Partial norms and the convergence of general products of matrices

Motivated by the theory of inhomogeneous Markov chains, we determine a sufficient condition for the convergence to 0 of a general product formed from a sequence of real or complex matrices. When the matrices have a common invariant subspace $H$, we give a sufficient condition for the convergence to 0 on $H$ of a general product. Our result is applied to obtain a condition for the weak ergodicity of an inhomogeneous Markov chain. We compare various types of contractions which may be defined for a single matrix, such as paracontraction, $l$--contraction, and $H$--contraction, where $H$ is an invariant subspace of the matrix

Stochastic topology optimisation with hierarchical tensor reconstruction

A novel approach for risk-averse structural topology optimization under uncertainties is presented which takes into account random material properties and random forces. For the distribution of material, a phase field approach is employed which allows for arbitrary topological changes during optimization. The state equation is assumed to be a high-dimensional PDE parametrized in a (finite) set of random variables. For the examined case, linearized elasticity with a parametric elasticity tensor is used. Instead of an optimization with respect to the expectation of the involved random fields, for practical purposes it is important to design structures which are also robust in case of events that are not the most frequent. As a common risk-aware measure, the Conditional Value at Risk (CVaR) is used in the cost functional during the minimization procedure. Since the treatment of such high-dimensional problems is a numerically challenging task, a representation in the modern hierarchical tensor train format is proposed. In order to obtain this highly efficient representation of the solution of the random state equation, a tensor completion algorithm is employed which only required the pointwise evaluation of solution realizations. The new method is illustrated with numerical examples and compared with a classical Monte Carlo sampling approach

Non-intrusive tensor reconstruction for high dimensional random PDEs

This paper examines a completely non-intrusive, sample-based method for the computation of functional low-rank solutions of high dimensional parametric random PDEs which have become an area of intensive research in Uncertainty Quantification (UQ). In order to obtain a generalized polynomial chaos representation of the approximate stochastic solution, a novel black-box rank-adapted tensor reconstruction procedure is proposed. The performance of the described approach is illustrated with several numerical examples and compared to Monte Carlo sampling

Die Steuerreformpläne der neuen Bundesregierung und das Bürgergeld: Eine Simulation von Risiken und Nebenwirkungen

Die in Aussicht gestellten Steuererleichterungen sind das beherrschende Thema der ersten Wochen nach Amtsantritt der neuen Bundesregierung. Die vorliegende Studie enthält eine Simulation der mutmaßlichen Effekte einer Umsetzung des von der FDP vorgeschlagenen Stufentarifs und Bürgergelds auf den Arbeitsmarkt, der damit verbundenen fiskalischen Effekte und der zu erwartenden Verteilungswirkungen. Unsere Analyse zeigt, dass eine Gesamtreform, die alle Einkommensgruppen besser stellt, zwangsläufig zu Lasten der öffentlichen Haushalte geht. Rechnerisch ergeben sich fiskalische Kosten in Höhe von 86 Milliarden Euro pro Jahr. Da auch die Beschäftigungswirkungen nicht uneingeschränkt positiv zu bewerten sind, ist von der Einführung eines dreistufigen Steuersystems in Kombination mit einem Bürgergeld aus ökonomischer Sicht abzuraten. Betrachtet man den Steuerreformvorschlag separat, so ergeben sich positive Beschäftigungseffekte und geringere fiskalische Kosten. Allerdings sind diese mit etwa 86.000 Euro pro neu geschaffenen Vollzeitarbeitsplatz dennoch unverhältnismäßig hoch. Hinzu kommen nicht unproblematische Verteilungswirkungen zugunsten höherer Einkommen. Die geschilderten Wirkungen sind nicht von der konkreten Ausgestaltung des Reformvorschlages abhängig, sondern eher konzeptioneller Art. Sie würden auch dann in ähnlicher Weise zutage treten, wenn der FDP-Vorschlag in veränderter Form umgesetzt würde.Steuerreform, Stufentarif, Bürgergeld, Mikrosimulation, Arbeitsangebot

An EOM-assisted wave-vector-resolving Brillouin light scattering setup

Brillouin light scattering spectroscopy is a powerful technique which incorporates several extensions such as space-, time-, phase- and wave-vector resolution. Here, we report on the improvement of the wave-vector resolution by including an electro-optical modulator. This provides a reference to calibrate the position of the diaphragm hole which is used for wave-vector selection. The accuracy of this calibration is only limited by the accuracy of the wave-vector measurement itself. To demonstrate the validity of the approach the wave vectors of dipole-dominated spin waves excited by a microstrip antenna were measured.Comment: 5 pages, 5 figure