Search
Results
-
Homogenization and Numerical Upscaling for Spectral Fractional Diffusion
(2024)SAM Research ReportWe consider two-scale, linear spectral fractional diffusion of order \(2s\in (0,2)\) with homogeneous Dirichlet boundary condition and locally periodic, two-scale coefficients in a bounded domain \(D \subset \mathbb{R}^d\), with fundamental period \(Y=(0,1)^d \subset \mathbb{R}^d\). We derive a local limiting two-scale homogenized equation for the so-called Caffarelli-Sylvestre (CS) extension in the tensorized domain \(D\times Y \times ...Report -
Expression Rates of Neural Operators for Linear Elliptic PDEs in Polytopes
(2024)SAM Research ReportWe study the approximation rates of a class of deep neural network approximations of operators, which arise as data-to-solution maps G † of linear elliptic partial differential equations (PDEs), and act between pairs X, Y of suitable infinite-dimensional spaces. We prove expression rate bounds for approximate neural operators G with the structure G = R ◦ A ◦ E, with linear encoders E and decoders R. The constructive proofs are via a ...Report -
Frequency-Explicit Shape Holomorphy in Uncertainty Quantification for Acoustic Scattering
(2024)SAM Research ReportWe consider frequency-domain acoustic scattering at a homogeneous star-shaped penetrable ob stacle, whose shape is uncertain and modelled via a radial spectral parameterization with random coefficients. Using recent results on the stability of Helmholtz transmission problems with piece wise constant coefficients from [A. Moiola and E. A. Spence, Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions, ...Report -
Deep ReLU Neural Network Emulation in High-Frequency Acoustic Scattering
(2024)SAM Research ReportWe obtain wavenumber-robust error bounds for the deep neural network (DNN) emulation of the solution to the time-harmonic, sound-soft acoustic scattering problem in the exterior of a smooth, convex obstacle in two physical dimensions. The error bounds are based on a boundary reduction of the scattering problem in the unbounded exterior region to its smooth, curved boundary Γ using the so-called combined field integral equation (CFIE), a ...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 -
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 -
Exponential expressivity of ${\rm ReLU}^k$ 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 -
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 -
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 -
The Gevrey class implicit mapping theorem with application to UQ of semilinear elliptic PDEs
(2023)SAM Research ReportThis article is concerned with a regularity analysis of parametric operator equations with a perspective on uncertainty quantification. We study the regularity of mappings between Banach spaces near branches of isolated solutions that are implicitly defined by a residual equation. Under \(s\)-Gevrey assumptions on on the residual equation, we establish \(s\)-Gevrey bounds on the Fréchet derivatives of the local data-to-solution mapping. ...Report