Edit Profile (opens in new tab) Jentzen, Arnulf Co-Author Distance 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 · theses.fr Awards: Felix Klein Prize (2020) Documents Indexed: 134 Publications since 2007, including 3 Books and 41 Additional arXiv Preprints 2 Contributions as Editor Biographic References: 2 Publications Co-Authors: 83 Co-Authors with 133 Joint Publications 1,740 Co-Co-Authors all top 5 Co-Authors 3 single-authored 31 Hutzenthaler, Martin 16 Kloeden, Peter Eris 14 Beck, Christian 13 Becker, Sebastian 13 Riekert, Adrian 12 Cheridito, Patrick 11 Kruse, Thomas 10 Kuckuck, Benno 10 von Wurstemberger, Philippe 9 Kurniawan, Ryan 9 Salimova, Diyora F. 9 Welti, Timo 8 Grohs, Philipp 7 E, Weinan 7 Lindner, Felix 7 Pušnik, Primož 5 Gonon, Lukas 5 Rossmannek, Florian 4 Cox, Sonja Gisela 4 Hefter, Mario 4 Hornung, Fabian 4 Neufeld, Ariel David 3 Andersson, Adam 3 Gess, Benjamin 3 Han, Jiequn 3 Ibragimov, Shokhrukh 3 Kröger, Timo 3 Müller-Gronbach, Thomas 3 Neuenkirch, Andreas 3 Nguyen, Tuan-Anh 3 Röckner, Michael 3 Yaroslavtseva, Larisa 2 Beneventano, Pierfrancesco 2 Eberle, Simon 2 Graeber, Robin 2 Jacobe de Naurois, Ladislas 2 Mazzonetto, Sara 2 Padgett, Joshua Lee 2 Weiss, Georg Sebastian 1 Ackermann, Julia 1 Beccari, Matteo 1 Bercher, Aritz 1 Berger, Arno 1 Berner, Julius 1 Blömker, Dirk 1 Boussange, Victor 1 Braunwarth, Ramon 1 Caraballo Garrido, Tomás 1 Conus, Daniel 1 Da Prato, Giuseppe 1 Dereich, Steffen 1 Elbrächter, Dennis 1 Fehrman, Benjamin J. 1 Gallon, Davide 1 Gerencsér, Máté 1 Giles, Michael B. 1 Hairer, Martin 1 Han, Xiaoying 1 Hudde, Anselm 1 Jaafari, Nor 1 Kassing, Sebastian 1 Kleinberg, Konrad 1 Kofler, David 1 Koppensteiner, Sarah 1 Leber, Frank 1 Magnani, Emilia 1 Müller, Marvin S. 1 Noll, Marco 1 Pellissier, Loïc 1 Pohl, Katharina 1 Scarpa, Luca 1 Schneisgen, Daniela 1 Schwab, Christoph 1 Shen, Hao 1 Siegmund, Stefan 1 Šiška, David 1 Stadtmüller, Uli 1 Stelzer, Robert 1 Tuan Anh Nguyen 1 van Neerven, Jan M. A. M. 1 Wang, Xiaojie 1 Winkel, Georg 1 Zimmermann, Philipp all top 5 Serials 7 Discrete and Continuous Dynamical Systems. Series B 5 Journal of Mathematical Analysis and Applications 4 IMA Journal of Numerical Analysis 4 Journal of Complexity 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 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 Memoirs of the American Mathematical Society 2 Journal of Scientific Computing 2 Journal of Nonlinear Science 2 Potential Analysis 2 Journal of Machine Learning Research (JMLR) 2 Journal of Numerical Mathematics 2 Stochastic and Partial Differential Equations. Analysis and Computations 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 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 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 Advances in Computational Mathematics 1 Electronic Journal of Probability 1 Finance and Stochastics 1 Mathematical Finance 1 Infinite Dimensional Analysis, Quantum Probability and Related Topics 1 Communications in Nonlinear Science and Numerical Simulation 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 SIAM Journal on Mathematics of Data Science all top 5 Fields 77 Numerical analysis (65-XX) 68 Probability theory and stochastic processes (60-XX) 38 Partial differential equations (35-XX) 33 Computer science (68-XX) 13 Ordinary differential equations (34-XX) 9 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 7 Approximations and expansions (41-XX) 5 Operations research, mathematical programming (90-XX) 4 Systems theory; control (93-XX) 3 Statistics (62-XX) 2 General and overarching topics; collections (00-XX) 2 Dynamical systems and ergodic theory (37-XX) 2 Integral equations (45-XX) 2 Operator theory (47-XX) 1 History and biography (01-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) Publications by Year all cited Publications top 5 cited Publications Citations contained in zbMATH Open 86 Publications have been cited 2,968 times in 1,344 Documents Cited by ▼ Year ▼ Solving high-dimensional partial differential equations using deep learning. Zbl 1416.35137 Han, Jiequn; Jentzen, Arnulf; E, Weinan 390 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 244 2017 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. 226 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. 202 2012 Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients. Zbl 1330.60084 Hutzenthaler, Martin; Jentzen, Arnulf 114 2015 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 106 2019 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. 