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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 15.10. | 10:15 Uhr | MA 376 | Vorbesprechung | |

Do 22.10. | 10:15 Uhr | MA 376 | Christian Schröder | Model reduction for linear time invariant boundary value problems[abstract] |

Andreas Steinbrecher | Numerical Integration of DAEs Based on the Structural Analysis [abstract] | |||

Do 29.10. | 10:15 Uhr | MA 376 | - kein Seminar - | |

Do 05.11. | 10:15 Uhr | MA 376 | - kein Seminar - | |

Do 12.11. | 10:15 Uhr | MA 376 | Daniel Bankmann | On Linear-Quadratic Control Theory of Implicit Difference Equations [abstract] |

Benjamin Unger | On the classification of delay differential-algebraic equations [abstract] | |||

Do 19.11. | 10:15 Uhr | MA 376 | Robert Altmann | Preservation of Operator Properties for the Time Discretization of ODEs [abstract] |

Michael Götte | On the MATLAB toolbox MPSSim [abstract] | |||

Do 26.11. | 10:15 Uhr | MA 376 | Christian Mehl | Hamiltonian Jacobi methods [abstract] |

Andres Gonzalez Zumba | Modeling and Stability Analysis of Stochastic Multi-Physical Systems Applied to Energy Systems [abstract] | |||

Do 03.12. | 10:15 Uhr | MA 376 | Viola Paschke | Multibody systems including time delay [abstract] |

Sofia Bikopoulou | Generalized eigenvalue problem [abstract] | |||

Do 10.12. | 10:15 Uhr | MA 376 | Philipp Schulze | On the Port-Hamiltonian Structure of the Navier-Stokes Equations for Reactive Flows [abstract] |

Juraj Mihalik | Design of cam profiles using optimal control methods [abstract] | |||

Do 17.12. | 10:15 Uhr | MA 376 | Jeroen Stolwijk | Sensitivity Analysis for Nonlinear Rootfinding Problems [abstract] |

Do 07.01. | 10:15 Uhr | MA 376 | Matthew Salewski | Symmetries in Model Reduction [abstract] |

Do 14.01. | 10:15 Uhr | MA 376 | Christoph Zimmer | Theory and Numerical Analysis of Linear Operator Differential-Algebraic Equations with Delay [abstract] |

Volker Mehrmann | The sign-characteristic of structured matrix polynomials [abstract] | |||

Do 21.01. | 10:15 Uhr | MA 376 | Matthias Voigt | Balanced Truncation Model Reduction of Boundary Value Problems with Application to Data Assimilation [abstract] |

Punit Sharma | Structured distance to instability for port-Hamiltonian systems [abstract] | |||

Do 28.01. | 10:15 Uhr | MA 376 | Marine Froidevaux | A structure preserving first order formulation for quadratic eigenvalue problems [abstract] |

Martin Schramm | A 2-by-2 Jacobi Algorithm for Complex Conjugated Hamiltonian Matrices [abstract] | |||

Do 04.02. | 10:15 Uhr | MA 376 | Christoph Conrads | Projection Methods for Generalized Eigenvalue Problems [abstract] |

Arbi Moses Badlyan | Thermal Relaxation in Geometrically Frustrated Magnets (Spin Ice) [abstract] | |||

Do 11.02. | 10:15 Uhr | MA 376 | - kein Seminar - | |

### Christian Schröder (TU Berlin)

Donnerstag, 22. Oktober 2015

**Model reduction for linear time invariant boundary value problems**

I will present an (unfinished) idea for the dimension reduction of an discrete LTI system with boundary value constraints, as they arise in our current ECMath/Matheon project in the context of data assimilation in weather prediction. Also the results of first numerical experiments will be shown.

### Andreas Steinbrecher (TU Berlin)

Donnerstag, 22. Oktober 2015

**Numerical Integration of DAEs Based on the Structural Analysis**

The complete virtual design of dynamical systems, e.g., mechanical systems, electrical circuits, flow problems, or whole production processes, plays a key role in our technological progress. The modeling of dynamical processes often leads to systems of differential-algebraic equations (DAEs) of the form

(1) E(x(t),t)x'(t)=k(t,x(t))

where x are the unknown variables.

