arithmetic and prove quadratic convergence under weak conditions, and cubic convergence if the inexactness is small enough. The talk is based on an extension of a recent paper by Zhong-Zhi Bai and Cun-Qiang Miao.

### Lena Scholz (TU Berlin)

Donnerstag, 19. Mai 2016

**The Sigma-method for electrical circuit equations**

The dynamical behavior of physical processes is often modeled by differential-algebraic equations (DAEs). Here, often one has to deal with large-scale problems that can be of higher index. In order to be able to apply numerical integration methods a regularization or remodeling of the model equation is required. In many simualtion packages such as Dymola, SimulationX or MapleSim this regularization is performed based on a structural analysis of the equations. In this talk we will consider the Sigma-method of Pryce and its use for the model equations of electrical circuits. We will present some first preliminary results and discuss some open problems.

### Viola Paschke (TU Berlin)

Donnerstag, 26. Mai 2016

**Numerical solution of delayed multibody systems **

In this talk I will present some results from my master thesis. Time-delayed multibody systems (DMBSs) arise in mechanical engineering when multibody systems (MBSs) are subject to delayed feedback. We investigate the numerical solution of causal DMBSs. Based on the method of steps and the solver QUALIDAES we present DelayedMBS, a solver for causal DMBSs. For illustration we apply the solver to several examples. In particular, we consider the simple pendulum on a moving cart modeling a gantry crane system, and show effects of delayed feedback control.

### Luka Grubišić (University of Zagreb)

Donnerstag, 26. Mai 2016

**Mixed finite element formulations for pde problems on graphs**

In this talk we will review standard mixed finite element formulations for the eigenvalue as well as the evolution problem. In particular we will illustrate techniques for establishing convergence rates for discretizations. We will then present a mixed formulation for a finite element model of an endovascular stent. We will establish convergence rates for discretized problems and present numerical experiments on geometries of several commercial stents.

### Stephan Rave (University of Münster)

Donnerstag, 02. Juni 2016

**Model Order Reduction for Electrochemistry Simulations**

Goal of the MULTIBAT research project is to gain a better understanding of degradation phenomena in rechargeable lithium-ion batteries through mathematical modeling and simulation at the micrometer scale. Since the discrete models arising from this workflow require large computational resources for their solution, model order reduction is required in order to allow parameter studies of these models.

In this talk I will give a very short introduction to the reduced basis method, a generic model order reduction scheme applicable to wide ranges of parametrized partial differential equations, and will then present our recent work on applying reduced basis techniques to microscale battery models. I will also discuss the question of integrating model reduction into existing simulation workflows and present a simple data compression algorithm for approximation space computation under limited resources.

### Benjamin Unger (TU Berlin)

Donnerstag, 02. Juni 2016

**Efficient simulations via model reduction enhanced with operator splitting**

The mathematical description of physical phenomenon is usually given in form of a (partial) differential equation. The phenomenon itself is often an accumulation of several different processes. In terms of the mathematical model this means that the time derivative of the physical quantity under investigation depends on a combination of multiple sub-operators, which are usually of different nature. The different nature of the sub-operators conspires against effective numerical methods for the phenomenon itself. In case of model order reduction, transport and diffusion are such *conflicting* processes. If the combination of the sub-operators is given by a sum then operator splitting techniques are a promising remedy. In this talk I apply the sequential (Gudonov) splitting and the Strang splitting to two sample problems and show how model reduction can benefit from the splitting approach. If time permits I comment on symmetry reduction within the presented framework.

### Arbi Moses Badlyan (TU Berlin)

Donnerstag, 09. Juni 2016

**On the Port-Hamiltonian Structure of the Navier-Stokes Equations for Reactive Flows – A General Consideration**

Thermodynamics of fluids is a field theory with the objective to determine a set of fields which represent the state of the thermodynamic system. These fields are the solutions of partial-differential equations (PDEs), also known as field equations. The PDEs are the mathematical model of the thermodynamic system and are based on the equations of balance of mechanics and thermodynamics, viz. the conservation laws of mass and momentum and the equation of balance of (internal) energy.

In this talk I will present some results of a (yet unfinished) joint work with Christoph Zimmer (TU Berlin) on a general port-Hamiltonian formulation for homogeneous mixtures of viscous heat-conducting fluids in which a finite number of independent chemical reactions may occur. I will discuss three different mathematical models, each of them given by a set of balance equations describing the dynamics of the homogeneous mixtures on different levels of complexity. I will show that the weak formulations of the balance equations with respect to one concrete model and vanishing boundary conditions can be written as a dynamically equivalent Hamiltonian system. For this I will use the balance equations without specifying any constitutive relations for the stress-tensor, the heat and the diffusion flux vector. Finally, I will briefly explain how and under which assumptions linear constitutive relations can be derived and how more realistic non-vanishing boundary terms can be incorporated resulting in a port-Hamiltonian system.

