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Invariant Set of the Lorenz System Show image information
Attractor of the Mackey-Glass Equation Show image information
Unstable manifold of the Kuramoto-Sivashinsky equation Show image information
Petersen Graph Show image information

Invariant Set of the Lorenz System

Attractor of the Mackey-Glass Equation

Unstable manifold of the Kuramoto-Sivashinsky equation

Petersen Graph

Prof. Dr. Stefan Volkwein (Konstanz): "POD Reduced-Order Methods in Optimal Control and Design"
16:45 – D2

Abstract: Optimal control problems for partial differential equations (PDEs) are often hard to tackle numerically because their discretization by, e.g., finite elements or finite differences leads to very large scale optimization problems. Therefore, different techniques of model reduction were developed to approximate these problems by smaller ones that are tractable with less effort. Currently, Proper orthogonal decomposition (POD) is probably the mostly used and most successful model reduction technique for nonlinear optimal control problems, where the basis functions contain information from the solutions of the dynamical system at pre-specified time-instances, so-called snapshots. Due to a possible linear dependence or almost linear dependence the snapshots themselves are not appropriate as a basis. Hence a singular value decomposition is carried out and the leading generalized eigenfunctions are chosen as a basis, referred to as the POD basis. Therefore, a useful application is the use in optimization problems, where a PDE solver is part of the function evaluation. Obviously, thinking of a gradient evaluation or even a step-size rule in the optimization algorithm, an expensive function evaluation leads to an enormous amount of computing time. Here, the reduced order model can replace the system given by a PDE in the objective function. It is quite common that a PDE can be replaced by a five- or ten-dimensional system of ordinary differential equations. This results computationally in a very fast method for optimization compared to the effort for the computation of a single solution of a PDE.

In the talk we consider different optimization problems for PDEs (involving multiobjective optimal control problems) and discuss a-priori as well as a-posteriori error analysis for the reduced-order approximations.

The presented results are joint work with Stefan Banholzer, Dennis Beermann, Michael Dellnitz, Luca Mechelli and Sebastian Peitz. 

Dr. Janosch Rieger (Melborne): "Recent advances in domain reconstruction from electrical impedance tomography data"
16:15 – D1

Abstract: Electrical impedance tomography is an emerging budget-priced, non-invasive medical imaging technique that is very likely to complement computerised tomography in important applications such as pulmonary function control and breast cancer screening in the future. The main difficulty associated with this technology is that the arising inverse problem is strongly ill-posed.

In this talk, I will discuss an alternative approach to domain reconstruction from electrical impedance tomography data, which is based on the concept of the convex source support introduced by Kusiak and Sylvester, as well as an appropriate numerical discretisation of the resulting problem.

Dr. Stefanie Hittmeyer (Auckland): "The geometry of blenders in a three-dimensional Hénon-like family"
15:30 - C2

Abstract: Blenders are a geometric tool to construct complicated dynamics in diffeomorphisms of dimension at least three and vector fields of dimension at least four. They admit invariant manifolds that behave like geometric objects which have dimensions higher than expected from the manifolds themselves. We consider an explicit family of three-dimensional Hénon-like maps that exhibit blenders in a specific regime in parameter space. Using advanced numerical techniques we compute stable and unstable manifolds in this system, enabling us to show one of the first numerical pictures of the geometry of blenders. We furthermore present numerical evidence suggesting that the regime of existence of the blenders extends to a larger region in parameter space.

Adrian Ziessler: "The Computation of Invariant Manifolds for Partial Differential Equations by Set Oriented Numerics"
16:00 - D1

Prof. Dr. Eyke Hüllermeier: Machine Learning: Ideas, Concepts, and Selected Problems
16:00 - D1

Abstract: The field of machine learning has developed quite dynamically during the last decades. Today, it constitutes one of the key pillars of contemporary artificial intelligence and is at the core of the emerging field of data science.  Not less importantly, machine learning is entering more and more application domains, from industry and politics to personalization of Internet search, online shopping, and medical treatment. This talk starts with a short introduction to the basic ideas and principles of machine learning. In the second part, a selection of concrete problems within the realm of online learning and preference learning will be presented and discussed in a slightly more detailed way.

Adrian Ziessler: "The Computation of Invariant Manifolds for Partial Differential Equations by Set Oriented Numerics"
16:00 - TP21

Raphael Gerlach: "Symmetries in Graphs"
16:00 - D1

Further information:

The University for the Information Society