In general, the direct numerical integration leads to instabilities, non-convergence, or an order reduction of the numerical methods. These difficulties in the numerical integration occur due to so-called hidden constraints which are contained in the DAE but not explicitly stated as equations. Therefore, a regularization or remodeling of the model equations is required which preserves the set of solutions and explicitly contains all formerly hidden constraints. In modern simulation environments often a structural analysis is used to obtain required information for a suitable regularization.

In this talk we discuss the efficient and robust numerical integration of DAEs (1) of high index. First, we present an approach for the regularization of DAEs that is based on the Signature method of Pryce. Such a regularization may be valid only locally since the state selection can vary with the dynamical behavior of the system. The obtained regularization can then be solved piecewisely using stiff ODE-solvers. To avoid this varying state selection, secondly, we will propose an approach which also uses the information obtained from the Signature method to construct a regularized overdetermined formulation.

Based on that regularization approaches we present the software package QUALIDAES, which is suited for the direct numerical integration of both of the proposed regularizations.

### Daniel Bankmann (TU Berlin)

Donnerstag, 12. November 2015

**On Linear-Quadratic Control Theory of Implicit Difference Equations**

In this talk I will present some results of my master thesis. We will talk about a discrete time version of the Kalman-Yakubovich-Popov inequality, Lur'e equations and some application to optimal control. Furthermore, we will compare these results to the continuous time case and see what discretizations are involved.

### Benjamin Unger (TU Berlin)

Donnerstag, 12. November 2015

**On the classification of delay differential-algebraic equations **

The analysis of the quality of a numerical algorithm for differential equations is often based on a Taylor series expansion of the solution. Hence, a rigorous study of the smoothness properties of the solution is required, which for delay differential equations (DDE) has led to three different types: retarded, neutral, and advanced. Recently, Ha and Mehrmann (2015) proposed a similar characterization for delay differential-algebraic equations (DDAEs), which is suitable for numerical algorithms. However, the types obtained by this classification do not reflect the smoothness properties of the solution. In this talk I address this issue by some examples, propose a new classification and show its relation to previous works.

### Robert Altmann (TU Berlin)

Donnerstag, 19. November 2015

**Preservation of Operator Properties for the Time Discretization of ODEs**

Following the tradition of Christian and Benny, I will present some unfinished work. As structure preserving time integration methods try to preserve geometric properties of the system, we study integration schemes which preserve properties of the corresponding operator equation. For this, we try to formulate the ODE (including the initial condition) as a self-conjugate operator and analyse how the discretization can preserve that property.

### Michael Götte (TU Berlin)

Donnerstag, 19. November 2015

**On the MATLAB toolbox MPSSim**

We implemented a MATLAB toolbox for the numerical treatment of DAES, including structural index reduction, equation enhancement and simulation. The simulation is based on overdetermined formulations suitable for our solvers QUALIDAES and OVDBDF. The aim of this talk is to give a practical insight to the toolbox in form of a live presentation by simulating some examples.

### Christian Mehl (TU Berlin)

Donnerstag, 26. November 2015

**Hamiltonian Jacobi methods**

Jacobi's method for the diagonalization of a symmetric matrix is a famous, successful, and easy to implement algorithm for the computation of eigenvalues of symmetric matrices. No wonder that this algorithm has been generalized or adapted to many other classes of eigenvalue problems. In particular, Hamiltonian Jacobi methods for the Hamiltonian eigenvalue problem have been investigated by Byers (generalizing a nonsymmetric Jacobi method proposed by Stewart) and Bunse-Gerstner/Faßbender (generalizing a nonsymmetric Jacobi method proposed by Greenstadt and Eberlein).

