### 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 [1] |

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 |
---|---|---|---|---|---|

3.05.16 | 16:15 | MA 313 | Thorsten Raasch (U Mainz) | Global convergence of generalized
Newton methods in nonsmooth optimization (Abstract) | G. Kutyniok |

10.05.16 | 16:15 | MA 313 | Alberto Valli (U Trento) | Finite elements in electromagnetism
and application to eddy current equations (Abstract) | F. Tröltzsch |

17.05.16 | 16:15 | MA 313 | Martin Schmidt (U Erlangen-Nürnberg) | Mixed-Integer Nonlinear
Optimization of Stationary Gas Transport Problems
(Abstract) | V. Mehrmann |

24.05.16 | 16:15 | MA 313 | Michal Wojtylak (Jagiellonian U Krakow) | Rank two perturbations of operators and matrices
(Abstract) | V. Mehrmann |

31.05.16 | 16:15 | MA 313 | Pierre Weiss (U Toulouse) | Compressed sensing with structured acquisition
(Abstract) | G. Kutyniok |

21.06.16 | 16:15 | MA 313 | Chandrajit Bajaj (U of Texas in Austin) | Scalable Geometric Optimization with Applications to
Prediction of Assemblies (Abstract) | G.
Kutyniok |

12.07.16 | 16:15 | MA 313 | Juergen Fuhrmann (WIAS) | Models and numerics for
Nernst-Planck-Poisson systems with volume constraints
(Abstract) | G. Baerwolff |

26.07.16 | 16:15 | MA 376 | Dmitrii Karp (Far Eastern Federal U, Russia) | Log-concavity and total positivity
for kernels defined by series (Abstract) | O.
Holtz |

1.09.16 | 14:15 | MA 313 | Wolfgang Hackbusch (MPI Leipzig) | Recursive low-rank truncation
of general matrices (Abstract) | H.
Yserentant |

## Rückblick

- Kolloquium ModNumDiff WS 2015/16 [2]
- Kolloquium ModNumDiff SS 2015 [3]
- Kolloquium ModNumDiff WS 2014/15 [4]
- Kolloquium ModNumDiff SS 2014 [5]
- Kolloquium ModNumDiff WS 2013/14 [6]
- Kolloquium ModNumDiff SS 2013 [7]
- Kolloquium ModNumDiff WS 2012/13 [8]
- Kolloquium ModNumDiff SS 2012 [9]
- Kolloquium ModNumDiff WS 2011/12 [10]
- Kolloquium ModNumDiff SS 2011 [11]

### Thorsten Raasch (U Mainz)

**Global
convergence of generalized Newton methods in nonsmooth
optimization**

Dienstag, den 3.05.2016, 16.15 Uhr in MA
313

Abstract

We are concerned with the design of robust numerical methods for optimization problems with nonsmooth objective functions and/or nonsmooth constraints, like those arising from l1 or nuclear norm regularization. In these cases, the associated first-order optimality conditions have an equivalent reformulation as a piecewise smooth system of equations and can in turn be solved by generalized Newton methods. Such schemes have the advantage of locally superlinear or even locally quadratic convergence, and they often permit globalization by exact or inexact line search with respect to a suitable merit functional. Generalized Newton methods have been proposed and investigated since the late 1980s in the context of optimal control with smooth cost functionals and inequality control constraints, both in a finite- and infinite-dimensional setting. However, the analysis of globally convergent generalized Newton methods for nonsmooth regularization problems could be addressed only recently. This is mainly due to the fact that the optimality conditions of l1-regularization problems, e.g., are structurally different from those corresponding to smooth optimal control problems.

In this talk, we review the history of generalized Newton methods, and we discuss the design of semismooth and Bouligand-semismooth Newton schemes for l1-regularized recovery problems. Global convergence of the iteration is achieved by inexact line search with respect to the residual norm of the first-order optimality conditions. Various numerical examples are presented, ranging form image processing applications to parameter identification problems for PDEs. We discuss how to incorporate nonsmooth constraints and how to apply generalized Newton schemes to nuclear norm regularization problems.

