## Jentzen, Arnulf

Compute Distance To:
 Author ID: jentzen.arnulf Published as: Jentzen, Arnulf; Jentzen, A. Homepage: http://www.ajentzen.de/ External Links: MGP · ORCID · Wikidata · Google Scholar · dblp · GND · IdRef Awards: Felix Klein Prize (2020)
 Documents Indexed: 84 Publications since 2007, including 2 Books 2 Contributions as Editor Biographic References: 2 Publications Co-Authors: 56 Co-Authors with 83 Joint Publications 1,342 Co-Co-Authors
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### Co-Authors

 3 single-authored 20 Hutzenthaler, Martin 15 Kloeden, Peter Eris 8 Becker, Sebastian 7 Beck, Christian 7 E, Weinan 7 Welti, Timo 6 Cheridito, Patrick 6 Kruse, Thomas 6 Kurniawan, Ryan 6 Pušnik, Primož 5 Grohs, Philipp 5 Salimova, Diyora F. 5 von Wurstemberger, Philippe 4 Kuckuck, Benno 3 Andersson, Adam 3 Han, Jiequn 3 Lindner, Felix 3 Müller-Gronbach, Thomas 3 Neuenkirch, Andreas 3 Riekert, Adrian 3 Röckner, Michael 3 Rossmannek, Florian 3 Yaroslavtseva, Larisa 2 Cox, Sonja Gisela 2 Gess, Benjamin 2 Gonon, Lukas 2 Hefter, Mario 2 Jacobe de Naurois, Ladislas 2 Neufeld, Ariel David 2 Nguyen, Tuan-Anh 1 Berger, Arno 1 Berner, Julius 1 Blömker, Dirk 1 Braunwarth, Ramon 1 Caraballo Garrido, Tomás 1 Conus, Daniel 1 Da Prato, Giuseppe 1 Elbrächter, Dennis 1 Fehrman, Benjamin J. 1 Gerencsér, Máté 1 Hairer, Martin 1 Han, Xiaoying 1 Hornung, Fabian 1 Jaafari, Nor 1 Kofler, David 1 Leber, Frank 1 Schneisgen, Daniela 1 Schwab, Christoph 1 Shen, Hao 1 Siegmund, Stefan 1 Šiška, David 1 Stadtmüller, Uli 1 Stelzer, Robert 1 van Neerven, Jan M. A. M. 1 Wang, Xiaojie 1 Winkel, Georg
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### Serials

 6 Discrete and Continuous Dynamical Systems. Series B 5 Journal of Mathematical Analysis and Applications 4 IMA Journal of Numerical Analysis 4 The Annals of Applied Probability 4 Foundations of Computational Mathematics 3 The Annals of Probability 3 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 3 SIAM Journal on Numerical Analysis 3 Journal of Complexity 3 Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 3 Communications in Mathematical Sciences 3 SN Partial Differential Equations and Applications 2 BIT 2 Journal of Scientific Computing 2 Journal of Nonlinear Science 2 Potential Analysis 2 Journal of Machine Learning Research (JMLR) 2 Proceedings of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences 1 Bulletin of the Australian Mathematical Society 1 Nonlinearity 1 ZAMP. Zeitschrift für angewandte Mathematik und Physik 1 Mathematics of Computation 1 Applied Mathematics and Optimization 1 IEEE Transactions on Automatic Control 1 Journal of Computational and Applied Mathematics 1 Journal of Differential Equations 1 Memoirs of the American Mathematical Society 1 Numerische Mathematik 1 Transactions of the American Mathematical Society 1 Constructive Approximation 1 European Journal of Applied Mathematics 1 Numerical Algorithms 1 Proceedings of the National Academy of Sciences of the United States of America 1 Stochastic Processes and their Applications 1 SIAM Journal on Scientific Computing 1 Electronic Journal of Probability 1 Finance and Stochastics 1 Infinite Dimensional Analysis, Quantum Probability and Related Topics 1 Journal of Numerical Mathematics 1 Stochastics and Dynamics 1 Milan Journal of Mathematics 1 CBMS-NSF Regional Conference Series in Applied Mathematics 1 Communications in Computational Physics 1 Communications in Mathematics and Statistics 1 Stochastic and Partial Differential Equations. Analysis and Computations 1 SIAM Journal on Mathematics of Data Science
all top 5

### Fields

 57 Probability theory and stochastic processes (60-XX) 57 Numerical analysis (65-XX) 34 Partial differential equations (35-XX) 19 Computer science (68-XX) 12 Ordinary differential equations (34-XX) 8 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 4 Approximations and expansions (41-XX) 3 Statistics (62-XX) 3 Operations research, mathematical programming (90-XX) 2 General and overarching topics; collections (00-XX) 2 Operator theory (47-XX) 2 Systems theory; control (93-XX) 1 History and biography (01-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Integral equations (45-XX) 1 Functional analysis (46-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Fluid mechanics (76-XX) 1 Biology and other natural sciences (92-XX)

### Citations contained in zbMATH Open

77 Publications have been cited 1,904 times in 933 Documents Cited by Year
Solving high-dimensional partial differential equations using deep learning. Zbl 1416.35137
Han, Jiequn; Jentzen, Arnulf; E, Weinan
2018
Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients. Zbl 1228.65014
Hutzenthaler, Martin; Jentzen, Arnulf; Kloeden, Peter E.