Surprisingly, the Hamiltonian Jacobi method proposed by Bunse-Gerstner/Faßbender only shows linear asymptotic convergence despite the fact that asymptotic quadratic convergence can be observed by the Greenstadt/Eberlein Jacobi method. In this talk, we give an explanation for this unexpected behavior and suggest a different strategy to tackle Hamiltonian eigenvalue problems via nonsymmetric Jacobi methods.

### Andres Gonzalez (Universidad Politecnica de Valencia)

Donnerstag, 26. November 2015

**Modeling and Stability Analysis of Stochastic Multi-Physical Systems Applied to Energy Systems **

Deterministic approach has been traditionally used for modeling, simulation and stability analysis of multi-physical systems in general. Nowadays, in the engineering area almost all stability analysis procedures are performed under this view. However, the deterministic analysis does not provide a realistic assessment of the system performance, especially when important variables with uncertainty should be taken into account. If we also consider other characteristics of these systems such as their highly nonlinear nature and large size, the picture becomes even more complex. Based on the modeling by the use of Stochastic Differential-Algebraic Equations (SDAE), my present research work has as a main objective the study, characterization, classification, and evaluation of the most important direct and indirect stability analysis methods which can be applied to the large-scale energy systems assessment considering the effects of random variables and uncertainties, e.g. non-dispatchable generators, electric loads, electricity prices, faults, etc. This work try to suggest new alternatives about Advanced Stability Concepts oriented to planning, market rules, policies, and operation areas in the energy sector.

### Viola Paschke (TU Berlin)

Donnerstag, 03. Dezember 2015

**Multibody systems including time delay**

In this talk I will consider multibody systems including time delay (DMBSs). We model the DMBS as a delay differential-algebraic equation (DDAE), and involve delay in the dynamical part as well as in the constraints of the system. Delayed multibody systems arise when applying delayed feedback to multibody systems (MBS). We show this by means of the gantry crane system, which can be modeled by the mathematical pendulum attached to a moving trolley. Since the DMBS is a DDAE, difficulties as the noncausality and advancedness may arise. Based on results from the theory of MBS we achieve sufficient conditions for the causality of the DMBS. Furthermore, we determine the type of smoothness propagation of a causal DMBS.

### Sofia Bikopoulou (TU Berlin)

Donnerstag, 03. Dezember 2015

**Generalized eigenvalue problem**

Efficient and accurate methods for calculating eigenvalues and eigenvectors of random matrices were always in the focus of the mathematical community. Not much was accomplished on the subject until the contrivance of the QR decomposition. The well-known QZ algorithm is also distinct because of its outstanding computational efficiency. It applies to the identification of eigenvalues for the generalized eigenvalue problem. In this talk I will present parts of my Master thesis and the afore-mentioned numerical methods.

### Philipp Schulze (TU Berlin)

Donnerstag, 10. Dezember 2015

**On the Port-Hamiltonian Structure of the Navier-Stokes Equations for Reactive Flows**

As part of the Collaborative Research Centre 1029, our goal is to derive reduced order models for the reactive flow in a gas turbine. The governing equations are the Navier-Stokes equations for compressible flow supplemented by additional equations accounting for the temporal change of chemical composition. The governing equations may be derived from conservation laws of mass, momentum, and energy. For control, optimization, and design purposes, usually a small ODE (or DAE) system is demanded, which can be obtained by spatial discretization and subsequent model reduction of the governing PDEs. However, in general, important properties of the physical system, as stability and conservation laws, are lost in these steps leading to a reduced order model which does not reflect these important physical properties. One way of preserving these properties is based on the port-Hamiltonian formulation of the governing equations. The port-Hamiltonian structure guarantees stability and passivity and, thus, these properties may be preserved by structure-preserving discretization and model reduction techniques.

In this talk I will present some (yet unfinished) work on the port-Hamiltonian dynamics of reactive flows. I will derive a Hamiltonian formulation of the governing equations which is based on periodic or vanishing boundary conditions corresponding to zero energy flow through the boundary. The incorporation of more realistic boundary conditions leads to an infinite-dimensional port-Hamiltonian system, which exchanges energy with its environment through boundary ports. I will address this extension and discuss open issues on the way to finite-dimensional port-Hamiltonian systems.

