### 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 14.04.15 | 16:15 | MA 313 | Felix Krahmer (TU München) | Sparsity models in compressed sensing (Abstract) | G. Kutyniok |

Di 28.04.15 | 16:15 | MA 313 | Sonja Cox (U Amsterdam) | Numerical simulations for stochastic PDEs (Abstract) | R. Kruse |

Di 5.05.15 | 16:15 | MA 313 | Massimo Fornasier (TU München) | Sparse Mean-Field Optimal Control (Abstract) | G. Kutyniok |

Di 12.05.15 | 16:15 | MA 313 | Elias Jarlebring (KTH Stockholm) | The infinite Arnoldi method and applications (Abstract) | A. Miedlar V. Mehrmann |

Di 19.05.15 | 16:15 | MA 313 | Robert Lipton (Louisiana State U) | Cohesive Dynamics and Fracture (Abstract) | E. Emmrich |

Di 2.06.15 | 16:15 | MA 313 | Zlatko Drmac (U Zagreb) | to be announced | V. Mehrmann |

Di 9.06.15 | 14:45 | MA 313 | Arye Nehorai (Washington U in St. Louis) | Computable Performance Analysis of Sparse Recovery (Abstract) | G. Kutyniok |

Di 9.06.15 | 16:15 | MA 313 | Martin Stoll (MPI Magdeburg) | to be announced | V. Mehrmann A. Miedlar |

Fr 12.06.15 | 11:00 | MA 313 | Enrique Zuazua (BCAM, Bilbao) | to be announced | V. Mehrmann |

Di 16.06.15 | 16:15 | MA 313 | Esther Klann (TU Berlin) | Geometric inverse problems with applications in tomography (Abstract) | -- |

Di 23.06.15 | 16:15 | MA 313 | Bernd Sturmfels (UC Berkeley) | Decomposing Tensors into Frames (Abstract) | G. Kutyniok |

Di 7.07.15 | 16:15 | MA 313 | Wolfgang Wendland (U Stuttgart) | On the Gauss minimal energy problem with Riesz potentials | F. Tröltzsch |

20.10.15 | 16:15 | MA 313 | geheim | Ehrenkolloquium aus Anlaß des 65. Geburtstags von Günter Bärwolff | F. Tröltzsch |

## Rückblick

- Kolloquium ModNumDiff WS 2014/15
- Kolloquium ModNumDiff SS 2014
- Kolloquium ModNumDiff WS 2013/14
- Kolloquium ModNumDiff SS 2013
- Kolloquium ModNumDiff WS 2012/13
- Kolloquium ModNumDiff SS 2012
- Kolloquium ModNumDiff WS 2011/12
- Kolloquium ModNumDiff SS 2011
- Kolloquium ModNumDiff WS 2009/10
- Kolloquium ModNumDiff SS 2009
- Kolloquium ModNumDiff WS 2008
- Kolloquium ModNumDiff SS 2008
- Kolloquium ModNumDiff WS 2007
- Kolloquium ModNumDiff SS 2007
- Kolloquium ModNumDiff WS 2006/07
- Kolloquium ModNumDiff SS 2006
- Kolloquium ModNumDiff WS 2005/06
- Kolloquium ModNumDiff SS 2005

## Abstracts zu den Vorträgen:

### Felix Krahmer (TU München)

**Sparsity models in compressed sensing**

Dienstag, den 14.04.2015, 16.15 Uhr in MA 313

Abstract

The theory of compressed sensing shows that sparse signals, that is, vectors with only a small number of non-vanishing entries, can be recovered from a number of measurements that is considerably less than the signal dimension. Typically the measurements to allow for such guarantees are chosen at random. Considerable efforts have been spent over the last years to study random measurements with additional structural properties imposed by the applications. However, in contrast to generic (for example Gaussian) measurements, such structured random measurement systems are no longer universal, i.e., without additional modifications, the guarantees only hold for certain sparsity bases. This talk discusses two approaches to obtain recovery guarantees when the sparsity basis under consideration does not allow the application of the standard theory. On the one hand one can adjust the sampling distribution (for example in Fourier imaging applications), and on the other hand, one can apply a suitable temporal transform to achieve sparsity in the standard basis for each time instance (for example in photoacoustic tomography applications).

