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## Absolventen-Seminar • Numerische Mathematik

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

Koordination: | Ann-Kristin Baum |

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 17.10. | 10:15 Uhr | MA 376 | Vorbesprechung | |

Do 24.10. | 10:15 Uhr | MA 376 | - kein Seminar - | |

Do 31.10. | 10:15 Uhr | MA 376 | Matthias Voigt Olga Markova | New Approaches to Compute the H-infinity-Norm of Large-Scale Systems Lengths of matrix subalgebras |

Do 07.11. | 10:15 Uhr | MA 376 | - kein Seminar - | |

Do 14.11. | 10:15 Uhr | MA 376 | Robert Altmann Subashish Datta | Multibody Systems of Index 3, 5, and 199 Controller Optimization with Regional Pole Placement. |

Do 21.11. | 10:15 Uhr | MA 376 | - kein Seminar - | |

Do 28.11. | 10:15 Uhr | MA 376 | - kein Seminar - | |

Do 05.12. | 10:15 Uhr | MA 376 | Christoph Conrads Federico Poloni | Improving AMLS with AMG Modelling queues and buffers --- Probabilistic interpretation and accurate algorithms |

Do 12.12. | 10:15 Uhr | MA 376 | Jan Heiland | Decoupling and Optimization of Differential-Algebraic Equations with Application in Optimal Flow Control |

Do 19.12. | 10:15 Uhr | MA 376 | Phi Ha | Numerical solutions to Delay Differential-Algebraic Equations |

Do 26.12. | 10:15 Uhr | MA 376 | - kein Seminar - | |

Do 02.01. | 10:15 Uhr | MA 376 | - kein Seminar- | |

Do 09.01. | 10:15 Uhr | MA 376 | Barbara Scherlein | Uniform Ensemble Controllability of Parameter-Dependent Linear Systems |

Do 16.01. | Sarosh Quraishi Agnieszka Miedlar | Model reduction for parameter dependent eigenvalue problems Adaptive path-following methods for nonlinear PDE eigenvalue problems | ||

Do 23.01. | Leo Batzke Philipp Petersen | Low-Rank Perturbations of Alternating Matrix Pencils Classification of Edges using Compactly Supported Shearlets | ||

Do 30.01. | Michal Wojtylak Vladimir Kostić | On a distance of a regular pencil to the set of singular pencils. Distance to dislocation: computing the robustness of a class of eigenvalue localization regions | ||

Do 06.02. | - kein Seminar - | |||

Do 13.02. | Ann-Kristin Baum | Towards a positivity characterization of DAEs - A flow formula based on projections |

### Ann-Kristin Baum (TU Berlin)

Donnerstag, 13. Februar 2014

**Towards a positivity characterization of DAEs - A flow formula based on projections**

Positivity is a property of many physical variables like energy, the concentration of chemicals or the density of biological species. Besides positivity, these quantities are often subject to algebraic constraints like conservation laws or balance equations. These constraints extend the governing dynamics to a Differential-Algebraic Equation (DAE).

Starting with an example, we point out the issues that occur if one looses these physical properties in a numerical simulation by loosing positivity or violating the constraints. To ensure a physically meaningful discretization, the task is twofold: First, finding conditions that allow to validate if a given DAE is positive and all constraints are explicitly given. Second, analyzing how these properties are preserved within a numerical simulation.

In this talk, we address the first question and work towards a positivity characterization of general nonlinear DAEs. Using a projection approach, we filter out the differential and algebraic components of the DAE without changing the coordinate system and derive a flow formula that is stated in original (positive) variables. The flow approach allows to study positivity of DAEs in the framework of invariant manifolds with corners.

### Michal Wojtylak (Jagiellonian University Krakau, TU Berlin)

Donnerstag, 30. Januar 2014

**On a distance of a regular pencil to the set of singular pencils.**

Partial results on the topic announced in the title will be presented. A special attention will be put to the case, when the distance is defined as the minimum of the norms of rank one perturbations that make the pencil singular. The talk will be based on joint work with Christian Mehl and Volker Mehrmann.

