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Exponential convergence of mixed hp-DGFEM for the incompressible Navier-Stokes equations in R²
(2020)SAM Research ReportIn a polygon Ω ⊂ R2, we consider mixed hp-discontinuous Galerkin approximations of the stationary, incompressible Navier-Stokes equations, subject to no-slip boundary conditions. We use geometrically corner-refined meshes and hp spaces with linearly increasing polynomial degrees. Based on recent results on analytic regularity of velocity field and pressure of Leray solutions in Ω, we prove exponential rates of convergence of the mixed ...Report -
Quantized tensor FEM for multiscale problems: diffusion problems in two and three dimensions
(2020)SAM Research ReportHomogenization in terms of multiscale limits transforms a multiscale problem with n+1 asymptotically separated microscales posed on a physical domain D⊂Rd into a one-scale problem posed on a product domain of dimension (n+1)d by introducing n so-called “fast variables”. This procedure allows to convert n+1 scales in d physical dimensions into a single-scale structure in (n+1)d dimensions. We prove here that both the original, physical ...Report -
Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models
(2020)SAM Research ReportWe study the expression rates of deep neural networks (DNNs for short) for option prices written on baskets of d risky assets, whose log-returns are modelled by a multivariate Lévy process with general correlation structure of jumps. We establish sufficient conditions on the characteristic triplet of the Lévy process X that ensure ε error of DNN expressed option prices with DNNs of size that grows polynomially with respect to O(ε−1), and ...Report -
Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularities
(2020)SAM Research ReportWe prove exponential expressivity with stable ReLU Neural Networks (ReLU NNs) in H1(Ω) for weighted analytic function classes in certain polytopal domains Ω, in space dimension d=2,3. Functions in these classes are locally analytic on open subdomains D⊂Ω, but may exhibit isolated point singularities in the interior of Ω or corner and edge singularities at the boundary ∂Ω. The exponential expression rate bounds proved here imply uniform ...Report -
Higher-order Quasi-Monte Carlo Training of Deep Neural Networks
(2020)SAM Research ReportWe present a novel algorithmic approach and an error analysis leveraging Quasi-Monte Carlo points for training deep neural network (DNN) surrogates of Data-to-Observable (DtO) maps in engineering design. Our analysis reveals higher-order consistent, deterministic choices of training points in the input data space for deep and shallow Neural Networks with holomorphic activation functions such as tanh. These novel training points are proved ...Report -
hp-FEM for reaction-diffusion equations. II: Robust exponential convergence for multiple length scales in corner domains
(2020)SAM Research ReportReport -
Deep ReLU neural network expression for elliptic multiscale problems
(2020)SAM Research ReportReport -
Extrapolated Lattice Rule Integration in Computational Uncertainty Quantification
(2020)SAM Research ReportReport -
Deep ReLU Neural Network Expression Rates for Data-to-QoI Maps in Bayesian PDE Inversion
(2020)SAM Research ReportFor Bayesian inverse problems with input-to-response maps given by well-posed partial differential equations (PDEs) and subject to uncertain parametric or function space input, we establish (under rather weak conditions on the ``forward'', input-to-response maps) the parametric holomorphy of the data-to-QoI map relating observation data 𝛿� to the Bayesian estimate for an unknown quantity of interest (QoI). We prove exponential expression ...Report -
Space–time discontinuous Galerkin approximation of acoustic waves with point singularities
(2020)SAM Research ReportWe develop a convergence theory of space--time discretizations for the linear, 2nd-order wave equation in polygonal domains Ω⊂ℝ2, possibly occupied by piecewise homogeneous media with different propagation speeds. Building on an unconditionally stable space--time DG formulation developed in~\cite{MoPe18}, we (a) prove optimal convergence rates for the space--time scheme with local isotropic corner mesh refinement on the spatial domain, ...Report