The talk is based on joint work with Esther Hans (JGU Mainz), Dirk Lorenz (TU Braunschweig) and Christian Clason (U Duisburg-Essen).

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

### Alberto Valli (U Trento)

**Finite elements in electromagnetism and
application to eddy current equations**

Dienstag, den
10.05.2016, 16.15 Uhr in MA 313

Abstract

One of the most successful methods used in numerical approximation of partial differential equations is without a doubt the finite element method. In this talk we first present which type of finite elements are employed in electromagnetism, and in which way the associated degrees of freedom are chosen. Then we briefly describe how the numerical approximation of Maxwell equations stems from the variational formulation of the problem. Finally, the eddy current approximation of Maxwell equations is introduced, and it is shown how topological concepts enter into play when looking for an efficient numerical solution of this problem.

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

### Martin Schmidt (U Erlangen-Nürnberg)

**Mixed-Integer Nonlinear Optimization of
Stationary Gas Transport Problems**

Dienstag, den
17.05.2016, 16.15 Uhr in MA 313

Abstract

Modeling, simulation, and optimization of gas transport through pipeline systems is currently a highly active field of research. Due to highly nonlinear models of gas physics and engineering as well as discrete-continuous models of controllable network devices, the optimization of pipeline system operation and planning leads to large-scale mixed-integer nonlinear and nonconvex optimization or feasibility problems.

In this talk we discuss a two-stage solution approach for the feasibility problems. The first stage decides on the discrete controls of all active network devices with respect to a coarse approximation of physics and the second stage afterwards validates these controls with respect to detailed gas physics and engineering models. The first stage still is a nonconvex MINLP that we solve using a tailored MIP-based penalty alternating direction method whereas the second stage is a purely continuous NLP that we solve using a sequential approach.

Algorithmic aspects of both stages are discussed and numerical results on large-scale real-world networks show the applicability of the approach.

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

### Michal Wojtylak (Jagiellonian U Krakow)

**Rank two perturbations of
operators and matrices**

Dienstag, den 24.05.2016, 16.15
Uhr in MA 313

Abstract

The rank one perturbations have gained recently much attention. In its simplest version, the problem can be formulated as follows. Suppose that A is a matrix with some given spectrum and Jordan structure. What is then the spectrum of a rank one perturbation A+uv^*?

The problem may be considered in many variations. Frequently it is assumed that the matrix A has a special structure of entries (e.g. real symmetric matrix or Hamiltonian matrix) or one may consider a linear pencils A+zE. Another step forward is to consider rank two perturbations.

The motivation for this research comes from many sources. For example, in numerical analysis many backward errors are modeled with rank one matrices. Furthermore, when modeling an electrical RCL circuit we get linear pencil with a special structure of entries. Changing one parameter of the circuit (e.g. one of the resistors, capacitors or inductors) leads usually to rank two perturbations of the pencil.

In the talk beside elaborating on the motivations we will discuss current results on rank two perturbations of matrices and operators (joint work with A. Kula and J. Wysoczanski, Wroclaw University), paying some attention to problems appearing in the infinite dimensional setting.

### Pierre Weiss (U Toulouse)

**Compressed sensing with structured
acquisition**

Dienstag, den 31.05.2016, 16.15 Uhr in MA
313

Abstract

How to optimally design a sampling pattern given a set of acquisition constraints? This question is mostly unsolved, despite being essential for many applications such as Magnetic Resonance Imaging, X-ray and electron tomography, surface scattering,...

In this talk, I will provide an overview of recent results that I obtained with my collaborators in an attempt to better understand this problem and provide practical solutions for the Iseult project: the largest MRI system in the world. From a theoretical point of view, our study yields a new perspective on the theory of compressed sensing. From a numerical point of view, it opens new challenges on the optimization over spaces of measures.