### Vladimir Kostić (University of Novi Sad, TU Berlin)

Donnerstag, 30. Januar 2014

**Distance to dislocation: computing the robustness of a class of eigenvalue localization regions **

Numerous problems in mechanics, mathematical physics, and engineering can be formulated as eigenvalue problems where the focus is, after determining that the eigenvalues are in the specific desirable domain, to detect admissible size of uncertainty that will not cause the eigenvalues to leave the domain. The most frequent of such domains in use are connected to stability of dynamical systems: open left half-plane of the complex plane (continuous dynamical systems) and open unit disk (discrete dynamical systems). Due to profound research in the past years on the robustness of eigenvalues, it is now well known that the concept of pseudospectra is more adequate tool then the classical spectra for the treatment of behaviour of nonnormal matrices. Because of this fact, many authors have considered quantities such as pseudospectral abscissa, pseudospectral radius, and in particular distance to stability in order to provide efficient tools for exploring phenomena of spectral change under small perturbations. In this talk, we will consider eigenvalue localization sets obtained by a class of Hermitian functions in the complex plane and present an algorithm that computes their robustness. In another words, given a matrix whose eigenvalues belong to a certain fixed domain (half-plane, circle, ellipse, ring, lemniscate, strip, cardioid, etc.), we will compute the maximal level of pseudospectra which is also contained in the same domain. The algorithm that we provide is based on Newton’s method and uses implicit determinant approach in order to optimise minimal singular value over an algebraic curve defined by a Hermitian function in the complex plane. So, the computed value can be considered as a distance to dislocation which is a more general version of distance to instability (where the domain is a left half-plane or a unit circle).

### Leo Batzke (TU Berlin)

Donnerstag, 23. Januar 2014

**Low-Rank Perturbations of Alternating Matrix Pencils **

Many applications give rise to alternating matrix pencils. In this talk, structure-preserving low-rank perturbations of T-alternating (i.e., symmetric / skew-symmetric) matrix pencils will be investigated.

While perturbations of structured matrices and unstructured pencils have already been investigated, in the case of structured matrix pencils different effects occur. Hence, the generic result will have features that arise due to the structure and ones that are characteristic to pencils.

In this talk, we will derive a generic canonical form for T-alternating matrix pencils under T-alternating rank-1 and rank-2 perturbations, which will also be generalized to the related class of T-(anti-)palindromic matrix pencils.

### Philipp Petersen (TU Berlin)

Donnerstag, 23. Januar 2014

**Classification of Edges using Compactly Supported Shearlets**

It has recently been established by Guo and Labate that the continuous shearlet transform can be used to detect features of images such as edges. Not only does the asymptotic behavior of the shearlet transform determine the position and orientation of singularities, it can also be utilized to classify such edges. In particular, given a characteristic function with a piecewise smooth singularity curve we can detect this curve as well as the non-smooth points on the curve. However, these results have only been established for band-limited shearlet transforms, which lack the spatial localization that compactly supported shearlets can provide. Therefore, it seems promising to examine the edge classification in the context of compactly supported shearlets. Indeed, within this talk we will present results, which significantly improve the classical results by utilizing compactly supported shearlets. These results will also be extended to the three dimensional case. Furthermore, we will show that we can extract information about curvature from the decay of the shearlet transform.

### Agnieszka Miedlar (TU Berlin)

Donnerstag, 16. Januar 2014

**Adaptive path-following methods for nonlinear PDE eigenvalue problems **

We introduce a new approach combining the adaptive fnite element method

with the homotopy method to determine the eigenvalues of the convection-

diffusion problem and problems from acoustic field computations with

proportional

damping. Some preliminary numerical examples will we presented to illustrate

the behavior of the method.

This is a joint work with C. Carstensen, J. Gedicke and V. Mehrmann.

