### 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 |
---|---|---|---|---|---|

Di 28.10.14 | 16:15 | MA 313 | Arnd Meyer (TU Chemnitz) | Coordinate free formulations of the classic shell models (Abstract) | F. Tröltzsch |

Di 4.11.14 | 16:15 | MA 313 | Andreas Prohl (U Tübingen) | Numerical Analysis of the stochastic cubic Schroedinger equation (Abstract) | E. Emmrich |

Di 18.11.14 | 16:15 | MA 313 | Michael L. Overton (Courant Institute, NYU) | Fast Approximation of the H_infinity Norm via Optimization over Spectral Value Sets (Abstract) | V. Mehrmann |

Di 2.12.14 | 16:15 | MA 313 | Ernst Stephan (U Hannover) | Time-domain BEM for acoustic problems - sound radiation from tyres (Abstrakt) | R. Schneider K. Schmidt |

Di 9.12.14 | 16:30 | MAR 0.009 | Sina Ober-Blöbaum (U Paderborn, FU Berlin) | Variational integrators in optimal control theory (Abstract) | V. Mehrmann |

Di 16.12.14 | 16:15 | MA 313 | Aneta Wróblewska-Kamińska (Polish Academy of Sciences) | Non-Newtonian fluids and abstract problems: applications of Orlicz spaces in the theory of nonlinear PDE (Abstract) | E. Emmrich |

Di 27.01.15 | 16:15 | MA 313 | Stefan Ulbrich (TU Darmstadt) | Multilevel methods for PDE-constrained optimization based on adaptive discretizations, reduced order models and error estimators (Abstract) | F. Tröltzsch |

Di 3.02.15 | 16:15 | MA 313 | Christian Engström (Umeå U, Sweden) | Rational operator functions and applications to photonic crystals (Abstract) | K. Schmidt |

Di 24.02.15 | 16:15 | MA 313 | Yuji Nakatsukasa (U Tokyo) | Solving nonconvex optimization problems via eigenvalues (Abstract) | V. Mehrmann A. Miedlar |

Di 17.03.15 | 16:15 | MA 313 | Roman Vershynin (U Michigan) | Mathematics of networks: advances and challenges (Abstract) | G. Kutyniok |

### Arnd Meyer (TU Chemnitz)

**Coordinate free formulations of the classic shell models**

Dienstag, den 28.10.2014, 16.15 Uhr in MA 313

Abstract:

We consider the classic linear shell equations of Koiter and Naghdi. These are given and investigated by using the tensor coefficients of the approximate linear strain tensors leading to rather complicate differential terms strongly dependent on the parametrization of the shell mid-surface. If these tensors are completely written with their tensor bases, we are able to find a coordinate free meaning of these complicate terms based on the metric and curvature tensor of the mid-surface and on the surface gradient operator. This notation has low effects on well established FE-calculations of shell deformations, but can be useful in new developments in discrete differential geometry.

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

### Andreas Prohl (U Tübingen)

**Numerical Analysis of the stochastic cubic Schroedinger equation **

Dienstag, den 4.11.2014, 16.15 Uhr in MA 313

Abstract:

In der vorgestellten stochastischen nichtlinearen Schroedingergleichung wechselwirken dispersive, nichtlineare, und Rauscheffekte, was die Konstruktion einer konvergenten numerischen Diskretisierung nichttrivial macht. Im Vortrag fasse ich aktuelle Resultate zu diesem Thema zusammen und zeige Probleme auf, die im Vergleich zur Numerik der deterministischen Schroedingergleichung zu loesen sind, um schliesslich ein optimal konvergentes Verfahren zu konstruieren. Ein wesentliches Hilfsmittel hierfür ist eine gleichmaessige Schranke fuer den Hamiltonian der Diskretisierung. Die vorgestellten Resultate sind einer aktuellen Arbeit mit C. Chen und J. Hong (Chinese Academy of Sciences, Beijing) entnommen.

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

### Michael L. Overton (Courant Institute, NYU)

**Fast Approximation of the H_infinity Norm via Optimization over Spectral Value Sets**

Dienstag, den 18.11.2014, 16.15 Uhr in MA 313

Abstract:

The H_infinity norm of a transfer matrix function for a continuous-time control system is the maximum of the norm of the transfer matrix on the imaginary axis, or equivalently, the reciprocal of the largest value of epsilon such that the associated epsilon-spectral value set is contained in the left half-plane. We start by defining spectral value sets and discussing some of their fundamental properties, including the intricate relationship between the singular vectors of the transfer matrix and the eigenvectors of the corresponding perturbed system matrix. We then introduce an iterative method for approximating the epsilon-spectral value set abscissa (the maximum of the real part of the points in the set), characterizing the fixed points of the iteration, and explain how the procedure can be combined with a Newton-bisection outer iteration to approximate the H_infinity norm. We then explain why this idealized algorithm sometimes breaks down and introduce a method called hybrid expansion-contraction to address this deficiency. Under reasonable assumptions, the new algorithm finds locally maximal values of the norm of the transfer matrix on the imaginary axis and although these are only lower bounds on the H_infinity norm, it typically finds good approximations in cases where we can test this. It is much faster than the standard Boyd-Balakrishnan-Bruinsma-Steinbuch algorithm to compute the H_infinity norm when the system matrices are large and sparse. The main work required by the algorithm is the computation of the rightmost eigenvalues of a sequence of matrices that are rank-one perturbations of a sparse matrix. Joint work with Nicola Guglielmi (L’Aquila), Mert Gürbüzbalaban (MIT) and Tim Mitchell (NYU).

