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Gratwanderung der Schweizer Wirtschaft
(2020)KOF StudiesDer Ausbruch der COVID19Pandemie führte in der Schweiz zu dem gravierendsten Einbruch der Wirtschaftstätigkeit seit über vier Jahrzehnten. Der achtwöchige Lockdown im Zeitraum zwischen März und Mai schlug sich stark auf die Schweizer Wirtschaft nieder und lähmte teilweise ganze Wirtschaftszweige. Zudem konnte sich die Schweiz als kleine, offene Volkswirtschaft der weltwirt¬schaftlichen Schockstarre und dem damit verbundenen Ausfall der ...Report 
On the conservation of energy in twodimensional incompressible flows
(2020)SAM Research ReportWe prove the conservation of energy for weak and statistical solutions of the twodimensional Euler equations, generated as strong (in an appropriate topology) limits of the underlying NavierStokes equations and a Monte CarloSpectral Viscosity numerical approximation, respectively. We characterize this conservation of energy in terms of a uniform decay of the socalled structure function, allowing us to extend existing results on energy ...Report 
Counterexamples to local Lipschitz and local Hölder continuity with respect to the initial values for additive noise driven stochastic differential equations with smooth drift coefficient functions with at most polynomially growing derivatives
(2020)SAM Research ReportIn the recent article [A. Jentzen, B. Kuckuck, T. MüllerGronbach, and L. Yaroslavtseva, arXiv:1904.05963 (2019)] it has been proved that the solutions to every additive noise driven stochastic differential equation (SDE) which has a drift coefficient function with at most polynomially growing first order partial derivatives and which admits a Lyapunovtype condition (ensuring the the existence of a unique solution to the SDE) depend in ...Report 
On the approximation of rough functions with deep neural networks
(2020)SAM Research ReportDeep neural networks and the ENO procedure are both efficient frameworks for approximating rough functions. We prove that at any order, the ENO interpolation procedure can be cast as a deep ReLU neural network. This surprising fact enables the transfer of several desirable properties of the ENO procedure to deep neural networks, including its highorder accuracy at approximating Lipschitz functions. Numerical tests for the resulting neural ...Report 
Space–time discontinuous Galerkin approximation of acoustic waves with point singularities
(2020)SAM Research ReportWe develop a convergence theory of spacetime discretizations for the linear, 2ndorder wave equation in polygonal domains Ω⊂ℝ2, possibly occupied by piecewise homogeneous media with different propagation speeds. Building on an unconditionally stable spacetime DG formulation developed in~\cite{MoPe18}, we (a) prove optimal convergence rates for the spacetime scheme with local isotropic corner mesh refinement on the spatial domain, ...Report 
Localized sensitivity analysis at highcurvature boundary points of reconstructing inclusions in transmission problems
(2020)SAM Research ReportIn this paper, we are concerned with the recovery of the geometric shapes of inhomogeneous inclusions from the associated far field data in electrostatics and acoustic scattering. We present a local resolution analysis and show that the local shape around a boundary point with a high magnitude of mean curvature can be reconstructed more easily and stably. In proving this, we develop a novel mathematical scheme by analyzing the generalized ...Report 
Mathematical modelling of plasmonic strain sensors
(2020)SAM Research ReportWe provide a mathematical analysis for a metasurface constructed of plasmonic nanoparticles mounted periodically on the surface of a microcapsule. We derive an effective transmission condition, which exhibits resonances depending on the interparticle distance. When the microcapsule is deformed, the resonances are shifted. We fully characterise the dependence of these resonances on the deformation of the microcapsule, enabling the detection ...Report 
Quantum ergodicity and localization of plasmon resonances
(2020)SAM Research ReportWe are concerned with the geometric properties of the surface plasmon resonance (SPR). SPR is a nonradiative electromagnetic surface wave that propagates in a direction parallel to the negative permittivity/dielectric material interface. It is known that the SPR oscillation is topologically very sensitive to the material interface. However, we show that the SPR oscillation asympotically localizes at places with high magnitude of curvature ...Report 
QuotientSpace Boundary Element Methods for Scattering at Complex Screens
(2020)SAM Research ReportA complex screen is an arrangement of panels that may not be even locally orientable because of junction lines. A comprehensive trace space framework for firstkind variational boundary integral equations on complex screens has been established in [ X. Claeys and R. Hiptmair, Integral equations on multiscreens, Integral Equations and Operator Theory, 77 (2013), pp. 167197] for the Helmholtz equation, and in [ X. Claeys and R. Hiptmair, ...Report 
Deep ReLU Neural Network Expression Rates for DatatoQoI Maps in Bayesian PDE Inversion
(2020)SAM Research ReportFor Bayesian inverse problems with inputtoresponse maps given by wellposed partial differential equations (PDEs) and subject to uncertain parametric or function space input, we establish (under rather weak conditions on the ``forward'', inputtoresponse maps) the parametric holomorphy of the datatoQoI map relating observation data 𝛿� to the Bayesian estimate for an unknown quantity of interest (QoI). We prove exponential expression ...Report