### Sarosh Quraishi (TU Berlin)

Donnerstag, 16. Januar 2014

**Model reduction for parameter dependent eigenvalue problems **

We present an approach for model reduction in discrete finite element (FE) models of parameter dependent mechanical systems e.g. disk brakes. The coefficient matrices from FEM are large, non-symmetric, and parameter dependent. Our approach is based on a proper orthogonal decomposition (POD) of the full scale finite element model for different parameter values to construct a reduced basis which works for other parameter combinations as well. To identify the role of relevant parameters for a brake-squeal, a detailed parameter study is necessary, which in-turn requires the solution of many large-scale quadratic eigenvalue problems (QEVP) for a variety of choices of the parameter. The state of the art modal-transformation approaches used in standard FE software converts the QEVP to a space of modal-coordinates. The modal-transformation matrices are typically constructed by solving a symmetric linear eigenvalue problem, which is obtained by dropping the non-symmetric, parameter dependent and damping terms in the QEVP i.e. by neglecting all the physical effects that are essential for self-excited vibrations. This simplistic approach works well only for the problems where an approximation of the imaginary part of the eigenvalue is required, but for studying the dynamical stability behavior of a brake with respect to squeal, a good approximation of the eigenvalues with positive real part is of crucial interest. Our POD based approach takes into account the parameter dependent nature of the damping and stiffness matrices. We obtain the model-order-reducing subspace by performing a POD on the matrix of dominant modes of the non-symmetric QEVP for a variety of parameter choices. Numerical experiments suggest that the new POD based approach is far more accurate for the brake squeal problem than state of the art algorithms used in FE programs so far.

### Barbara Scherlein

Donnerstag, 09. Januar 2014

**Uniform Ensemble Controllability of Parameter-Dependent Linear System**

We consider a time-invariant time-discrete single-input linear system which depends continuously on a parameter from a compact parameter set. The task is to steer it to a given target function which depends continuously on the parameter, too. The control, however, is not allowed to depend on the parameter. As it is not generally possible to reach the target function precisely, we only demand it to be approximated uniformly for all parameters. If any continuous target function can be approximated with arbitrary precision in this way we call the system uniformly ensemble controllable. Helmke and Schönlein have proved necessary conditions. They have also shown that a system is uniformly ensemble controllable if theses conditions are fulfilled and additionally, firstly, the parameter set is a real compact interval and secondly, the system matrix has only got simple eigenvalues. The main goal of the talk is to examine if these two additional conditions can be dropped. Especially, I give classes of systems with multiple eigenvalues which cannot be uniformly ensemble controlled as well as examples which are uniformly ensemble controllable; and I show that the set of uniformly ensemble controllable systems is neither open nor closed with respect to the supremum norm.

### Phi Ha (TU Berlin)

Donnerstag, 19. Dezember 2013

**Numerical solutions to Delay Differential-Algebraic Equations **

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

constraints due to conservation laws or surface conditions. On the other hand, time-delays occur naturally in various dynamical systems,

both physically, when the transfer phenomena (energy, signal, material) is not instantaneous, and artificially, when a time-delay is used

in the controller. The combination of differential-algebraic equations and time-delays leads to a new mathematical object: "delay differential-algebraic equations (Delay-DAEs)", which is a source of many complex behavior.

In this talk, we address the computational problem for numerical solutions to Delay-DAEs. First, we propose an algorithm, which extends

the classical Bellman method to compute the solution of general nonlinear Delay-DAEs. Second, we examine the validity and the effectiveness of

our algorithm by some illustrative examples.

### Jan Heiland (MPI Magdeburg)

Donnerstag, 12. Dezember 2013

**Decoupling and Optimization of Differential-Algebraic Equations with Application in Optimal Flow Control **

For the time-integration Navier-Stokes equations, most commonly used approximation scheme resort to the Pressure Poisson Equation (PPE) in order to decouple pressure and velocity of the flow. In the semi-discrete setting, the PPE is easily derived. However, in the continuous setting where the states of the velocity and the pressure are located in infinite-dimensional spaces, the PPE may not exist.