2011
Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. Zbl 1256.65003
Hutzenthaler, Martin; Jentzen, Arnulf; Kloeden, Peter E.
2012
Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Zbl 1382.65016
E, Weinan; Han, Jiequn; Jentzen, Arnulf
2017
Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients. Zbl 1330.60084
Hutzenthaler, Martin; Jentzen, Arnulf
2015
Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise. Zbl 1186.65011
Jentzen, Arnulf; Kloeden, Peter E.
2009
Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations. Zbl 1442.91116
Beck, Christian; E, Weinan; Jentzen, Arnulf
2019
Taylor approximations for stochastic partial differential equations. Zbl 1240.35001
Jentzen, Arnulf; Kloeden, Peter
2011
The numerical approximation of stochastic partial differential equations. Zbl 1205.60130
Jentzen, A.; Kloeden, P. E.
2009
Deep optimal stopping. Zbl 1495.60029
Becker, Sebastian; Cheridito, Patrick; Jentzen, Arnulf
2019
Galerkin approximations for the stochastic Burgers equation. Zbl 1267.60071
Blömker, Dirk; Jentzen, Arnulf
2013
Regularity analysis for stochastic partial differential equations with nonlinear multiplicative trace class noise. Zbl 1241.60030
Jentzen, Arnulf; Röckner, Michael
2012
Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coefficients. Zbl 1163.65003
Jentzen, A.; Kloeden, P. E.; Neuenkirch, A.
2009
Loss of regularity for Kolmogorov equations. Zbl 1322.35083
Hairer, Martin; Hutzenthaler, Martin; Jentzen, Arnulf
2015
Pathwise numerical approximations of SPDEs with additive noise under non-global Lipschitz coefficients. Zbl 1176.60051
Jentzen, Arnulf
2009
On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations. Zbl 1418.65149
E, Weinan; Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas
2019
Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations. Zbl 1283.60098
Hutzenthaler, Martin; Jentzen, Arnulf; Kloeden, Peter E.
2013
Efficient simulation of nonlinear parabolic SPDEs with additive noise. Zbl 1223.60050
Jentzen, Arnulf; Kloeden, Peter; Winkel, Georg
2011
Taylor expansions of solutions of stochastic partial differential equations with additive noise. Zbl 1220.35202
Jentzen, Arnulf; Kloeden, Peter
2010
On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients. Zbl 07206753
Hutzenthaler, Martin; Jentzen, Arnulf
2020
A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations. Zbl 1455.65200
Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas; Nguyen, Tuan Anh
2020
A Milstein scheme for SPDEs. Zbl 1318.60072
Jentzen, Arnulf; Röckner, Michael
2015
Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations. Zbl 1432.65011
Hutzenthaler, Martin; Jentzen, Arnulf; Wang, Xiaojie
2018
A random Euler scheme for Carathéodory differential equations. Zbl 1160.65003
Jentzen, A.; Neuenkirch, A.
2009
Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg-Landau equations. Zbl 1403.60052
Becker, Sebastian; Jentzen, Arnulf
2019
Convergence of the stochastic Euler scheme for locally Lipschitz coefficients. Zbl 1253.65006
Hutzenthaler, Martin; Jentzen, Arnulf
2011
On stochastic differential equations with arbitrary slow convergence rates for strong approximation. Zbl 1355.65012
Jentzen, Arnulf; Müller-Gronbach, Thomas; Yaroslavtseva, Larisa
2016
Solving the Kolmogorov PDE by means of deep learning. Zbl 1490.65006
Beck, Christian; Becker, Sebastian; Grohs, Philipp; Jaafari, Nor; Jentzen, Arnulf
2021
Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients. Zbl 1477.65020
Conus, Daniel; Jentzen, Arnulf; Kurniawan, Ryan
2019
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. Zbl 1480.60191
Berner, Julius; Grohs, Philipp; Jentzen, Arnulf
2020
Pathwise convergent higher order numerical schemes for random ordinary differential equations. Zbl 1140.65015
Kloeden, Peter E.; Jentzen, Arnulf
2007
Higher order pathwise numerical approximations of SPDEs with additive noise. Zbl 1228.65015
Jentzen, Arnulf
2011
Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations. Zbl 1472.65157
Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas; Nguyen, Tuan Anh; von Wurstemberger, Philippe
2020
Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks. Zbl 1469.60220
Hutzenthaler, Martin; Jentzen, Arnulf; von Wurstemberger, Philippe
2020
Pathwise Taylor schemes for random ordinary differential equations. Zbl 1162.65305
Jentzen, Arnulf; Kloeden, Peter E.
