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07/21/2023
Weighted analytic regularity for the integral fractional Laplacian in polyhedra
On polytopal domains in ℝ^3, we prove weighted analytic regularity of so...
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07/19/2023
Deep Operator Network Approximation Rates for Lipschitz Operators
We establish universality and expression rate bounds for a class of neur...
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02/01/2023
Multilevel Monte Carlo FEM for Elliptic PDEs with Besov Random Tree Priors
We develop a multilevel Monte Carlo (MLMC)-FEM algorithm for linear, ell...
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01/09/2023
A-posteriori QMC-FEM error estimation for Bayesian inversion and optimal control with entropic risk measure
We propose a novel a-posteriori error estimation technique where the tar...
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12/14/2022
Multilevel Domain Uncertainty Quantification in Computational Electromagnetics
We continue our study [Domain Uncertainty Quantification in Computationa...
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12/07/2022
Monte Carlo convergence rates for kth moments in Banach spaces
We formulate standard and multilevel Monte Carlo methods for the kth mom...
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09/23/2022
Exponential Convergence of hp FEM for the Integral Fractional Laplacian in Polygons
We prove exponential convergence in the energy norm of hp finite element...
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07/11/2022
Neural and gpc operator surrogates: construction and expression rate bounds
Approximation rates are analyzed for deep surrogates of maps between inf...
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04/08/2022
Exponential convergence of hp-FEM for the integral fractional Laplacian in 1D
We prove weighted analytic regularity for the solution of the integral f...
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03/22/2022
Exponential Convergence of hp-Time-Stepping in Space-Time Discretizations of Parabolic PDEs
For linear parabolic initial-boundary value problems with self-adjoint, ...
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01/14/2022
De Rham compatible Deep Neural Networks
We construct several classes of neural networks with ReLU and BiSU (Bina...
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01/06/2022
Analyticity and sparsity in uncertainty quantification for PDEs with Gaussian random field inputs
We establish summability results for coefficient sequences of Wiener-Her...
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12/15/2021
Weighted analytic regularity for the integral fractional Laplacian in polygons
We prove weighted analytic regularity of solutions to the Dirichlet prob...
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12/15/2021
Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations
We construct deep operator networks (ONets) between infinite-dimensional...
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11/13/2021
Deep Learning in High Dimension: Neural Network Approximation of Analytic Functions in L^2(ℝ^d,γ_d)
For artificial deep neural networks, we prove expression rates for analy...
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03/07/2021
Multilevel approximation of Gaussian random fields: Covariance compression, estimation and spatial prediction
Centered Gaussian random fields (GRFs) indexed by compacta such as smoot...
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02/23/2021
Deep ReLU Neural Network Approximation for Stochastic Differential Equations with Jumps
Deep neural networks (DNNs) with ReLU activation function are proved to ...
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01/28/2021
Deep ReLU Network Expression Rates for Option Prices in high-dimensional, exponential Lévy models
We study the expression rates of deep neural networks (DNNs for short) f...
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11/11/2020
Exponential Convergence of hp FEM for Spectral Fractional Diffusion in Polygons
For the spectral fractional diffusion operator of order 2s∈ (0,2) in bou...
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10/23/2020
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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04/23/2020
Analytic regularity for the incompressible Navier-Stokes equations in polygons
In a plane polygon P with straight sides, we prove analytic regularity o...
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04/22/2020
hp-FEM for reaction-diffusion equations II. Robust exponential convergence for multiple length scales in corner domains
In bounded, polygonal domains Ω⊂R^2 with Lipschitz boundary ∂Ω consistin...
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02/26/2020
Space-time discontinuous Galerkin approximation of acoustic waves with point singularities
We develop a convergence theory of space-time discretizations for the li...
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12/17/2019