### Inhalt des Dokuments

## Kolloquium der Arbeitsgruppe Modellierung • Numerik • Differentialgleichungen

Verantwortliche Dozenten: | Alle Professoren der Arbeitsgruppe Modellierung • Numerik • Differentialgleichungen |
---|---|

Koordination: | Dr. Christian Schröder, Dr. Hans-Christian Kreusler |

Termine: | Di 16-18 Uhr in MA 313 und nach Vereinbarung |

Inhalt: | Vorträge von Gästen und Mitarbeitern zu aktuellen Forschungsthemen |

## Beschreibung

Das Kolloquium der Arbeitsgruppe "Modellierung, Numerik, Differentialgleichungen" im Institut der Mathematik ist ein Kolloquium klassischer Art. Es wird also von einem breiten Kreis der Professoren und Mitarbeiter aus allen zugehörigen Lehrstühlen, insbesondere Angewandte Funktionalanalysis, Numerische Lineare Algebra, und Partielle Differentialgleichungen, besucht. Auch Studierende nach dem Bachelorabschluss zählen schon zu den Teilnehmern unseres Kolloquiums.

Aus diesen Gründen freuen wir uns insbesondere über Vorträge, die auf einen nicht spezialisierten Hörerkreis zugeschnitten sind und auch von Studierenden nach dem Bachelorabschuss bereits mit Profit gehört werden können.

Datum date | Zeit time | Raum room | Vortragende(r) speaker | Titel title | Einladender invited by |
---|---|---|---|---|---|

18.10.16 | 16:15 | MA 313 | Jose-Antonio Carrillo (Imperial College London) | Swarming models with repulsive-attractive effects (Abstract) | V. Mehrmann |

25.10.16 | 16:15 | MA 313 | Serkan Gugercin (Virginia Tech) | Data-driven reduced model construction with time-domain Loewner models (Abstract) | V. Mehrmann |

8.11.16 | 16:15 | MA 313 | Eskil Hansen (U Lund) | Splitting based integrators for parabolic problems (Abstract) | E. Emmrich |

22.11.16 | 16:15 | MA 313 | Zoltán Horváth (Széchenyi István U, Györ, Hungary) | Preservation of inequalities for initial value problems and their discretizations (Abstract) | V. Mehrmann |

29.11.16 | 16:15 | MA 313 | Herbert Egger (TU Darmstadt) | A variational discretization framework for compressible flow in pipeline networks (Abstract) | V. Mehrmann |

6.12.16 | 16:15 | MA 313 | Murat Manguoglu (METU, Ankara) | Parallel Solution of Sparse Linear Systems and Least Squares problems (Abstract) | V. Mehrmann |

17.01.17 | 16:15 | MA 313 | Kees Vuik (TU Delft) | Deflation with POD vectors for Porous Media Flow (Abstract) | R. Nabben |

14.02.17 | 16:15 | MA 313 | Fernando de Teran (UC3 Madrid) | Polynomial root-finding using companion matrices (Abstract) | V. Mehrmann |

## Rückblick

- Kolloquium ModNumDiff Sommer 2016
- Kolloquium ModNumDiff Winter 2015/16
- Kolloquium ModNumDiff Sommer 2015
- Kolloquium ModNumDiff Winter 2014/15
- Kolloquium ModNumDiff Sommer 2014
- Kolloquium ModNumDiff Winter 2013/14
- Kolloquium ModNumDiff Sommer 2013
- Kolloquium ModNumDiff Winter 2012/13
- Kolloquium ModNumDiff Sommer 2012
- Kolloquium ModNumDiff Winter 2011/12
- Kolloquium ModNumDiff Sommer 2011

### Jose-Antonio Carrillo (Imperial College London)

**Swarming models with repulsive-attractive effects**

Dienstag, den 18.10.2016, 16.15 Uhr in MA 313

Abstract

I give an overview of the different levels of description of collective behavior models highlighting some of the interesting mathematical open problems in the subject. Calculus of variations, dynamical systems, mean-field limits for PDEs, control theory, kinetic and aggregation-diffusion equations naturally show up as necessary tools to solve some of these questions.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