### Juraj Mihalik (TU Berlin)

Donnerstag, 10. Dezember 2015

**Design of cam profiles using optimal control methods**

The optimal design of the so called cam in the valve-train as part of a combustion engine is an important issue for its efficiency. The corresponding mathematical formulation results in an optimal control problem with linear system of ordinary differential equations, linear objective function, and non-linear constraints.

In this talk we attempt to solve this problem using a direct approach ("first discretize, then optimize"), and an indirect approach ("optimize with functions"), and compare the results. In the indirect approach, the Pontryagin's maximum principle is chosen to determine the necessary optimality conditions for systems including pure-state constraints. Furthermore, we discuss a strategy to solve for these conditions numerically.

### Jeroen Stolwijk (TU Berlin)

Donnerstag, 17. Dezember 2015

**Sensitivity Analysis for Nonlinear Rootfinding Problems**

Natural gas plays a crucial role in the energy supply of Europe and 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 objective of the speaker's current research is to keep the total simulation error, consisting of the modeling error, the discretisation error, data errors and rounding errors, below a prescribed tolerance while minimizing the computational cost. This is achieved by using temporal and spatial stepsize adaptivity and by switching both between models in a model hierarchy and between different discretisation schemes.

The gas flow through a pipeline network is modeled by the one-dimensional Euler equations, which are a system of partial differential equations consisting of the continuity equation, the impulse equation and the energy equation. In previous presentations of the speaker in the Absolventen-Seminar, he has carried out an error analysis for the Euler equations in purely algebraic and in stationary form. The current presentation focusses on a model consisting of two isothermal partial differential equations, which is called the "ISO2" model within the Collaborative Research Centre 154. The spatial and temporal discretisation of these equations lead to a nonlinear system of equations, which can be rewritten into a nonlinear rootfinding problem. This rootfinding problem has to be solved in each time integration step. As one step towards the determination of the error in the solution of these rootfinding problems, the condition number of the solution with respect to the input parameters is computed. The condition number is a measure for the amplification of errors in the (uncertain) input data towards the error in the solution.

The first part of the presentation consists of the derivation of (componentwise) absolute and relative condition numbers for general one-dimensional and n-dimensional nonlinear rootfinding problems. The second part applies the presented theory to two different spatial discretisation schemes for the ISO2 model. The presentation concludes with a discussion of the results, where it is shown that the values of the componentwise relative condition numbers of the two emerging nonlinear rootfinding problems can be reduced by decreasing the temporal stepsizes.

This is joint work with V. Mehrmann. It is supported by the German Research Foundation DFG.

### Matthew Salewski (TU Berlin)

Donnerstag, 07. Januar 2016

**Symmetries in Model Reduction**

Symmetries, characterized by the invariance of a system under some operation, are ubiquitous in nature and are an essential ingredient to preserve when constructing a model; for example, if a physical system is invariant to spatial translations, then any model of this system should also include this property. Furthermore, such considerations can be utilized in model reduction.

Here I discuss how symmetries can be used to reduce simple systems involving transport and diffusive phenomena. With some simple 1D systems, I will demonstrate what these invariances mean and how they may be used in reduction. We will see that when the symmetries are used in constructing a reduced-order model, they will manifest themselves explicitly in the reduced system.

### Christoph Zimmer (TU Berlin)

Donnerstag, 14. Januar 2016

**Theory and Numerical Analysis of Linear Operator Differential-Algebraic Equations with Delay**

Operator differential-algebraic equations (OpDAEs) are a helpful tool for the mathematical treatment of constrained partial differential equations. Furthermore, closed-loop control introduces delay in this systems. The resulting mathematical problem of an OpDAE with delay (OpDDAE) occurs for example in the control of incompressible flows. In this talk we will consider OpDDAEs. We will investigate the existence and uniqueness of solutions for differential-algebraic equations with delay in spaces of integrable functions as well as of solutions for OpDDAEs. In the second half we will study discretization of OpDDAEs by Galerkin schemes and verify the results with a numerical test calculation.

