1.8k citations
Counting papers with at most 25 authors, so collaboration author lists do not stand in for the people here.
- ETH ZurichCH99 papers
- University of MünsterDE18 papers
- University of PaviaIT9 papers
- University of ViennaAT9 papers
- University of Duisburg-EssenDE7 papers
- Chinese University of Hong Kong, ShenzhenCN6 papers
- Shenzhen Research Institute of Big DataCN6 papers
- University of Maryland, College ParkUS6 papers
- Princeton UniversityUS5 papers
- TU WienAT5 papers
- Austrian Academy of SciencesAT4 papers
- Chalmers University of TechnologySE4 papers
46 papers · 1 filter
Monte Carlo convergence rates for th moments in Banach spaces
Kristin Kirchner, Christoph Schwab
We formulate standard and multilevel Monte Carlo methods for the th moment of a Banach space valued random variable , interpreted as…
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 finite element discretizations for the integral fractional diffusion operator of order subject to homogene…
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…
De Rham compatible Deep Neural Network FEM
Marcello Longo, Joost A. A. Opschoor, Nico Disch +2
On general regular simplicial partitions of bounded polytopal domains , , we construct \emph{exact neural network (NN) emulations}…
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…
On the approximation of functions by tanh neural networks
Tim De Ryck, Samuel Lanthaler, Siddhartha Mishra
We derive bounds on the error, in high-order Sobolev norms, incurred in the approximation of Sobolev-regular as well as analytic functions by neural networks with the hyperbolic ta…
Exponential Convergence of FEM for Spectral Fractional Diffusion in Polygons
Lehel Banjai, Jens M. Melenk, Christoph Schwab
For the spectral fractional diffusion operator of order in bounded, curvilinear polygonal domains we prove exponential convergence of two classes of discreti…
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 for weighted analytic function classes in certain polytopal domains , in space dimensio…
Second-order accurate TVD numerical methods for nonlocal nonlinear conservation laws
Ulrik Skre Fjordholm, Adrian Montgomery Ruf
We present a second-order accurate numerical method for a class of nonlocal nonlinear conservation laws called the "nonlocal pair-interaction model" which was recently introduced b…
Well-balanced finite volume schemes for nearly steady adiabatic flows
Luc Grosheintz-Laval, Roger Käppeli
We present well-balanced finite volume schemes designed to approximate the Euler equations with gravitation. They are based on a novel local steady state reconstruction. The scheme…
Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations
Sebastian Becker, Ramon Braunwarth, Martin Hutzenthaler +2
One of the most challenging issues in applied mathematics is to develop and analyze algorithms which are able to approximately compute solutions of high-dimensional nonlinear parti…
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 , possibly occupied by piecewise ho…
Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities
Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse
Partial differential equations (PDEs) are a fundamental tool in the modeling of many real world phenomena. In a number of such real world phenomena the PDEs under consideration con…
Uniform error estimates for artificial neural network approximations for heat equations
Lukas Gonon, Philipp Grohs, Arnulf Jentzen +2
Recently, artificial neural networks (ANNs) in conjunction with stochastic gradient descent optimization methods have been employed to approximately compute solutions of possibly r…
Full error analysis for the training of deep neural networks
Christan Beck, Arnulf Jentzen, Benno Kuckuck
Deep learning algorithms have been applied very successfully in recent years to a range of problems out of reach for classical solution paradigms. Nevertheless, there is no complet…
Statistical solutions of the incompressible Euler equations
Samuel Lanthaler, Siddhartha Mishra, Carlos Parés-Pulido
We propose and study the framework of dissipative statistical solutions for the incompressible Euler equations. Statistical solutions are time-parameterized probability measures on…
Deep neural network approximations for Monte Carlo algorithms
Philipp Grohs, Arnulf Jentzen, Diyora Salimova
Recently, it has been proposed in the literature to employ deep neural networks (DNNs) together with stochastic gradient descent methods to approximate solutions of PDEs. There are…
Convergence rates of monotone schemes for conservation laws with discontinuous flux
Jayesh Badwaik, Adrian Montgomery Ruf
We prove that a class of monotone finite volume schemes for scalar conservation laws with discontinuous flux converge at a rate of in , whenever the flux…
Elastic energy regularization for inverse obstacle scattering problems
Julian Eckhardt, Ralf Hiptmair, Thorsten Hohage +2
By introducing a shape manifold as a solution set to solve inverse obstacle scattering problems we allow the reconstruction of general, not necessarily star-shaped curves. The bend…
On the convergence of the spectral viscosity method for the two-dimensional incompressible Euler equations with rough initial data
Samuel Lanthaler, Siddhartha Mishra
We propose a spectral viscosity method to approximate the two-dimensional Euler equations with rough initial data and prove that the method converges to a weak solution for a large…
A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations
Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse +1
Deep neural networks and other deep learning methods have very successfully been applied to the numerical approximation of high-dimensional nonlinear parabolic partial differential…
Low-rank Riemannian eigensolver for high-dimensional Hamiltonians
Maxim Rakhuba, Alexander Novikov, Ivan Oseledets
Such problems as computation of spectra of spin chains and vibrational spectra of molecules can be written as high-dimensional eigenvalue problems, i.e., when the eigenvector can b…
A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations
Philipp Grohs, Fabian Hornung, Arnulf Jentzen +1
Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems ranging from image classification/image recogniti…
Solving the Kolmogorov PDE by means of deep learning
Christian Beck, Sebastian Becker, Philipp Grohs +2
Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and…