
Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularities
We prove exponential expressivity with stable ReLU Neural Networks (ReLU...
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Modewise Tensor Decompositions: Multidimensional Generalizations of CUR Decompositions
Low rank tensor approximation is a fundamental tool in modern machine le...
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Approximation with Tensor Networks. Part I: Approximation Spaces
We study the approximation of functions by tensor networks (TNs). We sho...
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The Elasticity Complex
We investigate the Hilbert complex of elasticity involving spaces of sym...
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Analyticity and hp discontinuous Galerkin approximation of nonlinear Schrödinger eigenproblems
We study a class of nonlinear eigenvalue problems of Scrödinger type, wh...
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Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise
We consider the numerical approximation of the mild solution to a semili...
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On the approximation of dispersive electromagnetic eigenvalue problems in 2D
We consider timeharmonic electromagnetic wave equations in composites o...
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Tensor Rank bounds for Point Singularities in R^3
We analyze rates of approximation by quantized, tensorstructured representations of functions with isolated point singularities in R^3. We consider functions in countably normed Sobolev spaces with radial weights and analytic or Gevreytype control of weighted seminorms. Several classes of boundary value and eigenvalue problems from science and engineering are discussed whose solutions belong to the countably normed spaces. It is shown that quantized, tensorstructured approximations of functions in these classes exhibit tensor ranks bounded polylogarithmically with respect to the accuracy ϵ∈(0,1) in the Sobolev space H^1. We prove exponential convergence rates of three specific types of quantized tensor decompositions: quantized tensor train (QTT), transposed QTT and TuckerQTT. In addition, the bounds for the patchwise decompositions are uniform with respect to the position of the point singularity. An auxiliary result of independent interest is the proof of exponential convergence of hpfinite element approximations for Gevreyregular functions with point singularities in the unit cube Q=(0,1)^3. Numerical examples of function approximations and of Schrödingertype eigenvalue problems illustrate the theoretical results.
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