C. Schwab

activity

20032023

most citedIsotropic Gaussian random fields on the sphere: Regularity, fast simulation and stochastic partial differential equations

142 citations · 696 across the 46 of their papers we have counts for

collaborators

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40 papers · 1 filter

math.NA2023

Multilevel Monte Carlo FEM for Elliptic PDEs with Besov Random Tree Priors

Christoph Schwab, Andreas Stein

We develop a multilevel Monte Carlo (MLMC)-FEM algorithm for linear, elliptic diffusion problems in polytopal domain $D \subset R^{d}$ , with Besov-tree random coeffic…

math.NA2023

A-posteriori QMC-FEM error estimation for Bayesian inversion and optimal control with entropic risk measure

Marcello Longo, Christoph Schwab, Andreas Stein

We 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 Bayesi…

math.NA2022

Multilevel Domain Uncertainty Quantification in Computational Electromagnetics

Rubén Aylwin, Carlos Jerez-Hanckes, Christoph Schwab +1

We continue our study [Domain Uncertainty Quantification in Computational Electromagnetics, JUQ (2020), 8:301--341] of the numerical approximation of time-harmonic electromagnetic…

math.NA2022

Monte Carlo convergence rates for $k$ th moments in Banach spaces

Kristin Kirchner, Christoph Schwab

We formulate standard and multilevel Monte Carlo methods for the $k$ th moment $M_{ε}^{k} [ξ]$ of a Banach space valued random variable $ξ : Ω \to E$ , interpreted as…

math.NA2022

Multiresolution kernel matrix algebra

H. Harbrecht, M. Multerer, O. Schenk +1

We propose a sparse algebra for samplet compressed kernel matrices, to enable efficient scattered data analysis. We show the compression of kernel matrices by means of samplets pro…

math.NA2022★ 11 cited

Exponential Convergence of hp FEM for the Integral Fractional Laplacian in Polygons

Markus Faustmann, Carlo Marcati, Jens Markus Melenk +1

We prove exponential convergence in the energy norm of $h p$ finite element discretizations for the integral fractional diffusion operator of order $2 s \in (0, 2)$ subject to homogene…

math.NA2022★ 3 cited

Neural and spectral operator surrogates: unified construction and expression rate bounds

Lukas Herrmann, Christoph Schwab, Jakob Zech

Approximation rates are analyzed for deep surrogates of maps between infinite-dimensional function spaces, arising e.g. as data-to-solution maps of linear and nonlinear partial dif…

math.NA2022★ 4 cited

Exponential convergence of hp-FEM for the integral fractional Laplacian in 1D

Markus Faustmann, Carlo Marcati, Jens Markus Melenk +1

We prove weighted analytic regularity for the solution of the integral fractional Poisson problem on bounded intervals with analytic right-hand side. Based on this regularity resul…

math.NA2022

Exponential Convergence of $h p$ -Time-Stepping in Space-Time Discretizations of Parabolic PDEs

Ilaria Perugia, Christoph Schwab, Marco Zank

For linear parabolic initial-boundary value problems with self-adjoint, time-homogeneous elliptic spatial operator in divergence form with Lipschitz-continuous coefficients, and fo…

math.NA2022★ 12 cited

De Rham compatible Deep Neural Network FEM

Marcello Longo, Joost A. A. Opschoor, Nico Disch +2

On general regular simplicial partitions $T$ of bounded polytopal domains $Ω \subset R^{d}$ , $d \in {2, 3}$ , we construct \emph{exact neural network (NN) emulations}…

math.NA2022★ 3 cited

Analyticity and sparsity in uncertainty quantification for PDEs with Gaussian random field inputs

Dinh Dũng, Van Kien Nguyen, Christoph Schwab +1

We establish sparsity and summability results for coefficient sequences of Wiener-Hermite polynomial chaos expansions of countably-parametric solutions of linear elliptic and parab…

math.NA2021★ 26 cited

Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations

Carlo Marcati, Christoph Schwab

We construct and analyze approximation rates of deep operator networks (ONets) between infinite-dimensional spaces that emulate with an exponential rate of convergence the coeffici…

math.NA2021

Deep Learning in High Dimension: Neural Network Approximation of Analytic Functions in $L^{2} (R^{d}, γ_{d})$

Christoph Schwab, Jakob Zech

For artificial deep neural networks, we prove expression rates for analytic functions $f : R^{d} \to R$ in the norm of $L^{2} (R^{d}, γ_{d})$ where $d\in {\mathbb{N}}\…

math.NA2021

Deep ReLU neural networks overcome the curse of dimensionality for partial integrodifferential equations

Lukas Gonon, Christoph Schwab

Deep neural networks (DNNs) with ReLU activation function are proved to be able to express viscosity solutions of linear partial integrodifferental equations (PIDEs) on state space…

math.NA2021

Deep ReLU Network Expression Rates for Option Prices in high-dimensional, exponential Lévy models

Lukas Gonon, Christoph Schwab

We 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 Lé…

math.NA2020★ 15 cited

Exponential Convergence of $h p$ FEM for Spectral Fractional Diffusion in Polygons

