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Neural Networks for Singular Perturbations
(2024)SAM Research ReportWe prove deep neural network (DNN for short) expressivity rate bounds for solution sets of a model class of singularly perturbed, elliptic two-point boundary value problems, in Sobolev norms, on the bounded interval (−1,1). We assume that the given source term and reaction coefficient are analytic in [−1,1]. We establish expression rate bounds in Sobolev norms in terms of the NN size which are uniform with respect to the singular perturbation ...Report -
Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning
(2024)SAM Research ReportPhysics-informed neural networks (PINNs) and their variants have been very popular in recent years as algorithms for the numerical simulation of both forward and inverse problems for partial differential equations. This article aims to provide a comprehensive review of currently available results on the numerical analysis of PINNs and related models that constitute the backbone of physics-informed machine learning. We provide a unified ...Report -
Spectra and pseudo-spectra of tridiagonal k-Toeplitz matrices and the topological origin of the non-Hermitian skin effect
(2024)SAM Research ReportWe establish new results on the spectra and pseudo-spectra of tridiagonal k-Toeplitz operators and matrices. In particular, we prove the connection between the winding number of the eigenvalues of the symbol function and the exponential decay of the associated eigenvectors (or pseudo-eigenvectors). Our results elucidate the topological origin of the non-Hermitian skin effect in general one-dimensional polymer systems of subwavelength ...Report -
Scattering from time-modulated subwavelength resonators
(2024)SAM Research ReportWe consider wave scattering from a system of highly contrasting resonators with time-modulated material parameters. In this setting, the wave equation reduces to a system of coupled Helmholtz equations that models the scattering problem. We consider the one-dimensional setting. In order to understand the energy of the system, we prove a novel higher-order discrete, capacitance matrix approximation of the subwavelength resonant quasifrequencies. ...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 -
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 -
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 -
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 -
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