The talk is based on joint works with Rachel Ward and with Michael Sandbichler, Thomas Berer, Peter Burgholzer, and Markus Haltmeier.

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

### Sonja Cox (U Amsterdam)

**Numerical simulations for stochastic PDEs **

Dienstag, den 28.04.2015, 16.15 Uhr in MA 313

Abstract

Various models arising from finance and natural sciences involve stochastic partial differential equations (SPDEs). As the solutions to an SPDE generally cannot be given explicitly, numerical simulations are used to gain insight in their behaviour. In my talk I will explain the challenges encountered here. In particular, I will explain why generally one cannot expect large convergence rates and why non-linear equations pose difficulties that do not occur with deterministic PDEs. Finally, I will explain my recent results concerning approximations to non-linear S(P)DEs.

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

### Massimo Fornasier (TU München)

**Sparse Mean-Field Optimal Control**

Dienstag, den 5.05.2015, 16.15 Uhr in MA 313

Abstract

Starting with the seminal papers of Reynolds (1987), Vicsek et. al. (1995) Cucker-Smale (2007), there has been a flood of recent works on models of self-alignment and consensus dynamics. Self-organization has been so far the main driving concept. However, the evidence that in practice self-organization does not necessarily occur leads to the natural question of whether it is possible to externally influence the dynamics in order to promote the formation of certain desired patterns. Once this fundamental question is posed, one is also faced with the issue of defining the best way of obtaining the result, seeking for the most “economical” manner to achieve a certain outcome. The first part of this talk precisely addresses the issue of finding the sparsest control strategy for finite dimensional models in order to lead the dynamics optimally towards a given outcome. In the second part of the talk we introduce the rigorous limit process connecting finite dimensional sparse optimal control problems with ODE constraints to an infinite dimensional optimal control problem with a constraint given by a system of ODE for the leaders coupled with a PDE of Vlasov-type, governing the dynamics of the probability distribution of the followers. The technical derivation of the sparse mean-field optimal control is realized by the simultaneous development of the mean-field limit of the equations governing the followers dynamics together with the Gamma-limit of the finite dimensional sparse optimal control problems. Additionally we derive corresponding first order optimality conditions for the infinite dimensional optimal control problem in the form of Hamiltonian ﬂows in the Wasserstein space of probability measures, which correspond to natural limits of the finite dimensional Pontryagin Maximum principles. We conclude the talk by mentioning recent results in sparse optimal control of high-dimensional dynamical systems.

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

### Elias Jarlebring (KTH Stockholm)

**The infinite Arnoldi method and applications**

Dienstag, den 12.05.2015, 16.15 Uhr in MA 313

Abstract

The infinite Arnoldi method [A linear eigenvalue algorithm for the nonlinear eigenvalue problem, Numer. Math., 2012] is a recent approach which is equivalent to the Arnoldi method, but applicable to different types of nonlinear problems. It was originally developed for the nonlinear eigenvalue problem, with an analytic dependence on the spectral parameter, and we have recently shown that the construction can also be used to solve certain types of ODEs in the setting exponential integrators [An infinite Arnoldi exponential integrator for inhomogeneous linear ODEs, Koskela, J., arxiv preprint, 2015]. We present improvements, e.g., structure exploiting restarting, tensor representations, low-rank exploitation, for the infinite Arnoldi method, as well as different applications.

