Arnulf Jentzen

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Lower error bounds for the stochastic gradient descent optimization algorithm: Sharp convergence rates for slowly and fast decaying learning rates

Arnulf Jentzen, Philippe von Wurstemberger, Arnulf Jentzen, 2018, J. Complex..

Efficient approximation of high-dimensional functions with neural networks

Arnulf Jentzen, Patrick Cheridito, Florian Rossmannek, 2019 .

Convergence rates for the stochastic gradient descent method for non-convex objective functions pdf

Arnulf Jentzen, Benjamin J. Fehrman, Benjamin Gess, 2019, J. Mach. Learn. Res..

Analysis of the generalization error: Empirical risk minimization over deep artificial neural networks overcomes the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations

Arnulf Jentzen, Philipp Grohs, Julius Berner, 2018, SIAM J. Math. Data Sci..

Space-time error estimates for deep neural network approximations for differential equations pdf

Arnulf Jentzen, Philipp Grohs, Fabian Hornung, 2019, Advances in Computational Mathematics.

Solving high-dimensional partial differential equations using deep learning

Arnulf Jentzen, Weinan E, Jiequn Han, 2017, Proceedings of the National Academy of Sciences.

Divergence of the multilevel Monte Carlo method

Peter E. Kloeden, Arnulf Jentzen, Martin Hutzenthaler, 2011 .

Full error analysis for the training of deep neural networks pdf

Arnulf Jentzen, Benno Kuckuck, Christan Beck, 2019, ArXiv.

Overcoming the Curse of Dimensionality in the Numerical Approximation of Parabolic Partial Differential Equations with Gradient-Dependent Nonlinearities pdf

Arnulf Jentzen, Thomas Kruse, Martin Hutzenthaler, 2019, Foundations of Computational Mathematics.

Exponential moment bounds and strong convergence rates for tamed-truncated numerical approximations of stochastic convolutions

Arnulf Jentzen, Felix Lindner, Primoz Pusnik, 2018, Numerical Algorithms.

Pricing and Hedging American-Style Options with Deep Learning

Arnulf Jentzen, Patrick Cheridito, Sebastian Becker, 2019, Journal of Risk and Financial Management.

A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations pdf

Arnulf Jentzen, Philippe von Wurstemberger, Philipp Grohs, 2018, Memoirs of the American Mathematical Society.

Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations

E Weinan, Arnulf Jentzen, Christian Beck, 2017, J. Nonlinear Sci..

Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations pdf

Arnulf Jentzen, Philippe von Wurstemberger, Martin Hutzenthaler, 2020, ArXiv.

Overcoming the curse of dimensionality in the numerical approximation of Allen–Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations

Arnulf Jentzen, Thomas Kruse, Christian Beck, 2019, J. Num. Math..

Space-time deep neural network approximations for high-dimensional partial differential equations pdf

Arnulf Jentzen, Fabian Hornung, Diyora Salimova, 2020, ArXiv.

Deep neural network approximations for Monte Carlo algorithms pdf

Arnulf Jentzen, Philipp Grohs, Diyora Salimova, 2019, ArXiv.

Efficient approximation of high-dimensional functions with deep neural networks pdf

Arnulf Jentzen, Patrick Cheridito, Florian Rossmannek, 2019, ArXiv.

Pathwise approximation of stochastic differential equations on domains: higher order convergence rates without global Lipschitz coefficients

Peter E. Kloeden, Arnulf Jentzen, Andreas Neuenkirch, 2009, Numerische Mathematik.

High-dimensional approximation spaces of artificial neural networks and applications to partial differential equations pdf

Arnulf Jentzen, Philippe von Wurstemberger, Patrick Cheridito, 2020, ArXiv.

Foundations of Computational Mathematics, Hong Kong 2008: Pathwise Convergence of Numerical Schemes for Random and Stochastic Differential Equations

Peter E. Kloeden, Arnulf Jentzen, Andreas Neuenkirch, 2009 .

Generalised multilevel Picard approximations pdf

Arnulf Jentzen, Michael B. Giles, Timo Welti, 2019, ArXiv.

Higher Order Pathwise Numerical Approximations of SPDEs with Additive Noise

Arnulf Jentzen, Arnulf Jentzen, 2011, SIAM J. Numer. Anal..

Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities pdf

Tuan Anh Nguyen, Arnulf Jentzen, Martin Hutzenthaler, 2020, ArXiv.

Deep learning based numerical approximation algorithms for stochastic partial differential equations and high-dimensional nonlinear filtering problems pdf

Arnulf Jentzen, Patrick Cheridito, Ariel Neufeld, 2020, ArXiv.

A Milstein Scheme for SPDEs

Arnulf Jentzen, Michael Röckner, Arnulf Jentzen, 2010, Found. Comput. Math..

On arbitrarily slow convergence rates for strong numerical approximations of Cox–Ingersoll–Ross processes and squared Bessel processes

Arnulf Jentzen, Mario Hefter, Arnulf Jentzen, 2017, Finance Stochastics.

On deep learning based approximation algorithms for PDEs

Arnulf Jentzen, 2018 .

On Multilevel Picard Numerical Approximations for High-Dimensional Nonlinear Parabolic Partial Differential Equations and High-Dimensional Nonlinear Backward Stochastic Differential Equations

E Weinan, Arnulf Jentzen, Thomas Kruse, 2017, Journal of Scientific Computing.

A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations

Tuan Anh Nguyen, Arnulf Jentzen, Thomas Kruse, 2019, SN Partial Differential Equations and Applications.

Non-convergence of stochastic gradient descent in the training of deep neural networks pdf

Arnulf Jentzen, Patrick Cheridito, Florian Rossmannek, 2020, ArXiv.

Convergence of the Stochastic Euler Scheme for Locally Lipschitz Coefficients

Arnulf Jentzen, Martin Hutzenthaler, M. Hutzenthaler, 2009, Found. Comput. Math..

Solving stochastic differential equations and Kolmogorov equations by means of deep learning pdf

Sebastian Becker, Arnulf Jentzen, Philipp Grohs, 2018, ArXiv.

Towards a regularity theory for ReLU networks – chain rule and global error estimates

Arnulf Jentzen, Philipp Grohs, Julius Berner, 2019, 2019 13th International conference on Sampling Theory and Applications (SampTA).

A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients pdf

Arnulf Jentzen, Diyora Salimova, Timo Welti, 2018, Communications in Mathematical Sciences.

Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations

E Weinan, Arnulf Jentzen, Jiequn Han, 2017, Communications in Mathematics and Statistics.

An overview on deep learning-based approximation methods for partial differential equations pdf

Christian Beck, Arnulf Jentzen, Benno Kuckuck, 2020, ArXiv.

On deep learning based approximation algorithms for deterministic and stochastic PDEs

Arnulf Jentzen, 2018 .

Overcoming the curse of dimensionality: Solving high-dimensional partial differential equations using deep learning pdf

E Weinan, Arnulf Jentzen, Jiequn Han, 2017, ArXiv.

Galerkin Approximations for the Stochastic Burgers Equation

Arnulf Jentzen, Dirk Blömker, Arnulf Jentzen, 2013, SIAM J. Numer. Anal..

An Improved Maximum Allowable Transfer Interval for $L^{p}$-Stability of Networked Control Systems

Stefan Siegmund, Arnulf Jentzen, Arno Berger, 2010, IEEE Transactions on Automatic Control.

Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations pdf

Arnulf Jentzen, Lukas Gonon, Christian Beck, 2020, ArXiv.

Strong convergence rates on the whole probability space for space-time discrete numerical approximation schemes for stochastic Burgers equations pdf

Arnulf Jentzen, Felix Lindner, Martin Hutzenthaler, 2019, ArXiv.

Strong overall error analysis for the training of artificial neural networks via random initializations pdf

Arnulf Jentzen, Adrian Riekert, Arnulf Jentzen, 2020, ArXiv.

Weak error analysis for stochastic gradient descent optimization algorithms pdf

Arnulf Jentzen, Lukas Gonon, Diyora Salimova, 2020, ArXiv.

Nonlinear Monte Carlo methods with polynomial runtime for high-dimensional iterated nested expectations pdf

Arnulf Jentzen, Thomas Kruse, Christian Beck, 2020, ArXiv.

Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations

Sebastian Becker, Peter E. Kloeden, Arnulf Jentzen, 2018, BIT Numerical Mathematics.

Solving high-dimensional optimal stopping problems using deep learning pdf

Sebastian Becker, Arnulf Jentzen, Patrick Cheridito, 2019, European Journal of Applied Mathematics.

An Exponential Wagner-Platen Type Scheme for SPDEs

Sebastian Becker, Peter E. Kloeden, Arnulf Jentzen, 2013, SIAM J. Numer. Anal..

Deep Optimal Stopping

Sebastian Becker, Arnulf Jentzen, Patrick Cheridito, 2018, J. Mach. Learn. Res..

Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning pdf

E Weinan, Arnulf Jentzen, Jiequn Han, 2020, Nonlinearity.

Uniform error estimates for artificial neural network approximations for heat equations pdf

Arnulf Jentzen, Philipp Grohs, David Siska, 2019, ArXiv.

Overall error analysis for the training of deep neural networks via stochastic gradient descent with random initialisation pdf

Arnulf Jentzen, Timo Welti, Arnulf Jentzen, 2020, Appl. Math. Comput..

Deep splitting method for parabolic PDEs pdf

Arnulf Jentzen, Patrick Cheridito, Christian Beck, 2019, SIAM J. Sci. Comput..

Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations

Xiaojie Wang, Arnulf Jentzen, Martin Hutzenthaler, 2013, Math. Comput..

Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations

Arnulf Jentzen, Thomas Kruse, Martin Hutzenthaler, 2018, Proceedings of the Royal Society A.

Strong Lp-error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations pdf

Arnulf Jentzen, Benno Kuckuck, Martin Hutzenthaler, 2021, ArXiv.

Deep neural network approximation theory for high-dimensional functions pdf

Arnulf Jentzen, Patrick Cheridito, Benno Kuckuck, 2021, ArXiv.

A proof of convergence for the gradient descent optimization method with random initializations in the training of neural networks with ReLU activation for piecewise linear target functions pdf

Arnulf Jentzen, Adrian Riekert, 2021, ArXiv.

Weak Convergence Rates for Euler-Type Approximations of Semilinear Stochastic Evolution Equations with Nonlinear Diffusion Coefficients

Arnulf Jentzen, Ryan Kurniawan, Arnulf Jentzen, 2015, Found. Comput. Math..

Strong $L^p$-error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations

Arnulf Jentzen, Benno Kuckuck, Martin Hutzenthaler, 2021 .

Lower bounds for artificial neural network approximations: A proof that shallow neural networks fail to overcome the curse of dimensionality pdf

Arnulf Jentzen, Philipp Grohs, Shokhrukh Ibragimov, 2021, ArXiv.

Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations pdf

Arnulf Jentzen, Thomas Kruse, Martin Hutzenthaler, 2021, J. Num. Math..

Convergence rates for gradient descent in the training of overparameterized artificial neural networks with biases pdf

Arnulf Jentzen, Timo Kröger, Arnulf Jentzen, 2021, ArXiv.

Efficient Approximation of High-Dimensional Functions With Neural Networks

Arnulf Jentzen, Patrick Cheridito, Florian Rossmannek, 2019, IEEE Transactions on Neural Networks and Learning Systems.

On the existence of global minima and convergence analyses for gradient descent methods in the training of deep neural networks pdf

Arnulf Jentzen, Adrian Riekert, Arnulf Jentzen, 2021, ArXiv.

Non-convergence of stochastic gradient descent in the training of deep neural networks pdf

Arnulf Jentzen, Patrick Cheridito, Florian Rossmannek, 2020, J. Complex..

Existence, uniqueness, and convergence rates for gradient flows in the training of artificial neural networks with ReLU activation pdf

Arnulf Jentzen, Georg S. Weiss, Adrian Riekert, 2021, ArXiv.

Weak Convergence Rates for Spatial Spectral Galerkin Approximations of Semilinear Stochastic Wave Equations with Multiplicative Noise

Arnulf Jentzen, Timo Welti, Ladislas Jacobe de Naurois, 2015, Applied Mathematics & Optimization.