Our joint work is a generalization and extension of the previous work of Phillip Schulze (TU Berlin) and Robert Altmann (TU Berlin) on the Port-Hamiltonian Structure of the Navier-Stokes Equations for Reactive Flows.

### Volker Mehrmann (TU Berlin)

Donnerstag, 09. Juni 2016

**On computing the distance to the nearest singular pencil**

Given a regular matrix pencil A + \mu E, we consider the problem of determining the nearest singular matrix pencil with respect to the Frobenius norm. We present new approaches based on the solution of matrix differential equations for determining the nearest singular pencil A + \Delta A +\mu( E + \Delta E), one approach for general singular pencils and another one such that A+\Delta A and E+\Delta E have a common left/right null vector.

For the latter case the nearest singular pencil is shown to differ from the

original pencil by rank-one matrices \Delta A and \Delta E. In both cases we consider also the situation where only A is perturbed. The nearest singular pencil is approached by a two-level iteration, where a gradient flow is driven to a stationary point in the inner iteration and the outer level uses a fast iteration for the distance parameter. This approach extends also to structured pencils.

Joint work with Nicola Guglielmi and Christian Lubich.

### Jeroen Stolwijk (TU Berlin)

Donnerstag, 16. Juni 2016

**Perturbation Analysis for the Isothermal Semilinear Euler Equations**

Natural gas plays a crucial role in the energy supply of the world. After oil, it is the second most used energy supplier in Germany. The high and probably increasing demand for natural gas calls for an accurate, efficient and robust mathematical modeling, simulation and optimisation of the gas transport through the existing pipeline network.

The gas flow through a pipeline is modeled by the one-dimensional Euler equations, which can be simplified to an isothermal semilinear model. For the simulation and optimisation of large pipeline networks, simple finite difference schemes are typically used. We discuss two of these schemes in this talk, which lead to two systems of nonlinear equations. A perturbation analysis is performed for these two systems.

In the last talk I gave in this seminar, I discussed two types of condition numbers: normwise in both the output and input parameters as well as componentwise in the output parameters and normwise in the input parameters. In this talk, I will derive a condition number for nonlinear systems, which is componentwise in both the output and the input parameters. Surprisingly, this condition number has never been formulated in the literature before. It is shown that the componentwise condition number constitutes a significantly more accurate measure for the sensitivity of nonlinear systems than the normwise condition number.

This is joint work with V. Mehrmann. It is supported by the German Research Foundation DFG in the Collaborative Research Centre TRR 154, subproject B03.

### Punit Sharma (TU Berlin)

Donnerstag, 16. Juni 2016

**Structure preserving perturbations to port-Hamiltonian system**

Port-Hamiltonian (PH) systems are an important concept in energy based modeling of dynamical systems. Making use of the structure, they satisfy two important properties for control systems: passivity and stability. It is important to know that when a linear constant coefficient PH system is on the boundary of the region of asymptotic stability, i.e. when it has purely imaginary eigenvalues, or how much it has to be perturbed to be on this boundary. For unstructured systems this distance to instability (stability radius) is well-understood.

Similarly, it is also important to know that when a linear constant coefficient PH system is on the boundary of the region of strict passivity. This can be addressed by computing the norm of smallest structured perturbation that makes system to lose strict passivity. For unstructured systems this distance to passivity (passivity radius) is well-understood.

In the first half of the talk, I will present some structured eigenpair backward errors of matrix pencils that are important to check the passivity of port-Hamiltonian systems with respect to structure preserving perturbations. In the later half, I will briefly talk about the real stability radius for port-Hamiltonian systems and compare it with its complex counter part.

This is a joint work with Christian Mehl and Volker Mehrmann.

### Melina Simichanidou (TU Berlin)

Donnerstag, 23. Juni 2016

**Discontinuous Galerkin Finite Element Methods for the Reactive Euler Equations**

In this talk I will present some of the work regarding my Master Thesis. The nonreactive Euler equations of gas dynamics are nonlinear conservation laws of mass, momentum and energy. The aim of my work is to extend this set of equations with a reaction equation yielding the complete set of reactive Euler equations. In order to solve this enriched system the framework of the Discontinuous Galerkin Finite Element Method will be used. I will illustrate the behavior of the method using the so-called ZND model with the one-step chemical reaction A->B.