78 2009 Taylor approximations for stochastic partial differential equations. Zbl 1240.35001 Jentzen, Arnulf; Kloeden, Peter 74 2011 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 64 2019 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 60 2020 Deep optimal stopping. Zbl 1495.60029 Becker, Sebastian; Cheridito, Patrick; Jentzen, Arnulf 59 2019 The numerical approximation of stochastic partial differential equations. Zbl 1205.60130 Jentzen, A.; Kloeden, P. E. 59 2009 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 52 2020 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 51 2020 Galerkin approximations for the stochastic Burgers equation. Zbl 1267.60071 Blömker, Dirk; Jentzen, Arnulf 47 2013 Loss of regularity for Kolmogorov equations. Zbl 1322.35083 Hairer, Martin; Hutzenthaler, Martin; Jentzen, Arnulf 44 2015 Regularity analysis for stochastic partial differential equations with nonlinear multiplicative trace class noise. Zbl 1241.60030 Jentzen, Arnulf; Röckner, Michael 41 2012 Pathwise numerical approximations of SPDEs with additive noise under non-global Lipschitz coefficients. Zbl 1176.60051 Jentzen, Arnulf 40 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. 39 2009 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 39 2020 Solving the Kolmogorov PDE by means of deep learning. Zbl 1490.65006 Beck, Christian; Becker, Sebastian; Grohs, Philipp; Jaafari, Nor; Jentzen, Arnulf 38 2021 A Milstein scheme for SPDEs. Zbl 1318.60072 Jentzen, Arnulf; Röckner, Michael 36 2015 Efficient simulation of nonlinear parabolic SPDEs with additive noise. Zbl 1223.60050 Jentzen, Arnulf; Kloeden, Peter; Winkel, Georg 36 2011 Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations. Zbl 1283.60098 Hutzenthaler, Martin; Jentzen, Arnulf; Kloeden, Peter E. 35 2013 Taylor expansions of solutions of stochastic partial differential equations with additive noise. Zbl 1220.35202 Jentzen, Arnulf; Kloeden, Peter 35 2010 A mild Itô formula for SPDEs. Zbl 1423.35472 Da Prato, Giuseppe; Jentzen, Arnulf; Röckner, Michael 32 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 32 2021 DNN expression rate analysis of high-dimensional PDEs: application to option pricing. Zbl 1500.35009 Elbrächter, Dennis; Grohs, Philipp; Jentzen, Arnulf; Schwab, Christoph 32 2022 Deep splitting method for parabolic PDEs. Zbl 1501.65054 Beck, Christian; Becker, Sebastian; Cheridito, Patrick; Jentzen, Arnulf; Neufeld, Ariel 31 2021 Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning. Zbl 1490.60202 E, Weinan; Han, Jiequn; Jentzen, Arnulf 31 2022 Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations. Zbl 1432.65011 Hutzenthaler, Martin; Jentzen, Arnulf; Wang, Xiaojie 26 2018 Convergence of the stochastic Euler scheme for locally Lipschitz coefficients. Zbl 1253.65006 Hutzenthaler, Martin; Jentzen, Arnulf 26 2011 A random Euler scheme for Carathéodory differential equations. Zbl 1160.65003 Jentzen, A.; Neuenkirch, A. 25 2009 Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg-Landau equations. Zbl 1403.60052 Becker, Sebastian; Jentzen, Arnulf 24 2019 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 24 2020 Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients. Zbl 1477.65020 Conus, Daniel; Jentzen, Arnulf; Kurniawan, Ryan 23 2019 Multilevel Picard iterations for solving smooth semilinear parabolic heat equations. Zbl 1476.65273 E, Weinan; Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas 22 2021 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 21 2020 On stochastic differential equations with arbitrary slow convergence rates for strong approximation. Zbl 1355.65012 Jentzen, Arnulf; Müller-Gronbach, Thomas; Yaroslavtseva, Larisa 19 2016 Pathwise convergent higher order numerical schemes for random ordinary differential equations. Zbl 1140.65015 Kloeden, Peter E.; Jentzen, Arnulf 19 2007 Higher order pathwise numerical approximations of SPDEs with additive noise. Zbl 1228.65015 Jentzen, Arnulf 19 2011 Solving high-dimensional optimal stopping problems using deep learning. Zbl 1505.91413 Becker, Sebastian; Cheridito, Patrick; Jentzen, Arnulf; Welti, Timo 19 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 17 2020 A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations. Zbl 07679303 Grohs, Philipp; Hornung, Fabian; Jentzen, Arnulf; von Wurstemberger, Philippe 16 2023 Pathwise Taylor schemes for random ordinary differential equations. Zbl 1162.65305 Jentzen, Arnulf; Kloeden, Peter E. 14 2009 Convergence rates for the stochastic gradient descent method for non-convex objective functions. Zbl 1520.68143 Fehrman, Benjamin; Gess, Benjamin; Jentzen, Arnulf 13 2020 Taylor expansions of solutions of stochastic partial differential equations. Zbl 1201.60064 Jentzen, Arnulf 13 2010 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 12 2019 Renormalized powers of Ornstein-Uhlenbeck processes and well-posedness of stochastic Ginzburg-Landau equations. Zbl 