Solving the Kolmogorov PDE by Means of Deep Learning

Arnulf Jentzen, Philipp Grohs, Nor Jaafari, 2018, Journal of Scientific Computing.

Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning

Arnulf Jentzen, Weinan E, Jiequn Han, 2020, Nonlinearity.

Convergence analysis for gradient flows in the training of artificial neural networks with ReLU activation pdf

Arnulf Jentzen, Adrian Riekert, 2021, ArXiv.

Convergence proof for stochastic gradient descent in the training of deep neural networks with ReLU activation for constant target functions pdf

Luca Scarpa, Arnulf Jentzen, Martin Hutzenthaler, 2021, ArXiv.

Landscape analysis for shallow ReLU neural networks: complete classification of critical points for affine target functions pdf

Arnulf Jentzen, Patrick Cheridito, Florian Rossmannek, 2021, ArXiv.

A proof of convergence for gradient descent in the training of artificial neural networks for constant target functions pdf

Arnulf Jentzen, Patrick Cheridito, Florian Rossmannek, 2021, J. Complex..

A proof of convergence for stochastic gradient descent in the training of artificial neural networks with ReLU activation for constant target functions pdf

Arnulf Jentzen, Adrian Riekert, 2021, ArXiv.

Full history recursive multilevel Picard approximations for ordinary differential equations with expectations pdf

Arnulf Jentzen, Martin Hutzenthaler, Christian Beck, 2021, ArXiv.

Multilevel Picard iterations for solving smooth semilinear parabolic heat equations

Arnulf Jentzen, Thomas Kruse, Weinan E, 2021, Partial Differential Equations and Applications.

On nonlinear Feynman–Kac formulas for viscosity solutions of semilinear parabolic partial differential equations

Arnulf Jentzen, Martin Hutzenthaler, Christian Beck, 2020, Stochastics and Dynamics.

On full history recursive multilevel Picard approximations and numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations

E Weinan, Arnulf Jentzen, Thomas Kruse, 2016 .

DNN Expression Rate Analysis of High-Dimensional PDEs: Application to Option Pricing

Arnulf Jentzen, Philipp Grohs, Christoph Schwab, 2018, Constructive Approximation.

Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations

Peter E. Kloeden, Arnulf Jentzen, Martin Hutzenthaler, 2011, 1105.0226.

Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities

Arnulf Jentzen, Arnulf Jentzen, Primovz Puvsnik, 2015, IMA Journal of Numerical Analysis.

On existence and uniqueness properties for solutions of stochastic fixed point equations

Arnulf Jentzen, Lukas Gonon, Martin Hutzenthaler, 2019, Discrete & Continuous Dynamical Systems - B.

On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions

Arnulf Jentzen, Máté Gerencsér, Diyora Salimova, 2017, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients

Arnulf Jentzen, Martin Hutzenthaler, 2020 .

Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions

Arnulf Jentzen, Jan van Neerven, Sonja Cox, 2020, IMA Journal of Numerical Analysis.

Strong convergence of full-discrete nonlinearity-truncated accelerated exponential Euler-type approximations for stochastic Kuramoto-Sivashinsky equations

Arnulf Jentzen, Martin Hutzenthaler, Diyora Salimova, 2016, 1604.02053.

Loss of regularity for Kolmogorov equations

Martin Hairer, Arnulf Jentzen, Martin Hutzenthaler, 2012, 1209.6035.

Analysis and Numerical Approximations

Arnulf Jentzen, Arnulf Jentzen, 2015 .

Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients

Arnulf Jentzen, Daniel Conus, Ryan Kurniawan, 2014, The Annals of Applied Probability.

On the mild Itô formula in Banach spaces

Arnulf Jentzen, Sonja Cox, Ryan Kurniawan, 2016, 1612.03210.

Efficient simulation of nonlinear parabolic SPDEs with additive noise

Arnulf Jentzen, Peter Kloeden, P. Kloeden, 2011, 1210.8320.

Pathwise Taylor schemes for random ordinary differential equations

Peter E. Kloeden, Arnulf Jentzen, P. Kloeden, 2009 .

Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values

Arnulf Jentzen, Ryan Kurniawan, Adam Andersson, 2014, Journal of Mathematical Analysis and Applications.