2009
A mild Itô formula for SPDEs. Zbl 1423.35472
Da Prato, Giuseppe; Jentzen, Arnulf; Röckner, Michael
2019
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. Zbl 1475.65157
Jentzen, Arnulf; Salimova, Diyora; Welti, Timo
2021
Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations. Zbl 1471.65169
Beck, Christian; Hornung, Fabian; Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas
2020
Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities. Zbl 1466.65161
Jentzen, Arnulf; Pušnik, Primož
2020
Renormalized powers of Ornstein-Uhlenbeck processes and well-posedness of stochastic Ginzburg-Landau equations. Zbl 1342.60094
E, Weinan; Jentzen, Arnulf; Shen, Hao
2016
Deep splitting method for parabolic PDEs. Zbl 07395807
Beck, Christian; Becker, Sebastian; Cheridito, Patrick; Jentzen, Arnulf; Neufeld, Ariel
2021
Strong convergence of full-discrete nonlinearity-truncated accelerated exponential Euler-type approximations for stochastic Kuramoto-Sivashinsky equations. Zbl 1406.60102
Hutzenthaler, Martin; Jentzen, Arnulf; Salimova, Diyora
2019
An exponential Wagner-Platen type scheme for SPDEs. Zbl 1345.60068
Becker, Sebastian; Jentzen, Arnulf; Kloeden, Peter E.
2016
Taylor expansions of solutions of stochastic partial differential equations. Zbl 1201.60064
Jentzen, Arnulf
2010
Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations. Zbl 1403.60059
Jentzen, Arnulf; Salimova, Diyora; Welti, Timo
2019
Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning. Zbl 1490.60202
E, Weinan; Han, Jiequn; Jentzen, Arnulf
2022
Convergence rates for the stochastic gradient descent method for non-convex objective functions. Zbl 07255167
Fehrman, Benjamin; Gess, Benjamin; Jentzen, Arnulf
2020
Lower bounds for weak approximation errors for spatial spectral Galerkin approximations of stochastic wave equations. Zbl 1405.60094
Jacobe de Naurois, Ladislas; Jentzen, Arnulf; Welti, Timo
2018
Exponential moments for numerical approximations of stochastic partial differential equations. Zbl 1416.60072
Jentzen, Arnulf; Pušnik, Primož
2018
Solving high-dimensional optimal stopping problems using deep learning. Zbl 07441301
Becker, Sebastian; Cheridito, Patrick; Jentzen, Arnulf; Welti, Timo
2021
On the differentiability of solutions of stochastic evolution equations with respect to their initial values. Zbl 1373.34092
2017
Lower error bounds for the stochastic gradient descent optimization algorithm: sharp convergence rates for slowly and fast decaying learning rates. Zbl 1433.68353
Jentzen, Arnulf; von Wurstemberger, Philippe
2020
On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions. Zbl 1404.60082
Gerencsér, Máté; Jentzen, Arnulf; Salimova, Diyora
2017
On arbitrarily slow convergence rates for strong numerical approximations of Cox-Ingersoll-Ross processes and squared Bessel processes. Zbl 1425.91401
Hefter, Mario; Jentzen, Arnulf
2019
DNN expression rate analysis of high-dimensional PDEs: application to option pricing. Zbl 07493716
Elbrächter, Dennis; Grohs, Philipp; Jentzen, Arnulf; Schwab, Christoph
2022
Multilevel Picard iterations for solving smooth semilinear parabolic heat equations. Zbl 1476.65273
E, Weinan; Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas
2021
Strong error analysis for stochastic gradient descent optimization algorithms. Zbl 1460.65071
Jentzen, Arnulf; Kuckuck, Benno; Neufeld, Ariel; von Wurstemberger, Philippe
2021
Regularity properties for solutions of infinite dimensional Kolmogorov equations in Hilbert spaces. Zbl 1431.60051
2019
Non-convergence of stochastic gradient descent in the training of deep neural networks. Zbl 1494.65044
Cheridito, Patrick; Jentzen, Arnulf; Rossmannek, Florian
2021
Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations. Zbl 1473.65251
Becker, Sebastian; Braunwarth, Ramon; Hutzenthaler, Martin; Jentzen, Arnulf; von Wurstemberger, Philippe
2020
Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions. Zbl 1460.65054
Cox, Sonja; Hutzenthaler, Martin; Jentzen, Arnulf; van Neerven, Jan; Welti, Timo
2021
Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients. Zbl 1482.65143
Jentzen, Arnulf; Kurniawan, Ryan
2021
On existence and uniqueness properties for solutions of stochastic fixed point equations. Zbl 1462.60096
Beck, Christian; Gonon, Lukas; Hutzenthaler, Martin; Jentzen, Arnulf
2021
An improved maximum allowable transfer interval for $$L^p$$-stability of networked control systems. Zbl 1368.93642
Jentzen, Arnulf; Leber, Frank; Schneisgen, Daniela; Berger, Arno; Siegmund, Stefan
2010
A proof of convergence for gradient descent in the training of artificial neural networks for constant target functions. Zbl 07543650
Cheridito, Patrick; Jentzen, Arnulf; Riekert, Adrian; Rossmannek, Florian
2022
Pathwise convergence of numerical schemes for random and stochastic differential equations. Zbl 1179.65009
Jentzen, A.; Kloeden, P. E.; Neuenkirch, A.