### Volker Mehrmann (TU Berlin)

Donnerstag, 14. Januar 2016

**The sign-characteristic of structured matrix polynomials**

The sign characteristic is a special invariant associated with certain critical eigenvalues of structured matrix polynomials such as those with Hermitian or even and palindromic structure. It plays an essential role in the analysis of the stability of dynamical systems (with Hamiltonian systems being a special case). If the dynamical system is constrained, then the associated matrix polynomial has eigenvalues at \infty. We extend classical sign characteristic results by Gohberg, Lancaster, and Rodman to the case of infinite eigenvalues. We derive a systematic approach, studying how sign characteristics behave after an analytic change of variables, including the important special case of Möbius transformations, and we prove a signature constraint theorem. We also show that the sign characteristic at infinity stays invariant in a neighborhood under perturbations for even degree Hermitian matrix polynomials, while it may change for odd degree matrix polynomials. We argue that the non-uniformity can be resolved by introducing an extra zero leading matrix coefficient.

This is joint work with Vanni Noferini, Francoise Tisseur and Hongguo Xu.

### Matthias Voigt (TU Berlin)

Donnerstag, 21. Januar 2016

**Balanced Truncation Model Reduction of Boundary Value Problems with Application to Data Assimilation **

Data assimilation is a technique that allows the adaptation of a model in order decrease the mismatch to observations taken from the real physical process. In this way, the accuracy of the model-based forecast is increased. A typical application area is the weather forecast, where the model is supported by measurements from various weather stations. These models are usually very large and demand a lot of computing resources, not only for the simulation but also for the data assimilation process. Thus model reduction is of great importance to speed-up computations.

The goal of this talk is to introduce variational data assimilation which can be formulated as the solution of a certain boundary value problem. I will present some first unpolished ideas to reduce this boundary value problem and discuss some of the occurring problems such as the optimal matching of the boundary conditions.

### Punit Sharma (TU Berlin)

Donnerstag, 21. Januar 2016

**Structured distance to instability for port-Hamiltonian systems**

Port-Hamiltonian systems are highly structured and as a consequence they satisfy two important properties for control systems: passivity and stability. It is important to know the distance of a linear time-invariant port-Hamiltonian system from the boundary of the set of systems that are not port-Hamiltonian. This can be partially addressed by computing the structured stability radius, i.e., the norm of smallest structure preserving perturbation that makes the system not stable, for port-Hamiltonian systems.

In this talk, we discuss the structured distance to instability for linear time-invariant port-Hamiltonian systems. This talk describes joint work with Christian Mehl and Volker Mehrmann.

### Marine Froidevaux (École Polytechnique Fédérale de Lausanne)

Donnerstag, 28. Januar 2016

**A structure preserving first order formulation for quadratic eigenvalue problems**

Quadratic eigenvalue problems (QEPs) arise from a wide range of applications, a typical example of which is the analysis of structural dynamics. Very often in practice, the matrix polynomial underlying a QEP has a special structure (hermitian, even, odd, palindromic,…) which results in a symmetry in the spectrum. In order to solve such structured eigenvalue problems with a satisfying accuracy, it is sometimes essential to use numerical methods which preserve the structure of the matrix polynomial. The most common approach in the resolution of QEPs is to proceed via linearization. However, linearizations that are used in practice typically destroy the structure of the problem and, when the matrix polynomial is singular, elongate some of the singular chains. This generally leads to a loss of efficiency and accuracy in the computed results.

In this talk, I will present some of the conclusions of my Master thesis. I will introduce a structure preserving first order formulation for QEPs, as well as a complete algorithm for the computation of eigenvalues in structured and in general QEPs.