### Chandrajit Bajaj (U of Texas in Austin)

**Scalable Geometric
Optimization with Applications to Prediction of
Assemblies**

Dienstag, den 21.06.2016, 16.15 Uhr in MA
313

Abstract

Geometric optimization is the computational reduction technique of choice for a wide variety of model selection, ranking and assembly prediction problems. Moreover, optimization occurs naturally for solutions to rigid and flexible geometric shape similarity, complementarity matching problems (e.g. predicting multi-component assemblies, disaster reconstructions etc). The optimization functional is often a multi-dimensional correlation integral while the search space is the product of transformations groups with dimension growth exponential in the number of movable components (e.g. O(3^n) for an n-residue torsionally flexible molecule).

In this talk, I shall dwell on solution of geometric optimization methods that combat the curse of high dimensionality, and also achieve adequate tradeoffs between speed and accuracy. Fast approximate estimations to the geometric similarity or complementarity matching optimization problem take advantageof a new scheme of generating low-discrepancy samplings of the n-product configuration spaces, as well as utilization of approximate non-uniform fast Fourier transforms. As an application, I shall present a general approximation algorithm to predict multi-piece 3D assemblies, and then describe a provably polynomial time approximation scheme (PTAS) for the special case of predicting symmetric 3D spherical shell assemblies, given a constant number of primitive component molecules that make up the asymmetric unit. This prediction optimization is based on a method for generation of congruently tiled spherical arrangements using a new generative class of polyhedra.

### Juergen Fuhrmann (WIAS)

**Models and numerics for
Nernst-Planck-Poisson systems with volume constraints**

Dienstag, den 12.07.2016, 16.15 Uhr in MA 313

Abstract

The Nernst-Planck-Poisson system describes ion transport in electrolytes in a self-consistent electrical field. It allows for numerical simulations of electrolytic systems which are able to resolve the polarization boundary layer close to electrode surfaces. Classically, the Nernst-Planck-Poisson system assumes vanishing molar volumes for ionic species, leading to simulation results which contradict experimental evidence.

The talk reviews a number of past and recent attempts on model improvements which take into account finite molar volumes of ionic species and solvation effects. It presents a a successful finite volume discretization strategy from semiconductor analysis and discusses problem reformulations which allow for its application in the context of electrolyte modeling based on improved Nernst-Planck equations. Special emphasis is made on the proper reflection of qualitative properties of the physical model at the discrete level. Synergies with numerical approaches to semiconductor modeling with generalized carrier statistic functions are discussed. Along with calculation results for benchmark examples, the influence of various model improvements is demonstrated.

### Dmitrii Karp (Far Eastern Federal U, Russia)

**Log-concavity and total
positivity for kernels defined by series**

Dienstag, den
26.07.2016, 16.15 Uhr in MA 376

Abstract

In his celebrated book Karlin proved that certain generating functions of a Polya frequency sequences form Polya frequency kernels. In a series of papers between 2010 and 2015 we observed that various generating functions of log-concave sequences are log-concave in the properly chosen parameter. In the talk we discuss Karlin's theorem and its potential analogues and generalizations. The construction in some particular cases involves some generic polynomials defined in terms of Toeplitz determinants. We also formulate several conjectures regarding location of zero of such polynomials.

Preceding this talk there will be coffee, tea, and biscuits at 15:50 in room MA 376 - everybody's welcome.

### Wolfgang Hackbusch (MPI Leipzig)

**Recursive low-rank truncation of general
matrices**

Dienstag, den 1.09.2016, 14.15 Uhr in MA 313

Abstract

The best approximation of a matrix by a low-rank matrix can be obtained by the singular value decomposition. For large-sized matrices this approach is too costly. Instead one may use a block decomposition. Approximating the small submatrices by low-rank matrices and agglomerating them into a new, coarser block decomposition, one obtains a recursive method. The required computation work is O(rnm) where r is the desired rank and nxm is the size of the matrix. The paper discusses the new error estimates for A-B and M-A where A is the result of the recursive truncation applied to M, while B is the best approximation.

Preceding this talk there will be coffee, tea, and biscuits from 13:45 - everybody's welcome.

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