2009
On the mild Itô formula in Banach spaces. Zbl 1402.60067
Cox, Sonja; Jentzen, Arnulf; Kurniawan, Ryan; Pušnik, Primož
2018
Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values. Zbl 1473.35670
2021
Deep neural network approximations for solutions of PDEs based on Monte Carlo algorithms. Zbl 1490.65232
Grohs, Philipp; Jentzen, Arnulf; Salimova, Diyora
2022
Landscape analysis for shallow neural networks: complete classification of critical points for affine target functions. Zbl 1491.68179
Cheridito, Patrick; Jentzen, Arnulf; Rossmannek, Florian
2022
A unified existence and uniqueness theorem for stochastic evolution equations. Zbl 1200.60051
Jentzen, A.; Kloeden, P. E.
2010
Weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noise. Zbl 1485.60061
Jacobe de Naurois, Ladislas; Jentzen, Arnulf; Welti, Timo
2021
Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations. Zbl 1472.60111
Becker, Sebastian; Gess, Benjamin; Jentzen, Arnulf; Kloeden, Peter E.
2020
On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear parabolic partial differential equations. Zbl 1490.60200
Beck, Christian; Hutzenthaler, Martin; Jentzen, Arnulf
2021
Full error analysis for the training of deep neural networks. Zbl 1492.65132
Beck, Christian; Jentzen, Arnulf; Kuckuck, Benno
2022
Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities. Zbl 07568620
Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas
2022
A proof of convergence for stochastic gradient descent in the training of artificial neural networks with ReLU activation for constant target functions. Zbl 07575573
2022
Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning. Zbl 1490.60202
E, Weinan; Han, Jiequn; Jentzen, Arnulf
2022
DNN expression rate analysis of high-dimensional PDEs: application to option pricing. Zbl 07493716
Elbrächter, Dennis; Grohs, Philipp; Jentzen, Arnulf; Schwab, Christoph
2022
A proof of convergence for gradient descent in the training of artificial neural networks for constant target functions. Zbl 07543650
Cheridito, Patrick; Jentzen, Arnulf; Riekert, Adrian; Rossmannek, Florian
2022
Deep neural network approximations for solutions of PDEs based on Monte Carlo algorithms. Zbl 1490.65232
Grohs, Philipp; Jentzen, Arnulf; Salimova, Diyora
2022
Landscape analysis for shallow neural networks: complete classification of critical points for affine target functions. Zbl 1491.68179
Cheridito, Patrick; Jentzen, Arnulf; Rossmannek, Florian
2022
Full error analysis for the training of deep neural networks. Zbl 1492.65132
Beck, Christian; Jentzen, Arnulf; Kuckuck, Benno
2022
Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities. Zbl 07568620
Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas
2022
A proof of convergence for stochastic gradient descent in the training of artificial neural networks with ReLU activation for constant target functions. Zbl 07575573
2022
Solving the Kolmogorov PDE by means of deep learning. Zbl 1490.65006
Beck, Christian; Becker, Sebastian; Grohs, Philipp; Jaafari, Nor; Jentzen, Arnulf
2021
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. Zbl 1475.65157
Jentzen, Arnulf; Salimova, Diyora; Welti, Timo
2021
Deep splitting method for parabolic PDEs. Zbl 07395807
Beck, Christian; Becker, Sebastian; Cheridito, Patrick; Jentzen, Arnulf; Neufeld, Ariel
2021
Solving high-dimensional optimal stopping problems using deep learning. Zbl 07441301
Becker, Sebastian; Cheridito, Patrick; Jentzen, Arnulf; Welti, Timo
2021
Multilevel Picard iterations for solving smooth semilinear parabolic heat equations. Zbl 1476.65273
E, Weinan; Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas
2021
Strong error analysis for stochastic gradient descent optimization algorithms. Zbl 1460.65071
Jentzen, Arnulf; Kuckuck, Benno; Neufeld, Ariel; von Wurstemberger, Philippe
2021
Non-convergence of stochastic gradient descent in the training of deep neural networks. Zbl 1494.65044
Cheridito, Patrick; Jentzen, Arnulf; Rossmannek, Florian
2021
Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions. Zbl 1460.65054
Cox, Sonja; Hutzenthaler, Martin; Jentzen, Arnulf; van Neerven, Jan; Welti, Timo
2021
Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients. Zbl 1482.65143
Jentzen, Arnulf; Kurniawan, Ryan
2021
On existence and uniqueness properties for solutions of stochastic fixed point equations. Zbl 1462.60096
Beck, Christian; Gonon, Lukas; Hutzenthaler, Martin; Jentzen, Arnulf
2021
Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values. Zbl 1473.35670
2021
Weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noise. Zbl 1485.60061
Jacobe de Naurois, Ladislas; Jentzen, Arnulf; Welti, Timo
2021
On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear parabolic partial differential equations. Zbl 1490.60200
Beck, Christian; Hutzenthaler, Martin; Jentzen, Arnulf
2021
On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients. Zbl 07206753
Hutzenthaler, Martin; Jentzen, Arnulf
2020
A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations. Zbl 1455.65200
Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas; Nguyen, Tuan Anh
2020
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. Zbl 1480.60191
Berner, Julius; Grohs, Philipp; Jentzen, Arnulf
2020
Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations. Zbl 1472.65157
Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas; Nguyen, Tuan Anh; von Wurstemberger, Philippe
2020
Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks. Zbl 1469.60220
Hutzenthaler, Martin; Jentzen, Arnulf; von Wurstemberger, Philippe
2020
Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations. Zbl 1471.65169
Beck, Christian; Hornung, Fabian; Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas
2020
Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities. Zbl 1466.65161
Jentzen, Arnulf; Pušnik, Primož
2020
Convergence rates for the stochastic gradient descent method for non-convex objective functions. Zbl 07255167
Fehrman, Benjamin; Gess, Benjamin; Jentzen, Arnulf
2020
Lower error bounds for the stochastic gradient descent optimization algorithm: sharp convergence rates for slowly and fast decaying learning rates. Zbl 1433.68353
Jentzen, Arnulf; von Wurstemberger, Philippe
2020
Numerical simulations for full history recursive multilevel Picard approximations for systems of high-dimensional partial differential equations. Zbl 1473.65251
Becker, Sebastian; Braunwarth, Ramon; Hutzenthaler, Martin; Jentzen, Arnulf; von Wurstemberger, Philippe
2020
Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations. Zbl 1472.60111
Becker, Sebastian; Gess, Benjamin; Jentzen, Arnulf; Kloeden, Peter E.
2020
Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations. Zbl 1442.91116
Beck, Christian; E, Weinan; Jentzen, Arnulf
2019
Deep optimal stopping. Zbl 1495.60029
Becker, Sebastian; Cheridito, Patrick; Jentzen, Arnulf
2019
On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations. Zbl 1418.65149
E, Weinan; Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas
2019
Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg-Landau equations. Zbl 1403.60052
Becker, Sebastian; Jentzen, Arnulf
2019
Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients. Zbl 1477.65020
Conus, Daniel; Jentzen, Arnulf; Kurniawan, Ryan
2019
A mild Itô formula for SPDEs. Zbl 1423.35472
Da Prato, Giuseppe; Jentzen, Arnulf; Röckner, Michael
2019
Strong convergence of full-discrete nonlinearity-truncated accelerated exponential Euler-type approximations for stochastic Kuramoto-Sivashinsky equations. Zbl 1406.60102
Hutzenthaler, Martin; Jentzen, Arnulf; Salimova, Diyora
2019
Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations. Zbl 1403.60059
Jentzen, Arnulf; Salimova, Diyora; Welti, Timo
2019
On arbitrarily slow convergence rates for strong numerical approximations of Cox-Ingersoll-Ross processes and squared Bessel processes. Zbl 1425.91401
Hefter, Mario; Jentzen, Arnulf
2019
Regularity properties for solutions of infinite dimensional Kolmogorov equations in Hilbert spaces. Zbl 1431.60051
2019
Solving high-dimensional partial differential equations using deep learning. Zbl 1416.35137
Han, Jiequn; Jentzen, Arnulf; E, Weinan
2018
Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations. Zbl 1432.65011
Hutzenthaler, Martin; Jentzen, Arnulf; Wang, Xiaojie
2018