### Martin Schramm (TU Berlin)

Donnerstag, 28. Januar 2016

**A 2-by-2 Jacobi Algorithm for Complex Conjugated Hamiltonian Matrices**

My talk will follow Christian's presentation from November 2015. I will talk about the current status of my Master Thesis. The Jacobi algorithm is an iterative method for finding the eigenvalues of (small) symmetric matrices, Hamiltonian matrices are block-structured and could, for example, arise from algebraic differential equations. I will present a Jacobi-like, structure preserving algorithm for complex conjugated Hamiltonian matrices. Basic sub-algorithms, convergence behavior and possible problems will be discussed.

The second part of the presentation will be about an application of the algorithm: We consider the matrix sum H0 + k*t*H1, where H0 and H1 are complex conjugated Hamiltonian matrices and t a small step. I will introduce the Laub Trick, a variation of the QR algorithm, which in many cases is cheaper than Jacobi. Depending on t, when applied to the matrix sum above, the opposite might be the case.

### Christoph Conrads (TU Berlin)

Donnerstag, 04. Februar 2016

**Projection Methods for Generalized Eigenvalue Problems**

In this talk, we will discuss backward errors, dense solvers, and a new projection method for generalized eigenvalue problems (GEPs) with Hermitian positive semidefinite (HPSD) matrices.

We will present properties and origins of such GEPs. We will also address quickly computable and structure preserving backward error bounds for these kinds of GEPs. There is an abundance of literature on backward error measures possessing one of these features but only recently, the author came across a backward error providing both.

We will elaborate on dense solvers for GEPs with HPSD matrices. The standard solver for GEPs with Hermitian matrices is fast but requires positive definite mass matrices and is only conditionally backward stable; the QZ algorithm for general GEPs is backward stable but it is also magnitudes slower and does not preserve any problem properties. In the talk, we will present two new backward stable and structure preserving solvers, one using deflation, the other one using the generalized singular value decomposition (GSVD). In comparison to the QZ algorithm, both solvers are competitive with the standard solver in our tests.

Finally, we will touch on a new solver for large, sparse GEPs. The talk is based on the preliminary results of my Master's thesis. The slides will be available on my website.

### Arbi Moses Badlyan (TU Berlin)

Donnerstag, 04. Februar 2016

**Thermal Relaxation in Geometrically Frustrated Magnets (Spin Ice)**

Geometrical frustration is a common feature of condensed matter systems in which the lattice geometry inhibits the formation of a single ground state configuration. Linus Pauling's model of the proton position in water ice, with its characteristic property of a residual zero point entropy, stands as the paradigm for geometrically frustrated systems. In magnetic systems geometrical frustration arises when the topology of the spin lattice leads to frustration of the spin-spin interaction. Of particular recent interest are certain rare-earth pyrochlores such as Dysprosium-Titanate Dy_{2}Ti_{2}O_{7}, in which the spin configurations are analogous to that of the proton positions in water ice. The Dysprosium ions Dy^{3+} in Dysprosium-Titantate are located on a lattice of corner sharing tetrahedra and their Ising-like magnetic moments (spins) are constrained by crystal field interactions to point radially into or out of the tetrahedra. Interactions between the spins require that on each tetrahedron two spins point inward and two point outward, a constraint that is similar to ordering of protons in water ice (ice rules). Violating the ice rules by flipping a spin out of a ground-state configuration leads to a pair of point-like topological defects. Continuation of the spin flips over neighbouring tetrahedra separates the defect and creates a string with a monopole and an anti-monopole at its ends (magnetic excitations).

In this talk I will present some results of my diploma thesis, which I wrote at the National Metrology Institute of Germany (PTB). The purpose of my thesis was to investigate the thermal properties of spin ice material and in particular to determine the specific heat capacity of Dysprosium-Titanate at temperatures below 1 Kelvin and in constant magnetic fields.

The determination of the heat capacity of a solid body for low temperatures using relaxation calorimetry requires the thermodynamic modelling of measured temperature responses. The necessary field equations are derived from the first law of thermodynamics and the constitutive relation for the internal energy. In geometrically frustrated magnets (spin ice) the internal energy is represented by a linear functional of not only the current temperature but also of the entire temperature history.