### Robert Lipton (Louisiana State U)

**Cohesive Dynamics and Fracture**

Dienstag, den 19.05.2015, 16.15 Uhr in MA 313

Abstract

Dynamic brittle fracture is a multiscale phenomena operating across a wide range of length and time scales. Apply enough stress or strain to a sample of brittle material and one eventually snaps bonds at the atomistic scale leading to fracture of the macroscopic specimen. At present there is a growing demand for new fracture models capable of predicting complex fracture patterns inside materials used in modern infrastructure. The peridynamic formulation introduced in the work of Silling 2000 is a promising method for modeling free crack propagation. Here we work in the peridynamic formulation and introduce a new type of nonlocal, nonlinear, cohesive continuum model for assessing the deformation state inside a cracking body. In this model short-range forces between material points are initially elastic and then become unstable and soften beyond a critical relative displacement. The dynamics inside the deforming body selects whether a material point lies inside or outside the "process zone" associated with nonlinear behavior corresponding to softening. This is in contrast to a classic cohesive zone fracture model that collapses the process zone onto predetermined surfaces and assumes linear elastic deformation away from these surfaces. Here the natural length scale that controls the size of the process zone is the radius of nonlocal interaction between material points. An explicit inequality is identified showing how the length scale of nonlocal interaction controls the volume of the process zone. The volume of the process zone is shown to vanish as the nonlocal interaction distance is decreased to zero. We apply Gamma convergence arguments coming from the theory of image processing to find that the limiting dynamics has an energy density associated with a process zone confined to a surface. Distinguished limits of cohesive evolutions are identified and are found to have both bounded linear elastic energy and Griffith surface energy. The limit dynamics corresponds to the simultaneous evolution of linear elastic displacement described by the classic wave equation together with a fracture set across which the displacement is discontinuous.

### Arye Nehorai (Washington U in St. Louis)

**Computable Performance Analysis of Sparse Recovery**

Dienstag, den 9.06.2015, 14.45 Uhr in MA 313

Abstract

The last decade has witnessed burgeoning developments in the reconstruction of signals based on exploiting their low-dimensional structures, particularly their sparsity, block-sparsity, and low-rankness. The reconstruction performance of these signals is heavily dependent on the structure of the operating matrix used in sensing. The quality of these matrices in the context of signal recovery is usually quantified by the restricted isometry constant and its variants. However, the restricted isometry constant and its variants are extremely difficult to compute.

We present a framework for analytically computing the performance of the recovery of signals with sparsity structures. We define a family of incoherence measures to quantify the goodness of arbitrary sensing matrices. Our primary contribution is the design of efficient algorithms, based on linear programming and second order cone programming, to compute these incoherence measures. As a by-product, we implement efficient algorithms to verify sufficient conditions for exact signal recovery in the noise-free case. The utility of the proposed incoherence measures lies in their relationship to the performance of reconstruction methods. We derive closed-form expressions of bounds on the recovery errors of convex relaxation algorithms in terms of these measures.

Preceding this talk there will be coffee, tea, and biscuits at 14:30 in room MA 315 - everybody's welcome.

### Esther Klann (TU Berlin)

**Geometric inverse problems with applications in tomography**

Dienstag, den 16.06.2015, 16.15 Uhr in MA 313

Abstract

In inverse problems one wants to find unknown parameters or structures by indirect measurements. Typical inverse problems occur in medical imaging, e.g., tomography. The task is to find the inner structure of an object, e.g., a human torso, from line integrals of some form of radiation that traveled through the object. In geometric inverse problems one wants to recover this inner structure also in terms of geometric information, i.e., the number of different interior objects (bone, liver, lung, etc.), their location and their shape. We present a Mumford-Shah type functional for the simultaneous reconstruction and segmentation of an object from tomography data. The minimization of the functional is realized using shape sensitivity analysis and carried out in the level-set framework. We show reconstructions and/or segmentations for several tomographic applications (SPECT/CT, limited angle tomography, region of interest CT). We also give a brief theoretical analysis regarding existence and convergence of minimizers of the Mumford-Shah type functional.

### Bernd Sturmfels (UC Berkeley)

**Decomposing Tensors into Frames**

Dienstag, den 23.06.2015, 16.15 Uhr in MA 313

Abstract

A symmetric tensor of small rank decomposes into a configuration of only few vectors. We study the variety of tensors for which this configuration is a unit norm tight frame. This is joint work with Luke Oeding and Elina Robeva, aimed at bridging the gap between frame theory and applied algebraic geometry.