Lower bounds for weak approximation errors for spatial spectral Galerkin approximations of stochastic wave equations. Zbl 1405.60094
Jacobe de Naurois, Ladislas; Jentzen, Arnulf; Welti, Timo
2018
Exponential moments for numerical approximations of stochastic partial differential equations. Zbl 1416.60072
Jentzen, Arnulf; Pušnik, Primož
2018
On the mild Itô formula in Banach spaces. Zbl 1402.60067
Cox, Sonja; Jentzen, Arnulf; Kurniawan, Ryan; Pušnik, Primož
2018
Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Zbl 1382.65016
E, Weinan; Han, Jiequn; Jentzen, Arnulf
2017
On the differentiability of solutions of stochastic evolution equations with respect to their initial values. Zbl 1373.34092
2017
On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions. Zbl 1404.60082
Gerencsér, Máté; Jentzen, Arnulf; Salimova, Diyora
2017
On stochastic differential equations with arbitrary slow convergence rates for strong approximation. Zbl 1355.65012
Jentzen, Arnulf; Müller-Gronbach, Thomas; Yaroslavtseva, Larisa
2016
Renormalized powers of Ornstein-Uhlenbeck processes and well-posedness of stochastic Ginzburg-Landau equations. Zbl 1342.60094
E, Weinan; Jentzen, Arnulf; Shen, Hao
2016
An exponential Wagner-Platen type scheme for SPDEs. Zbl 1345.60068
Becker, Sebastian; Jentzen, Arnulf; Kloeden, Peter E.
2016
Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients. Zbl 1330.60084
Hutzenthaler, Martin; Jentzen, Arnulf
2015
Loss of regularity for Kolmogorov equations. Zbl 1322.35083
Hairer, Martin; Hutzenthaler, Martin; Jentzen, Arnulf
2015
A Milstein scheme for SPDEs. Zbl 1318.60072
Jentzen, Arnulf; Röckner, Michael
2015
Galerkin approximations for the stochastic Burgers equation. Zbl 1267.60071
Blömker, Dirk; Jentzen, Arnulf
2013
Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations. Zbl 1283.60098
Hutzenthaler, Martin; Jentzen, Arnulf; Kloeden, Peter E.
2013
Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. Zbl 1256.65003
Hutzenthaler, Martin; Jentzen, Arnulf; Kloeden, Peter E.
2012
Regularity analysis for stochastic partial differential equations with nonlinear multiplicative trace class noise. Zbl 1241.60030
Jentzen, Arnulf; Röckner, Michael
2012
Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients. Zbl 1228.65014
Hutzenthaler, Martin; Jentzen, Arnulf; Kloeden, Peter E.
2011
Taylor approximations for stochastic partial differential equations. Zbl 1240.35001
Jentzen, Arnulf; Kloeden, Peter
2011
Efficient simulation of nonlinear parabolic SPDEs with additive noise. Zbl 1223.60050
Jentzen, Arnulf; Kloeden, Peter; Winkel, Georg
2011
Convergence of the stochastic Euler scheme for locally Lipschitz coefficients. Zbl 1253.65006
Hutzenthaler, Martin; Jentzen, Arnulf
2011
Higher order pathwise numerical approximations of SPDEs with additive noise. Zbl 1228.65015
Jentzen, Arnulf
2011
Taylor expansions of solutions of stochastic partial differential equations with additive noise. Zbl 1220.35202
Jentzen, Arnulf; Kloeden, Peter
2010
Taylor expansions of solutions of stochastic partial differential equations. Zbl 1201.60064
Jentzen, Arnulf
2010
An improved maximum allowable transfer interval for $$L^p$$-stability of networked control systems. Zbl 1368.93642
Jentzen, Arnulf; Leber, Frank; Schneisgen, Daniela; Berger, Arno; Siegmund, Stefan
2010
A unified existence and uniqueness theorem for stochastic evolution equations. Zbl 1200.60051
Jentzen, A.; Kloeden, P. E.
2010
Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise. Zbl 1186.65011
Jentzen, Arnulf; Kloeden, Peter E.
2009
The numerical approximation of stochastic partial differential equations. Zbl 1205.60130
Jentzen, A.; Kloeden, P. E.
2009
Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coefficients. Zbl 1163.65003
Jentzen, A.; Kloeden, P. E.; Neuenkirch, A.
2009
Pathwise numerical approximations of SPDEs with additive noise under non-global Lipschitz coefficients. Zbl 1176.60051
Jentzen, Arnulf
2009
A random Euler scheme for Carathéodory differential equations. Zbl 1160.65003
Jentzen, A.; Neuenkirch, A.
2009
Pathwise Taylor schemes for random ordinary differential equations. Zbl 1162.65305
Jentzen, Arnulf; Kloeden, Peter E.