Lehel Banjai, Jens M. Melenk, Christoph Schwab

For the spectral fractional diffusion operator of order $2 s \in (0, 2)$ in bounded, curvilinear polygonal domains $Ω$ we prove exponential convergence of two classes of $h p$ discreti…

math.NA2020★ 20 cited

Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularities

Carlo Marcati, Joost A. A. Opschoor, Philipp C. Petersen +1

We prove exponential expressivity with stable ReLU Neural Networks (ReLU NNs) in $H^{1} (Ω)$ for weighted analytic function classes in certain polytopal domains $Ω$ , in space dimensio…

math.NA2020★ 3 cited

Higher-order Quasi-Monte Carlo Training of Deep Neural Networks

M. Longo, S. Mishra, T. K. Rusch +1

We 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…

math.NA2020★ 1 cited

Quantized tensor FEM for multiscale problems: diffusion problems in two and three dimensions

V. Kazeev, I. Oseledets, M. Rakhuba +1

Homogenization in terms of multiscale limits transforms a multiscale problem with $n + 1$ asymptotically separated microscales posed on a physical domain $D \subset R^{d}$ int…

math.NA2020

$h p$ -FEM for reaction-diffusion equations II. Robust exponential convergence for multiple length scales in corner domains

Lehel Banjai, Jens M. Melenk, Christoph Schwab

In bounded, polygonal domains $Ω \subset R^{2}$ with Lipschitz boundary $\partial Ω$ consisting of a finite number of Jordan curves admitting analytic parametrizations, we ana…

math.NA2020★ 2 cited

Space-time discontinuous Galerkin approximation of acoustic waves with point singularities

Pratyuksh Bansal, Andrea Moiola, Ilaria Perugia +1

We develop a convergence theory of space-time discretizations for the linear, 2nd-order wave equation in polygonal domains $Ω \subset R^{2}$ , possibly occupied by piecewise ho…

math.NA2019★ 1 cited

Tensor Rank bounds for Point Singularities in $R^{3}$

Carlo Marcati, Maxim Rakhuba, Christoph Schwab

We analyze rates of approximation by quantized, tensor-structured representations of functions with isolated point singularities in $R^{3}$ . We consider functions in counta…

math.NA2018

Improved Efficiency of a Multi-Index FEM for Computational Uncertainty Quantification

Josef Dick, Michael Feischl, Christoph Schwab

We propose a multi-index algorithm for the Monte Carlo (MC) discretization of a linear, elliptic PDE with affine-parametric input. We prove an error vs. work analysis which allows…

math.NA2017★ 47 cited

Tensor FEM for spectral fractional diffusion

Lehel Banjai, Jens M. Melenk, Ricardo H. Nochetto +3

We design and analyze several Finite Element Methods (FEMs) applied to the Caffarelli-Silvestre extension that localizes the fractional powers of symmetric, coercive, linear ellipt…

Multilevel Monte Carlo FEM for Elliptic PDEs with Besov Random Tree Priors

A-posteriori QMC-FEM error estimation for Bayesian inversion and optimal control with entropic risk measure

Multilevel Domain Uncertainty Quantification in Computational Electromagnetics

Monte Carlo convergence rates for kth moments in Banach spaces

Multiresolution kernel matrix algebra

Exponential Convergence of hp FEM for the Integral Fractional Laplacian in Polygons

Neural and spectral operator surrogates: unified construction and expression rate bounds

Exponential convergence of hp-FEM for the integral fractional Laplacian in 1D

Exponential Convergence of hp-Time-Stepping in Space-Time Discretizations of Parabolic PDEs

De Rham compatible Deep Neural Network FEM

Analyticity and sparsity in uncertainty quantification for PDEs with Gaussian random field inputs

Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations

Deep Learning in High Dimension: Neural Network Approximation of Analytic Functions in L2(Rd,γd​)

Deep ReLU neural networks overcome the curse of dimensionality for partial integrodifferential equations

Deep ReLU Network Expression Rates for Option Prices in high-dimensional, exponential Lévy models

Exponential Convergence of hp FEM for Spectral Fractional Diffusion in Polygons

Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularities

Higher-order Quasi-Monte Carlo Training of Deep Neural Networks

Quantized tensor FEM for multiscale problems: diffusion problems in two and three dimensions

hp-FEM for reaction-diffusion equations II. Robust exponential convergence for multiple length scales in corner domains

Space-time discontinuous Galerkin approximation of acoustic waves with point singularities

Tensor Rank bounds for Point Singularities in R3

Improved Efficiency of a Multi-Index FEM for Computational Uncertainty Quantification

Tensor FEM for spectral fractional diffusion

Monte Carlo convergence rates for $k$ th moments in Banach spaces

Exponential Convergence of $h p$ -Time-Stepping in Space-Time Discretizations of Parabolic PDEs

Deep Learning in High Dimension: Neural Network Approximation of Analytic Functions in $L^{2} (R^{d}, γ_{d})$

Exponential Convergence of $h p$ FEM for Spectral Fractional Diffusion in Polygons

$h p$ -FEM for reaction-diffusion equations II. Robust exponential convergence for multiple length scales in corner domains

Tensor Rank bounds for Point Singularities in $R^{3}$