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

### Ernst Stephan (U Hannover)

**Time-domain BEM for acoustic problems - sound radiation from tyres
**

Dienstag, den 2.12.2014, 16:15 Uhr in MA 313

Abstract:

We investigate a time-domain Galerkin boundary element (TDBEM) method for the wave equation outside a Lipschitz obstacle in an (absorbing) half-space, with application to the sound radiation of tyres. A priori estimates are presented for both closed surfaces and screens, and we discuss the relevant properties of anisotropic Sobolev spaces and the boundary integral operators between them. Based on the mapping properties of these retarded potentials (weakly singular/hypersingular), we derive a posteriori error estimates for the space-time Galerkin scheme for Dirichlet, Neumann and acoustic problems together with adaptive boundary element schemes. The applicability of the TDBEM is demonstrated with numerical simulations for the sound radiation of car and truck tyres showing physical relevant features like Horn and Doppler effect.

### Sina Ober-Blöbaum (U Paderborn, FU Berlin)

**Variational integrators in optimal control theory**

Dienstag, den 9.12.2014, 16:30 Uhr in MAR 0.009

Abstract:

This talk focuses on different aspects of geometric integration and their use for numerical optimal control methods. Geometric integrators are structure-peserving integrators with the goal to capture the dynamical system's behavior in a most realistic way. Using structure-preserving methods for the simulation of mechanical systems, specific properties of the underlying system are handed down to the numerical solution, for example, the energy of a conservative system shows no numerical drift or momentum maps induced by symmetries are preserved exactly. One particular class of geometric integrators is the class of variational integrators. They are derived from a discrete variational principle based on a discrete action function that approximates the continuous one. The resulting schemes are symplectic-momentum conserving and exhibit good energy behaviour.

For the numerical solution of optimal control problems, direct methods are based on a discretization of the underlying differential equations which serve as equality constraints for the resulting finite dimensional nonlinear optimization problem. For the case of mechanical systems, the presented method, denoted by DMOC (Discrete Mechanics and Optimal Control), is based on the discretization of the variational structure of the system directly. Besides the structure preserving properties that are handed down to the optimal control algorithm, we derive approximation and convergence properties of the algorithm. For example, it can be shown, that the approximation order of the adjoint equations resulting from the necessary optimality conditions is the same as for the state equations due to the symplecticity of the discretization scheme.

Various extensions such as higher order variational methods, time-adaptive schemes and multirate integrators are discussed and demonstrated by different examples from mechanical and electrical engineering.

Preceding this talk there will be coffee, tea, and biscuits at 16:00 in room MAR 0.009 - everybody's welcome.

### Aneta Wróblewska-Kamińska (Polish Academy of Sciences)

**Non-Newtonian fluids and abstract problems: applications of Orlicz spaces in the theory of nonlinear PDE**

Dienstag, den 16.12.2014, 16:15 Uhr in MA 313

Abstract:

We are interested in the existence of solutions to strongly nonlinear partial differential equations. We concentrate mainly on problems which come from dynamics of non-Newtonian fluids of a nonstandard rheology, more general then of power-law type, and also on some abstract theory of elliptic and parabolic equations. In considered problems the nonlinear highest order term (stress tensor) is monotone and its behaviour – coercivity/growth condition – is given with help of some general convex function. In our research we would like to cover both cases: sub- and super-linear growth of nonlinearity (shear thickening and shear tinning fluids) as well its anisotropic and non-homogenous behaviour. Such a formulation requires a general framework for the function space setting, therefore we work with non-reflexive and non-separable anisotropic Orlicz and Musielak-Orlicz spaces. Within the presentation we would like to emphasise problems we have met during our studies, their reasons and methods which allow us to achieve existence results.

References

[1] A. Wroblewska-Kaminska. Unsteady flows of non-Newtoniana fluids in generalized Orlicz spaces, Discrete and Continuous Dynamical Systems - A, 33 (2013), no 6, 2565-2592.

[2] E. Emmrich, A. Wroblewska-Kaminska. Convergence of a full discretization of quasilinear parabolic equations in isotropic and anisotropic Orlicz spaces. SIAM Journal on Numerical Analysis, 2013.