### Serkan Gugercin (Virginia Tech)

**Data-driven reduced model construction with time-domain Loewner models**

Dienstag, den 25.10.2016, 16.15 Uhr in MA 313

Abstract

We presents a data-driven nonintrusive model reduction approach for large-scale systems with linear state dependence. Traditionally, model reduction is performed in an intrusive projection-based framework, where the operators of the full model are required either explicitly in an assembled form or implicitly through a routine that returns the action of the operators on a vector. Our nonintrusive approach constructs reduced models directly from trajectories of the inputs and outputs of the full model, without requiring the full-model operators. These trajectories are generated by running a simulation of the full model; the method then infers frequency-response data from these simulated time-domain trajectories and uses the data-driven Loewner framework to derive a reduced model. Only a single time-domain simulation is required to derive a reduced model with the new data-driven nonintrusive approach. We demonstrate the propose methodology on benchmark examples and a finite element model of a cantilever beam. This is joint work with Benjamin Peherstorfer and Karen Willcox.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

### Eskil Hansen (U Lund)

**Splitting based integrators for parabolic problems**

Dienstag, den 8.11.2016, 16.15 Uhr in MA 313

Abstract

The aim of this talk is to give an overview of some recent progress when analyzing splitting schemes applied to nonlinear parabolic problems. Such equations are frequently encountered in biology, chemistry and physics, as they e.g. describe reaction-diffusion systems. More precisely, we will consider splitting based integrators for evolution equations governed by m-dissipative vector fields. The dissipative property yields a rather general framework which serves as the foundation of our numerical analysis. First, we recapitulate some of the classical approximation results by Brezis and Pazy, which yield convergence of several splitting schemes. Secondly, we present a new strategy for deriving convergence orders for a family of (formally) first-order schemes when, e.g., applied to degenerate parabolic equations with delay source terms and the abstract Riccati equation.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

### Zoltán Horváth (Széchenyi István U, Györ, Hungary)

**Preservation of inequalities for initial value problems and their discretizations**

Dienstag, den 22.11.2016, 16.15 Uhr in MA 313

Abstract

Many initial value problems (IVPs) for ODE, DAE, PDE possess certain closed subsets C of the state space with the property that any trajectories of the IVP starting in C remain in C forward in time, in which case we call C forward invariant w.r.t. the IVP. Often C is defined by a bunch of inequalities for the state variables; then forward invariance of C means that the inequalities along the solutions will hold true whenever they are satisfied at the initial time. Besides being a stability property of the IVP, forward invariance of a set C often reflects some physical properties of the underlying physical process modelled by the IVP (e.g. positivity preservation of species concentrations for reaction systems, energy diminishing property of dissipative systems or feasibility of compressible flow variables).

In the talk we derive necessary and sufficient conditions for linear nonnegativity preserving DAEs, which can be checked by linear optimization codes. We do not apply projectors nor generalized inverses of matrices. Previously in the literature only sufficient conditions existed which appear to be too strong in many cases.

Concerning similar constructive results for nonlinear ODEs we shall point out a methodology based on theorems of alternatives that for ODEs and C sets defined by some nonlinear equalities the forward invariance of C can be characterized by equivalent conditions that can be checked by optimization programs.

Then we consider numerical methods applied to IVPs having a forward invariant set C. We aim at finding conditions on the step sizes in terms of the method parameters, the IVP operators and C that enable the numerical preservation of the forward invariance of C.

Concerning numerical methods for nonnegativity preserving DAEs we shall present characterization of time step sizes of the method under which nonnegativity preservation of the states are preserved, resulting a complete generalization of the Bolley-Crouzeix theory given for linear ODEs and Runge-Kutta and multistep methods.

Finally, we consider IVPs for which even the explicit Euler method does not preserve the invariance property. This holds true typically when FEM is applied for PDEs without lumping. We shall point out a theorem that applies to many of these IVPs and implicit time steppings like BDF or DIRK methods.

