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Tunable Localisation in Parity-Time-Symmetric Resonator Arrays with Imaginary Gauge Potentials
(2024)SAM Research ReportThe aim of this paper is to illustrate both analytically and numerically the interplay of two fundamentally distinct non-Hermitian mechanisms in a deep subwavelength regime. Considering a parity-time symmetric system of one-dimensional subwavelength resonators equipped with two kinds of non-Hermiticity — an imaginary gauge potential and on-site gain and loss — we prove that all but two eigenmodes of the system pass through exceptional ...Report -
Regularized dynamical parametric approximation
(2024)SAM Research ReportThis paper studies the numerical approximation of evolution equations by nonlinear parametrizations \(u(t)=\Phi(q(t))\) with time-dependent parameters \(q(t)\), which are to be determined in the computation. The motivation comes from approximations in quantum dynamics by multiple Gaussians and approximations of various dynamical problems by tensor networks and neural networks. In all these cases, the parametrization is typically irregular: ...Report -
Exponential Convergence of hp-ILGFEM for semilinear elliptic boundary value problems with monomial reaction
(2024)SAM Research ReportWe study the fully explicit numerical approximation of a semilinear elliptic boundary value model problem, which features a monomial reaction and analytic forcing, in a bounded polygon \(\Omega\subset\mathbb{R}^2\) with a finite number of straight edges. In particular, we analyze the convergence of \(hp\)-type iterative linearized Galerkin (\(hp\)-ILG) solvers. Our convergence analysis is carried out for conforming \(hp\)-finite element ...Report -
Exponential Expressivity of ReLU^k Neural Networks on Gevrey Classes with Point Singularities
(2024)SAM Research ReportWe analyze deep Neural Network emulation rates of smooth functions with point singularities in bounded, polytopal domains \(\mathrm{D} \subset \mathbb{R}^d\), \(d=2,3\). We prove exponential emulation rates in Sobolev spaces in terms of the number of neurons and in terms of the number of nonzero coefficients for Gevrey-regular solution classes defined in terms of weighted Sobolev scales in \(\mathrm{D}\), comprising the countably-normed ...Report -
Banach lattices with upper p-estimates: Free and injective objects
(2024)SAM Research ReportWe study the free Banach lattice FBL(p,∞)[E] with upper p-estimates generated by a Banach space E. Using a classical result of Pisier on factorization through Lp,∞(μ) together with a finite dimensional reduction, it is shown that the spaces ℓp,∞(n) witness the universal property of FBL(p,∞)[E] isomorphically. As a consequence, we obtain a functional representation for FBL(p,∞)[E], answering a previously open question. More generally, our ...Report -
A Two-Scale Effective Model for Defect-Induced Localization Transitions in Non-Hermitian Systems
(2024)SAM Research ReportWe illuminate the fundamental mechanism responsible for the transition between the non-Hermitian skin effect and defect-induced localization in the bulk. We study a Hamiltonian with non-reciprocal couplings that exhibits the skin effect (the localization of all eigenvectors at one edge) and add an on-site defect in the center. Using a two-scale asymptotic method, we characterize the long-scale growth and decay of the eigenvectors and ...Report -
Coupled Boundary and Volume Integral Equations for Electromagnetic Scattering
(2024)SAM Research ReportWe study frequency domain electromagnetic scattering at a bounded, penetrable, and inhomogeneous obstacle \(\Omega \subset \mathbb{R}^3\). From the Stratton-Chu integral representation, we derive a new representation formula when constant reference coefficients are given for the interior domain. The resulting integral representation contains the usual layer potentials, but also volume potentials on \(\Omega\). Then it is possible to ...Report -
Wavelet compressed, modified Hilbert transform in the space-time discretization of the heat equation
(2024)SAM Research ReportOn a finite time interval \((0,T)\), we consider the multiresolution Galerkin discretization of a modified Hilbert transform \((H_T)\) which arises in the space-time Galerkin discretization of the linear diffusion equation. To this end, we design spline-wavelet systems in \((0,T)\) consisting of piecewise polynomials of degree \(\geq 1\) with sufficiently many vanishing moments which constitute Riesz bases in the Sobolev spaces \( ...Report -
Time-dependent electromagnetic scattering from dispersive materials
(2024)SAM Research ReportThis paper studies time-dependent electromagnetic scattering from metamaterials that are described by dispersive material laws. We consider the numerical treatment of a scattering problem in which a dispersive material law, for a causal and passive homogeneous material, determines the wave-material interaction in the scatterer. The resulting problem is nonlocal in time inside the scatterer and is posed on an unbounded domain. Well-posedness ...Report -
The Language of Hyperelastic Materials
(2024)SAM Research ReportThe automated discovery of constitutive laws forms an emerging area that focuses on automatically obtaining symbolic expressions describing the constitutive behavior of solid materials from experimental data. Existing symbolic/sparse regression methods rely on availability of libraries of material models, which are typically hand-designed by a human expert relying on known models as reference, or deploy generative algorithms with exponential ...Report