[3] A. Wroblewska-Kaminska. Existence result for the motion of several rigid bodies in an incompressible non-Newtonian fluid with growth conditions in Orlicz spaces. Nonlinearity 27 (2014) 685-716.

[4] P. Gwiazda, P. Wittbold, A. Wroblewska, A. Zimmermann. Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces. Journal of Differential Equations, 253 (2012) 635–666.

[5] P. Gwiazda, P. Minakowski, A. Wroblewska-Kaminska. Elliptic problems in generalized Orlicz-Musielak spaces. Central European Journal or Mathematics, 10, no. 6 (2012), 2019-2032.

[6] P. Gwiazda, A. Swierczewska-Gwiazda, A. Wroblewska. Generalized Stokes system in Orlicz spaces, Discrete and Continuous Dynamical Systems A, 32 (2012), Issue 6, 2125-2146.

[7] A. Wroblewska. Steady flow of non-Newtonian fluids - monotonicity methods in generalized Orlicz spaces. Nonlinear Analysis, 72 (2010), 4136-4147.

### Stefan Ulbrich (TU Darmstadt)

**Multilevel methods for PDE-constrained optimization based on adaptive discretizations, reduced order models and error estimators**

Dienstag, den 27.01.2015, 16.15 Uhr in MA 313

Abstract

We consider optimization problems governed by partial differential equations with control and/or state constraints. In recent years there has been a significant effort to develop multilevel optimization methods that combine adaptive discretization techniques, reduced order models and a posteriori error estimation in an efficient way. These methods offer the potential to carry out most optimization iterations on comparably cheap discretizations and to solve the optimization problem at computational costs of only a few PDE-simulations.

In this talk we discuss recent developments for multilevel optimization methods. In particular, we consider a multilevel optimization approach that generates a hierarchy of adaptive discretizations during the optimization iteration using adaptive finite-element approximations and reduced order models such as POD. The adaptive refinement strategy is based on a posteriori error estimators for the PDE-constraint, the adjoint equation and the reduced objective function. State constraints are handled by Moreau-Yosida regularization. The resulting optimization method allows to use existing adaptive PDE-solvers and error estimators in a modular way. We demonstrate the efficiency of the approach by numerical examples.

### Christian Engström (Umeå U, Sweden)

**Rational operator functions and applications to photonic crystals**

Dienstag, den 3.02.2015, 16.15 Uhr in MA 313

Abstract

Operator functions with a nonlinear dependence of a spectral parameter describe a large number of processes in science. In particular, when dispersion is present the operator function is nonlinear. This nonlinearity is rational in several areas of physics and here we study functions that have applications in electromagnetic field theory. We establish variational principles and provide two-sided estimates for all the eigenvalues of a class of rational operator functions. Moreover, we apply the new theory to an unbounded operator function used to model photonic crystals. These nano-sized structures can be used to control the flow of light and are for example used in integrated optics. The operator function is discretised with a high order finite element method and several examples illustrate the general theory. In particular, we show the connection between eigenvalue accumulation at the poles and a numerical approximation of the corresponding singular sequence. This talk is based on a joint work with Heinz Langer and Christiane Tretter.

### Yuji Nakatsukasa (U Tokyo)

**Solving nonconvex optimization problems via eigenvalues **

Dienstag, den 24.02.2015, 16.15 Uhr in MA 313

Abstract

While nonconvex optimization problems are generally regarded as difficult, matrix eigenvalue problems form an important class of problems that can be solved efficiently. In this work we advocate solving some nonconvex optimization problems via an eigenvalue problem.

This talk focuses on the trust region subproblem (TRS), which arises in an algorithm for nonlinear programming. TRS is usually solved via an iterative process of solving linear systems or eigenvalue problems. An alternative approach is to use semidefinite programming, but this also involves solving linear systems iteratively. In this work we introduce an algorithm that solves just one generalized eigenvalue problem, or more specifically just one eigenpair. Our algorithm allows for a non-standard norm without a change of variables and is suited both to the dense and the large-sparse cases. Experiments suggest that our algorithm is superior to existing ones both in accuracy and especially efficiency, particularly for large-sparse problems. Time permitting, we discuss how the approach can be extended to problems with general quadratic constraints.

Based on joint work with Satoru Adachi, Satoru Iwata and Akiko Takeda.

### Roman Vershynin (U Michigan)

**Mathematics of networks: advances and challenges**

Dienstag, den 17.03.2015, 16.15 Uhr in MA 313

Abstract

Real-world networks (social, technological, biological) present us with new challenges across many disciplines, including mathematics, statistics, and computer science. Networks are often modeled as random graphs. We will discuss the classical Erdos-Renyi model of random graphs as well as its inhomogeneous version, which allows to model the emergence of clusters in networks. We will then focus on recent advances in the community detection problem for sparse networks. The role of fundamental mathematical concepts and tools will be highlighted, in particular graph Laplacians, matrix-valued concentration inequalities, Grothendieck's inequality and semidefinite programming.