The results of the talk are partly joint with V. Mehrmann, T. Terlaky, Y. Song, V. Thomée and P. Chatzipantelidis.

### Herbert Egger (TU Darmstadt)

**A variational discretization framework for compressible flow in pipeline networks**

Dienstag, den 29.11.2016, 16.15 Uhr in MA 313

Abstract

We consider the flow of compressible fluids through pipes and pipe networks. As a starting point, a thermodynamically consistent variational characterization of solutions to the one dimensional Euler equations is presented which very directly encodes the conservation of mass, energy, and entropy. This variational principle is suitable for a conforming Galerkin approximation in space which automatically inherits the basic physical conservation laws. A mixed finite element method is briefly discussed as a particular choice. We also investigate the discretization in time by a problem adapted implicit time stepping scheme for which we prove exact conservation of mass and a slight dissipation of energy and negative entropy. These deviations from the strict conservation laws are due to numerical dissipation of the implicit time discretization. The resulting fully discrete method can be extended naturally to more general flow models and also to pipe networks and is therefore well suited for the simulation of gas transport in pipelines.

### Murat Manguoglu (METU, Ankara)

**Parallel Solution of Sparse Linear Systems and Least Squares problems**

Dienstag, den 6.12.2016, 16.15 Uhr in MA 313

Abstract

Many applications in science and engineering give rise to sparse linear systems of equations. Typically sparse linear systems of equations arise from the discretization of Partial Differential Equations (PDEs) while some sparse systems are not governed by PDEs. Sparse algorithms are well-known for their poor utilization of the cache due to irregular memory access. In addition, traditional algorithms that are designed for sequential platforms usually have inherited limitations for parallelism.

In the first part of this talk, we will present a new class of sparse solvers that use the generalized parallel DS factorization as opposed to the classical sparse LU factorization. In the second part, we will talk about a new parallel algorithm for finding the minimum norm solution of sparse underdetermined linear least squares problems.

We will provide the parallel scalability of these solvers compared to other well-known algorithms on various computing platforms. The results are obtained for large sparse systems that arise in a variety of applications.

### Kees Vuik (TU Delft)

**Deflation with POD vectors for Porous Media Flow**

Dienstag, den 17.1.2017, 16.15 Uhr in MA 313

Abstract

We consider systems originating from the simulation of multi-phase flow through porous media. The spatially discretized coupled nonlinear equations are solved with a two-stage preconditioner. We consider the first step of this procedure, i.e. solving of the pressure equation. We develop preconditioners based on deflation and a selection of deflation vectors motivated by proper orthogonal decomposition (POD) to a number of pre-computed solutions. We investigate alternatives using different varieties of reduced-order modelling. Furthermore we explore the connection between POD-based preconditioning and deflation methods. One of the difficulties for deflation methods is to find the right deflation vectors for general problems. The combination of deflation with the POD methods looks very promising in this respect.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 313 - everybody's welcome.

### Fernando De Teran (UC3 Madrid)

**Polynomial root-finding using companion matrices**

Dienstag, den 14.2.2017, 16.15 Uhr in MA 313

Abstract

The use of companion matrices to compute the roots of scalar polynomials is a standard approach (it is for instance, the one followed by the command 'roots' in MATLAB). It essentially consists in computing the roots of a scalar polynomial as the eigenvalues of the companion matrix. In this talk, I will review several numerical and theoretical issues on this topic. I will pay special attention to the backward stability of solving the polynomial root-finding problem using companion matrices. More precisely, to this question: Even if the computed eigenvalues of the companion matrix are the eigenvalues of a nearby matrix, does this guarantee that they are the roots of a nearby polynomial? Typically, the companion matrix methods focus on monic polynomials, since one can always divide by the leading coefficient, if necessary. But, is it enough for the backward stability issue to focus on monic polynomials? I will also pay attention to some other (more theoretical) questions like: How many companion matrices are there and what do they look like?

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 313 - everybody's welcome.