2009
Pathwise convergence of numerical schemes for random and stochastic differential equations. Zbl 1179.65009
Jentzen, A.; Kloeden, P. E.; Neuenkirch, A.
2009
Pathwise convergent higher order numerical schemes for random ordinary differential equations. Zbl 1140.65015
Kloeden, Peter E.; Jentzen, Arnulf
2007
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### Cited by 1,350 Authors

 57 Jentzen, Arnulf 24 Hutzenthaler, Martin 22 Kloeden, Peter Eris 21 Wang, Xiaojie 20 Mao, Xuerong 14 Gan, Siqing 14 Liu, Wei 12 E, Weinan 12 Hong, Jialin 11 Kruse, Thomas 11 Szpruch, Lukasz 11 Tambue, Antoine 10 Karniadakis, George Em 10 Kruse, Raphael 10 Ngo, Hoang-Long 10 Przybyłowicz, Paweł 10 Reisinger, Christoph 9 Cui, Jianbo 9 Kovács, Mihály 9 Kumar, Chaman 9 Lang, Annika 9 Larsson, Stig 9 Redmann, Martin 8 Han, Jiequn 8 Hu, Junhao 8 Lord, Gabriel James 8 Neuenkirch, Andreas 8 Sabanis, Sotirios 8 Schwab, Christoph 7 Andersson, Adam 7 Beck, Christian 7 Becker, Sebastian 7 Kamrani, Minoo 7 Mishra, Siddhartha 7 Warin, Xavier 7 Yang, Hongfu 7 Yuan, Chenggui 6 Blömker, Dirk 6 Cao, Yanzhao 6 Grohs, Philipp 6 Guo, Qian 6 Kuznetsov, Dmitriĭ Feliksovich 6 Li, Xiaoyue 6 Liu, Mingzhu 6 Liu, Zhihui 6 Mukam, Jean Daniel 6 Müller-Gronbach, Thomas 6 Nguyen, Tuan-Anh 6 Petersen, Philipp Christian 6 Qi, Ruisheng 6 Song, Minghui 6 Taguchi, Dai 6 Yaroslavtseva, Larisa 6 Ying, Lexing 6 Zhang, Qimin 6 Zhao, Weidong 5 Bachouch, Achref 5 Bréhier, Charles-Edouard 5 Duan, Jinqiao 5 Gao, Shuaibin 5 Huang, Chengming 5 Kühn, Christian 5 Kurniawan, Ryan 5 Lindner, Felix 5 Lu, Jianfeng 5 Luong, Duc-Trong 5 Oh, Tadahiro 5 Perdikaris, Paris G. 5 Pham, Huyên 5 Schneider, Reinhold 5 Šiška, David 5 Tan, Li 5 Wu, Fuke 5 Yan, Zhenya 5 Yang, Xu 5 Yin, George Gang 5 Zhang, Wei 4 Asai, Yusuke 4 Bao, Feng 4 Bayer, Christian 4 Bessaih, Hakima 4 Cheridito, Patrick 4 Deya, Aurélien 4 dos Reis, Gonçalo 4 Flandoli, Franco 4 Giles, Michael B. 4 Hairer, Martin 4 Halidias, Nikolaos 4 Han, Xiaoying 4 Hefter, Mario 4 Jerez, Silvia 4 Kutyniok, Gitta 4 Matoussi, Anis 4 Milošević, Marija 4 Mirzaee, Farshid 4 Morkisz, Paweł M. 4 Oosterlee, Cornelis Willebrordus 4 Raslan, Mones 4 Rößler, Andreas 4 Salimova, Diyora F. ...and 1,250 more Authors
all top 5

### Cited in 205 Serials

 63 Journal of Computational and Applied Mathematics 61 Journal of Computational Physics 40 Applied Mathematics and Computation 28 SIAM Journal on Numerical Analysis 27 Journal of Scientific Computing 26 Applied Numerical Mathematics 26 Communications in Computational Physics 23 Computer Methods in Applied Mechanics and Engineering 23 Journal of Mathematical Analysis and Applications 22 Stochastic and Partial Differential Equations. Analysis and Computations 19 BIT 18 SIAM Journal on Scientific Computing 16 The Annals of Applied Probability 16 International Journal of Computer Mathematics 16 Discrete and Continuous Dynamical Systems. Series B 15 Stochastic Processes and their Applications 14 Numerical Algorithms 13 Mathematics of Computation 11 Quantitative Finance 10 Stochastic Analysis and Applications 10 SIAM Journal on Financial Mathematics 9 Journal of Differential Equations 9 Monte Carlo Methods and Applications 8 Numerische Mathematik 7 Physica D 7 Journal of Complexity 7 Foundations of Computational Mathematics 7 Stochastics and Dynamics 7 Multiscale Modeling & Simulation 7 Advances in Difference Equations 7 Research in the Mathematical Sciences 6 Calcolo 6 Mathematics and Computers in Simulation 6 Computational and Applied Mathematics 6 Electronic Journal of Probability 6 Differentsial’nye Uravneniya i Protsessy Upravleniya 6 SN Partial Differential Equations and Applications 5 Computers & Mathematics with Applications 5 Chaos, Solitons and Fractals 5 Applied Mathematics and Optimization 5 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 5 SIAM Journal on Control and Optimization 5 Statistics & Probability Letters 5 Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 5 