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Weighted analytic regularity for the integral fractional Laplacian in polyhedra
(2023)SAM Research ReportOn polytopal domains in 3D, we prove weighted analytic regularity of solutions to the Dirichlet problem for the integral fractional Laplacian with analytic right-hand side. Employing the Caffarelli-Silvestre extension allows to localize the problem and to decompose the regularity estimates into results on vertex, edge, face, vertex-edge, vertex-face, edge-face and vertex-edge-face neighborhoods of the boundary. Using tangential differentiability ...Report -
Deep Operator Network Approximation Rates for Lipschitz Operators
(2023)SAM Research ReportWe establish universality and expression rate bounds for a class of neural Deep Operator Networks (DON) emulating Lipschitz (or Hölder) continuous maps \(\mathcal G:\mathcal X\to\mathcal Y\) between (subsets of) separable Hilbert spaces \(\mathcal X\), \(\mathcal Y\). The DON architecture considered uses linear encoders \(\mathcal E\) and decoders \(\mathcal D\) via (biorthogonal) Riesz bases of \(\mathcal X\), \(\mathcal Y\), and an ...Report -
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Close-to-touching acoustic subwavelength resonators: eigenfrequency separation and gradient blow-up
(2020)SAM Research ReportIn this paper, we study the behaviour of the coupled subwavelength resonant modes when two high-contrast acoustic resonators are brought close together. We consider the case of spherical resonators and use bispherical coordinates to derive explicit representations for the capacitance coefficients which, we show, capture the system's resonant behaviour at leading order. We prove that the pair of resonators has two subwavelength resonant ...Report -
Science in the Swiss Public. The State of Science Communication and Public Engagement with Science in Switzerland
(2021)Swiss Academies ReportsScience communication and public engagement with science have repeatedly been called for in recent years, particularly during the COVID-19 pandemic. Therefore, die Swiss Academies of the Arts and Sciences have set up an expert group to assess the state of science communication in Switzerland, and to provide recommendations for how to improve it. The expert group report is based on a comprehensive review of the available interdisciplinary ...Report -
Force Computation for Dielectrics using Shape Calculus
(2022)SAM Research ReportWe are concerned with the numerical computation of electrostatic forces/torques in only piecewise homogeneous materials using the boundary element method (BEM). Conventional force formulas based on the Maxwell stress tensor yield functionals that fail to be continuous on natural trace spaces. Thus their use in conjunction with BEM incurs slow convergence and low accuracy. We employ the remedy discovered in {\sc P.~Panchal and R.~Hiptmair}, ...Report -