### Philipp Schulze (TU Berlin)

Donnerstag, 23. Juni 2016

**The shifted proper orthogonal decomposition**

Transport-dominated phenomena occur in numerous applications, for instance in combustion engines. For efficient model-based control and optimization of these systems, models are needed which can be evaluated very fast. However, transport-dominated phenomena are a major challenge for common model reduction techniques, as the proper orthogonal decomposition (POD).

In this talk we introduce the shifted proper orthogonal decomposition (sPOD) which is an extension of the POD by making use of co-moving frames. This allows the sPOD to describe even transport-dominated phenomena with a small number of modes. Some features and results are presented and open questions and challenges are discussed.

### Igor Pontes Duff Pereira (Onera)

Donnerstag, 30. Juni 2016

**H2-optimal model approximation by structured time-delay reduced order models**

In the engineering area (e.g. aerospace, automotive, biology, circuits), dynamical systems are the basic framework used for modeling, controlling and analyzing a large variety of systems and phenomena. Due to the increasing use of dedicated computer-based modeling design software, numerical simulation turns to be more and more used to simulate a complex system or phenomenon and shorten both development time and cost. However, the need of an enhanced model accuracy inevitably leads to an increasing number of variables and resources to manage at the price of a high numerical cost. This counterpart is the justification for model reduction.

For linear time-invariant systems, several model reduction approaches have been effectively developed since the 60's. Among these, interpolation-based methods stand out due to their flexibility and low computational cost, making them a predestined candidate in the reduction of truly large-scale systems. Recent advances demonstrate ways to find reduction parameters that locally minimize the H2-norm of the error system.

### Michael Dellnitz (Universität Paderborn)

Donnerstag, 30. Juni 2016

**Set Oriented Numerical Methods for Dynamical System and Optimization Problems**

Over the last two decades so-called *set oriented* numerical methods have been developed in the context of the numerical treatment of dynamical systems. The basic idea is to cover the objects of interest - for instance *invariant sets* or *invariant measures* - by outer approximations which are created via multilevel subdivision techniques. At the beginning of this century these methods have been modified in such a way that they are also applicable to the numerical treatment of *multiobjective optimization problems*. Due to the fact that they are set oriented in nature these techniques allow for the direct computation of the entire so-called *Pareto set*.

In this talk recent developments in the area of set oriented numerics will be presented both for dynamical systems and optimization problems. The reliability of these methods will be demonstrated by several applications such as the approximation of transport processes in ocean dynamics, or the optimization of a cruise control with respect to energy consumption and travel distance. Moreover a new algorithmic approach will be described which allows to extend these techniques to the context of infinite dimensional dynamical systems.

### Marine Froidevaux (TU Berlin)

Donnerstag, 07. Juli 2016

**Towards a spectral analysis of photonic crystals **

A photonic crystal is a periodic material that affects the propagation of electromagnetic waves. By adjusting its dielectric parameter and its dimensions, it is possible to allow light at certain frequencies to propagate through the crystal, while other frequencies in a certain range are refrained from travelling through. This range of forbidden frequencies is usually called band gap. Being able to tune the crystal’s parameters so as to obtain an optimal band gap is essential for applications such as optical wave guides, filters or optical transistors.

In this talk, we are concerned with deriving an adequate eigenvalue problem from the Maxwell’s equations in order to perform the spectral analysis of parameter-dependant photonic crystals. In particular, we will discuss open problems arising when considering a 3-dimensional crystal.

### Andres Gonzalez Zumba (TU Berlin)

Donnerstag, 21. Januar 2016

**Energy Based Modeling and Stability Analysis of Multi-Physical Systems with DAEs, Applied to Electric Power Systems**

Multi-physical systems are commonly modeled using Differential Equations. But, considering the different time-scales in the dynamics of their subsystems (stiffness) and their physical restrictions, these are rather conveniently modeled as a combination of ordinary differential equations with algebraic constraints known as Differential-Algebraic Equations (DAE). In other hand, the theory of Port-Hamiltonian systems (PH) provides a new geometric framework based on the combination of Hamiltonian mechanics and networks theories. PH approach formalizes the basic interconnection laws together with the power-conserving elements by a geometric (interconnection) structure, and defines the Hamiltonian function as the total energy stored in a system. This gives a direct physical interpretation since the physical balance equations are directly derived from the interconnection structure and the physical energy of the system.

