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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 -
Perturbed Block Toeplitz matrices and the non-Hermitian skin effect in dimer systems of subwavelength resonators
(2023)SAM Research ReportThe aim of this paper is fourfold: (i) to obtain explicit formulas for the eigenpairs of perturbed tridiagonal block Toeplitz matrices; (ii) to make use of such formulas in order to provide a mathematical justification of the non-Hermitian skin effect in dimer systems by proving the condensation of the system's bulk eigenmodes at one of the edges of the system; (iii) to show the topological origin of the non-Hermitian skin effect for dimer ...Report -
An Antithetic Multilevel Monte Carlo-Milstein Scheme for Stochastic Partial Differential Equations
(2023)SAM Research ReportWe present a novel multilevel Monte Carlo approach for estimating quantities of interest for stochastic partial differential equations (SPDEs). Drawing inspiration from [Giles and Szpruch: Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation, Annals of Appl. Prob., 2014], we extend the antithetic Milstein scheme for finite-dimensional stochastic differential equations to Hilbert space-valued ...Report -
Coupled Surface and Volume Integral Equations for Electromagnetism
(2023)SAM Research ReportWe study frequency domain electromagnetic scattering at a bounded, penetrable, and inhomogeneous obstacle. By defining constant reference coefficients, a new representation formula for interior and exterior vector fields is proposed, based on the general form of the Stratton-Chu integral representation. The final integral equation system consists of surface integral operators arising from a Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) ...Report -
Deep ReLU networks and high-order finite element methods II: Chebyshev emulation
(2023)SAM Research ReportExpression rates and stability in Sobolev norms of deep ReLU neural networks (NNs) in terms of the number of parameters defining the NN for continuous, piecewise polynomial functions, on arbitrary, finite partitions \(\mathcal{T}\) of a bounded interval \((a,b)\) are addressed. Novel constructions of ReLU NN surrogates encoding the approximated functions in terms of Chebyshev polynomial expansion coefficients are developed. Chebyshev ...Report -
An operator preconditioning perspective on training in physics-informed machine learning
(2023)SAM Research ReportIn this paper, we investigate the behavior of gradient descent algorithms in physics-informed machine learning methods like PINNs, which minimize resid uals connected to partial differential equations (PDEs). Our key result is that the difficulty in training these models is closely related to the conditioning of a specific differential operator. This operator, in turn, is associated to the Hermi tian square of the differential operator ...Report -
Stability of the non-Hermitian skin effect
(2023)SAM Research ReportThis paper shows that the skin effect in systems of non-Hermitian subwavelength resonators is robust with respect to random imperfections in the system. The subwavelength resonators are highly contrasting material inclusions that resonate in a low-frequency regime. The non-Hermiticity is due to the introduction of an imaginary gauge potential, which leads to a skin effect that is manifested by the system's eigenmodes accumulating at one ...Report -
Edge modes in subwavelength resonators in one dimension
(2023)SAM Research ReportWe present the mathematical theory of one-dimensional infinitely periodic chains of subwavelength resonators. We analyse both Hermitian and non-Hermitian systems. Subwavelength resonances and associated modes can be accurately predicted by a finite dimensional eigenvalue problem involving a capacitance matrix. We are able to compute the Hermitian and non-Hermitian Zak phases, showing that the former is quantised and the latter is not. ...Report -
A-posteriori QMC-FEM error estimation for Bayesian inversion and optimal control with entropic risk measure
(2023)SAM Research ReportWe propose a novel a-posteriori error estimation technique where the target quantities of interest are ratios of high-dimensional integrals, as occur e.g. in PDE constrained Bayesian inversion and PDE constrained optimal control subject to an entropic risk measure. We consider in particular parametric, elliptic PDEs with affine-parametric diffusion coefficient, on high-dimensional parameter spaces. We combine our recent a-posteriori ...Report