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Spurious Quasi-Resonances in Boundary Integral Equationsfor the Helmholtz Transmission Problem
(2021)SAM Research ReportWe consider the Helmholtz transmission problem with piecewise-constant material coefficients, and the standard associated direct boundary integral equations. For certain coefficients and geometries, the norms of the inverses of the boundary integral operators grow rapidly through an increasing sequence of frequencies, even though this is not the case for the solution operator of the transmission problem; we call this phenomenon that of ...Report -
Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies
(2021)SAM Research ReportAs a continuation of the previous works [13, 4, 15], this paper provides several contributions to the mathematical analysis of subwavelength resonances in a high-contrast medium containing N acoustic obstacles. Our approach is based on an exact decomposition formula which reduces the solution of the sound scattering problem to that of a N dimensional linear system, and characterizes resonant frequencies as the solutions to a N-dimensional ...Report -
On the validity of the tight-binding method for describing systems of subwavelength resonators
(2021)SAM Research ReportThe goal of this paper is to relate the capacitance matrix formalism to the tight-binding approximation. By doing so, we open the way to the use of mathematical techniques and tools from condensed matter theory in the mathematical and numerical analysis of metamaterials, in particular for the understanding of their topological properties. We firstly study how the capacitance matrix formalism, both when the material parameters are static ...Report -
Analytic regularity for the Navier-Stokes equations in polygons with mixed boundary conditions
(2021)SAM Research ReportWe prove weighted analytic regularity of Leray-Hopf variational solutions for the stationary, incompressible Navier-Stokes Equations (NSE) in plane polygonal domains, subject to analytic body forces. We admit mixed boundary conditions which may change type at each vertex, under the assumption that homogeneous Dirichlet (''no-slip'') boundary conditions are prescribed on at least one side at each vertex of the domain. The weighted analytic ...Report -
Finite volume methods for the computation of statistical solutions of the incompressible Euler equations
(2021)SAM Research ReportWe present an efficient numerical scheme based on Monte Carlo integration to approximate statistical solutions of the incompressible Euler equations. The scheme is based on finite volume methods, which provide a more flexible framework than previously existing spectral methods for the computation of statistical solutions for incompressible flows. This finite volume scheme is rigorously proven to, under experimentally verifiable assumptions, ...Report -
Long Expressive Memory for Sequence Modeling
(2021)SAM Research ReportWe propose a novel method called Long Expressive Memory (LEM) for learning long-term sequential dependencies. LEM is gradient-based, it can efficiently process sequential tasks with very long-term dependencies, and it is sufficiently expressive to be able to learn complicated input-output maps. To derive LEM, we consider a system of multiscale ordinary differential equations, as well as a suitable time-discretization of this system. For ...Report -
Understanding neural networks with reproducing kernel Banach spaces
(2021)SAM Research ReportCharacterizing the function spaces corresponding to neural networks can provide a way to understand their properties. In this paper we discuss how the theory of reproducing kernel Banach spaces can be used to tackle this challenge. In particular, we prove a representer theorem for a wide class of reproducing kernel Banach spaces that admit a suitable integral representation and include one hidden layer neural networks of possibly infinite ...Report