This example of the Pressure Poisson Equation points to the general issue about the conditions under which a transformation of the discrete approximation commutes with a discrete approximation of a related transformation of the continuous problem. In this talk on decoupling and optimization of differential-algebraic equations, we consider two such transformations, namely

a.) formulation of the optimal control problem as a system of optimality conditions and

b.) decoupling of the differential and algebraic equations

and their interaction with numerical approximations. We will discuss, when do these transformations, whose finite-dimensional counterparts are well understood, apply in the continuous setting and whether they commute with discretizations.

### Christoph Conrads (TU Berlin)

Donnerstag, 05. Dezember 2013

**Improving AMLS with AMG**

We look at the current status of an existing Automated Multilevel Substructuring (AMLS) implementation, explain in which way Algebraic Multgrid (AMG) is interesting for us, and in which direction we can learn from AMG to further the development of AMLS.

### Federico Poloni (Università di Pisa)

Donnerstag, 05. Dezember 2013

**Modelling queues and buffers --- Probabilistic interpretation and accurate algorithms**

We present a problem in queuing theory that results in an equation formally very similar to the boundary-value problems encountered in optimal control theory. We consider a common algorithm for its solution, the structured doubling algorithm. Relying on the sign structure of the matrix involved and the applicative source of the problem, we (i) give a probabilistic interpretation of the algorithm and the matrices appearing in it, which highlights the link to numerical methods for ODEs, and (ii) show that this interpretation can be used to produce so-called \emph{triplet representations}, which are the key to obtaining an accurate algorithm (in the componentwise sense).

### Robert Altmann (TU Berlin)

Donnerstag, 14. November 2013

**Multibody Systems of Index 3, 5, and 199**

In this talk, I introduce the topic of control constraints. A typical example is a crane model for which we aim to find an input such that the working load follows a prescribed trajectory. Although such problems look like constrained multibody systems, the resulting differential-algebraic equations are of index 5 or higher. In order to see the difference to standard multibody systems, we analyse several examples and look at numerical results.

### Subashish Datta

Donnerstag, 14. November 2013

**Controller Optimization with Regional Pole Placement.**

Cost is a major concern in modern control systems, and components that

influence the cost are often the actuators, sensors and hardware used to

implement the controller. Hence, efficient designs should reduce cost and

complexity in implementation. Simultaneously, the closed loop system

should meet the design specifications (settling time/damping ratio) to

ensure safe operation. In this talk, the optimization of several control

objectives like: i) minimize norm of the controller which directly

influences the amplitude of control signal and hence the actuators rating

and ii) minimize order of the controller which helps in reducing the

complexity of hardware, will be discussed. In addition, how the resulting

controller guarantees that the pre-defined design specifications are

achieved, will be presented.

### Olga Markova

Donnerstag, 31. Oktober 2013

**Lengths of matrix subalgebras**

By the length of a finite system of generators for a finite-dimensional associative algebra over an arbitrary field we mean the least non-negative integer k such that the words in these generators of lengths not exceeding k span this algebra (as a vector space). The maximum length for the systems of generators of an algebra is referred to as the length of the algebra. Following Paz (1984) we consider the question when the length of a matrix subalgebra can be bounded by a linear function in the order of matrices. We provide linear upper bounds for the lengths of upper triangular and commutative matrix subalgebras over arbitrary fields and give examples of the direct length evaluation for certain matrix algebras of these types.

### Matthias Voigt

Donnerstag, 31. Oktober 2013

**New Approaches to Compute the H-infinity-Norm of Large-Scale Systems **

In this talk two approaches for computing the H-infinity-norm of large-scale descriptor systems will be presented. Both methods are based on the computation of dominant poles and a subsequent optimization procedure. The first method exploits a relation to the structured complex stability radius. Then an optimization over structured pseudospectra is performed in a nested iteration. The inner iteration computes the rightmost point of an epsilon-pseudospectrum, whereas the outer iteration varies epsilon in order to find the pseudospectrum that touches the imaginary axis. The second method is an extension of a well-known algorithm based on even matrix pencils.. The even IRA algorithm is used to get some desired eigenvalues, where the dominant poles are employed to determine appropriate shifts. If time permits, an open problem will be briefly introduced.