Considering the previous statements, the work which is being performed in the present project is the study of the key aspects of Port-Hamiltonian Differential-Algebraic Systems (PHDAE). As well as, the examination and discussion of strategies for the dynamic stability analysis of the systems in this formulation. This new approach is being applied in the modeling and stability analysis of Electric Power Systems (EPS), over all oriented to the study of the Swing Equation, the main mathematical model which describes the dynamical behavior of synchronous machines.

### Christoph Zimmer (TU Berlin)

Donnerstag, 14. Juli 2016

**Model approximation with multiplicative isometric operators in H_2 **

The mathematical description of physical, chemical, or biological processes often leads to control problems with a large number of degrees of freedom and therefore to systems with high computational costs. The aim of model order reduction is on the one hand to lower the computational time as well as the complexity of the model, and on the other hand to maintain the input-output-behavior. One approach for this task is to search for a linear time-invariant system (LTI-system) with a minimal error in the Hardy-space H_2.

In this talk we will investigate the approximation behavior of LTI-systems with additional multiplicative isometric operators of H_2, e.g., the operators of input-output-delays in the time domain. The advantage of this approach is that it maintains the energy of impulse responses of the reduced order model and leads to a bigger class of possible models for the approximation at the same time. For this class we will derive a representation of the error, calculate optimality conditions and show numerical examples with parameter dependent operators.

This is a joint work with Igor Pontes Duff Pereira.

### Sofia Bikopoulou (TU Berlin)

Donnerstag, 14. Juli 2016

**Fault Tolerant Methods for Solving Linear Systems in High-Performance Computing**

High-Performance Computing (HPC) systems were initially concerned with executing code on parallel and distributed architectures, providing unique opportunities to industry and academia. Heavy demands were placed on systems by many simultaneous requests. The advent of exascale computers has shifted the focus on efficient computational models able to support fault tolerant operations. The challenge of the last decades is to build systems that are both inexpensive and highly reliable. Given the current state-of-the-practice, fewer errors are introduced, but not all errors are prevented.

In this talk I will present recent techniques for tolerating faults when solving linear systems on large-scale platforms. Our priority is to deliver acceptable level of service and to enable the system's continued operation, even though faults are present. The aforementioned techniques will be applied to the well-known algorithm GMRES.

### Tobias Breiten (Karl-Franzens-Universität Graz)

Donnerstag, 28. Juli 2016

**Control strategies for the Fokker-Planck equation**

The probability distribution function of a dragged Brownian particle can be characterized by the Fokker-Planck equation. By means of an optical tweezer, interaction with the particle is possible and leads to a bilinear control system. It is known that the uncontrolled system converges to the stationary distribution. However, depending on the parameters of the system, this convergence can be inadequately slow. Projection-based decoupling of the Fokker-Planck equation allows to design certain feedback control laws that locally increase the rate of convergence to the stationary distribution. Different strategies based on Lyapunov and projected Riccati equations are presented and the performance is studied by means of local and global Lyapunov functions. Numerical examples are shown to verify the theoretical findings.

### Kevin Carlberg (Sandia)

Donnerstag, 28. Juli 2016

**Nonlinear model reduction: discrete optimality and time parallelism**

Large-scale models of nonlinear dynamical systems arise in applications ranging from compressible fluid dynamics to structural dynamics. Due to the large computational cost incurred by these models, it is impractical to use them in time-critical scenarios such as control, design, uncertainty quantification. Model reduction aims to mitigate this computational burden. To date, reduced-order models (ROMs) for relatively simple models (e.g., linear-time-invariant systems; elliptic, parabolic, and linear hyperbolic PDEs) have been widely adopted, as researchers have developed methods that are accurate, reliable, and certified. In contrast, model reduction for nonlinear dynamical systems lacks these assurances and thus remains in its infancy; the most common method - POD-Galerkin - is often unstable.

This talk will describe several advances that have made nonlinear model reduction viable for a new frontier of problems. However, doing so has required fundamentally new data-driven approaches, as the critical tools leveraged for simpler models (e.g., Gramians, coercivity constants) are no longer available. First, I will introduce the notion of least-squares Petrov–Galerkin (LSPG) projection—and the associated GNAT method—which enables accuracy via discrete optimality: it performs optimal projection after the dynamical system has been discretized in time. Comparative theoretical and numerical studies will highlight the benefits of LSPG projection over Galerkin projection.

Second, I will introduce a new approach for data-driven time parallelism. Because the GNAT model incurs a small computational footprint, parallelizing the computation in the spatial domain quickly saturates; this limits the realizable wall-time speedup for the GNAT ROM. To address this, we introduce a new method for parallelizing the simulation in time. The technique relies on a coarse propagator that ensures rapid convergence by leveraging previously available time-domain data.