Abstract and Applied Analysis 5 Analysis and Applications (Singapore) 5 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 5 Proceedings of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences 4 Constructive Approximation 4 European Journal of Applied Mathematics 4 Advances in Computational Mathematics 4 Finance and Stochastics 4 Communications in Nonlinear Science and Numerical Simulation 4 Methodology and Computing in Applied Probability 4 SIAM/ASA Journal on Uncertainty Quantification 4 SIAM Journal on Mathematics of Data Science 3 Journal of the Franklin Institute 3 Journal of Statistical Physics 3 Mathematical Methods in the Applied Sciences 3 Nonlinearity 3 Physics Letters. A 3 ZAMP. Zeitschrift für angewandte Mathematik und Physik 3 The Annals of Probability 3 Memoirs of the American Mathematical Society 3 Systems & Control Letters 3 Computational Mechanics 3 Journal of Nonlinear Science 3 Potential Analysis 3 Applied Mathematical Finance 3 Engineering Analysis with Boundary Elements 3 Discrete and Continuous Dynamical Systems 3 Nonlinear Dynamics 3 Infinite Dimensional Analysis, Quantum Probability and Related Topics 3 Journal of Applied Mathematics and Computing 3 Mathematical Biosciences and Engineering 3 Discrete and Continuous Dynamical Systems. Series S 3 Statistics and Computing 3 Journal of Mathematical Modeling 3 Probability, Uncertainty and Quantitative Risk 2 Computers and Fluids 2 Communications in Mathematical Physics 2 Inventiones Mathematicae 2 Journal of Applied Probability 2 Transactions of the American Mathematical Society 2 Insurance Mathematics & Economics 2 Chinese Annals of Mathematics. Series B 2 Journal of Computational Mathematics 2 Journal of Theoretical Probability 2 Applied Mathematics Letters 2 MCSS. Mathematics of Control, Signals, and Systems 2 Communications in Statistics. Theory and Methods 2 SIAM Review 2 SIAM Journal on Optimization 2 Filomat 2 Journal of Difference Equations and Applications 2 Electronic Communications in Probability 2 Bernoulli 2 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 2 Vietnam Journal of Mathematics 2 Chaos ...and 105 more Serials
all top 5

### Cited in 43 Fields

 640 Numerical analysis (65-XX) 583 Probability theory and stochastic processes (60-XX) 240 Partial differential equations (35-XX) 184 Computer science (68-XX) 94 Ordinary differential equations (34-XX) 79 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 72 Systems theory; control (93-XX) 52 Biology and other natural sciences (92-XX) 43 Statistics (62-XX) 43 Fluid mechanics (76-XX) 37 Calculus of variations and optimal control; optimization (49-XX) 35 Dynamical systems and ergodic theory (37-XX) 31 Operations research, mathematical programming (90-XX) 28 Approximations and expansions (41-XX) 27 Statistical mechanics, structure of matter (82-XX) 18 Integral equations (45-XX) 13 Operator theory (47-XX) 13 Quantum theory (81-XX) 12 Functional analysis (46-XX) 12 Mechanics of deformable solids (74-XX) 7 Linear and multilinear algebra; matrix theory (15-XX) 5 Harmonic analysis on Euclidean spaces (42-XX) 5 Information and communication theory, circuits (94-XX) 3 General and overarching topics; collections (00-XX) 3 Real functions (26-XX) 3 Special functions (33-XX) 2 Functions of a complex variable (30-XX) 2 Convex and discrete geometry (52-XX) 2 Global analysis, analysis on manifolds (58-XX) 2 Optics, electromagnetic theory (78-XX) 2 Classical thermodynamics, heat transfer (80-XX) 2 Geophysics (86-XX) 1 History and biography (01-XX) 1 Mathematical logic and foundations (03-XX) 1 Number theory (11-XX) 1 Commutative algebra (13-XX) 1 Category theory; homological algebra (18-XX) 1 Measure and integration (28-XX) 1 Difference and functional equations (39-XX) 1 Integral transforms, operational calculus (44-XX) 1 General topology (54-XX) 1 Mechanics of particles and systems (70-XX) 1 Mathematics education (97-XX)

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