1342.60094 E, Weinan; Jentzen, Arnulf; Shen, Hao 12 2016 An exponential Wagner-Platen type scheme for SPDEs. Zbl 1345.60068 Becker, Sebastian; Jentzen, Arnulf; Kloeden, Peter E. 12 2016 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 12 2020 Uniform error estimates for artificial neural network approximations for heat equations. Zbl 1502.65171 Gonon, Lukas; Grohs, Philipp; Jentzen, Arnulf; Kofler, David; Šiška, David 12 2022 Exponential moments for numerical approximations of stochastic partial differential equations. Zbl 1416.60072 Jentzen, Arnulf; Pušnik, Primož 11 2018 Deep neural network approximations for solutions of PDEs based on Monte Carlo algorithms. Zbl 1490.65232 Grohs, Philipp; Jentzen, Arnulf; Salimova, Diyora 11 2022 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 10 2019 Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations. Zbl 1403.60059 Jentzen, Arnulf; Salimova, Diyora; Welti, Timo 10 2019 Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations. Zbl 1517.60081 Becker, Sebastian; Gess, Benjamin; Jentzen, Arnulf; Kloeden, Peter E. 10 2023 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 10 2021 Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities. Zbl 1501.65084 Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas 10 2022 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 9 2018 An overview on deep learning-based approximation methods for partial differential equations. Zbl 07675828 Beck, Christian; Hutzenthaler, Martin; Jentzen, Arnulf; Kuckuck, Benno 9 2023 Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients. Zbl 1482.65143 Jentzen, Arnulf; Kurniawan, Ryan 9 2021 Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values. Zbl 1473.35670 Andersson, Adam; Jentzen, Arnulf; Kurniawan, Ryan 9 2021 On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear parabolic partial differential equations. Zbl 1490.60200 Beck, Christian; Hutzenthaler, Martin; Jentzen, Arnulf 9 2021 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 8 2020 Strong error analysis for stochastic gradient descent optimization algorithms. Zbl 1460.65071 Jentzen, Arnulf; Kuckuck, Benno; Neufeld, Ariel; von Wurstemberger, Philippe 8 2021 Non-convergence of stochastic gradient descent in the training of deep neural networks. Zbl 1494.65044 Cheridito, Patrick; Jentzen, Arnulf; Rossmannek, Florian 7 2021 On the differentiability of solutions of stochastic evolution equations with respect to their initial values. Zbl 1373.34092 Andersson, Adam; Jentzen, Arnulf; Kurniawan, Ryan; Welti, Timo 7 2017 Regularity properties for solutions of infinite dimensional Kolmogorov equations in Hilbert spaces. Zbl 1431.60051 Andersson, Adam; Hefter, Mario; Jentzen, Arnulf; Kurniawan, Ryan 5 2019 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 5 2017 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 5 2010 Full error analysis for the training of deep neural networks. Zbl 1492.65132 Beck, Christian; Jentzen, Arnulf; Kuckuck, Benno 5 2022 On the mild Itô formula in Banach spaces. Zbl 1402.60067 Cox, Sonja; Jentzen, Arnulf; Kurniawan, Ryan; Pušnik, Primož 4 2018 On existence and uniqueness properties for solutions of stochastic fixed point equations. Zbl 1462.60096 Beck, Christian; Gonon, Lukas; Hutzenthaler, Martin; Jentzen, Arnulf 4 2021 A proof of convergence for gradient descent in the training of artificial neural networks for constant target functions. Zbl 1502.65037 Cheridito, Patrick; Jentzen, Arnulf; Riekert, Adrian; Rossmannek, Florian 4 2022 Space-time error estimates for deep neural network approximations for differential equations. Zbl 07649348 Grohs, Philipp; Hornung, Fabian; Jentzen, Arnulf; Zimmermann, Philipp 4 2023 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. 3 2020 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 3 2021 Pathwise convergence of numerical schemes for random and stochastic differential equations. Zbl 1179.65009 Jentzen, A.; Kloeden, P. E.; Neuenkirch, A. 2 2009 On the strong regularity of degenerate additive noise driven stochastic differential equations with respect to their initial values. Zbl 1471.34116 Jentzen, Arnulf; Kuckuck, Benno; Müller-Gronbach, Thomas; Yaroslavtseva, Larisa 2 2021 Landscape analysis for shallow neural networks: complete classification of critical points for affine target functions. Zbl 1491.68179 Cheridito, Patrick; Jentzen, Arnulf; Rossmannek, Florian 2 2022 Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations. Zbl 1515.65025 Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas; Tuan Anh Nguyen 2 2023 Lower bounds for artificial neural network approximations: a proof that shallow neural networks fail to overcome the curse of dimensionality. Zbl 07693253 Grohs, Philipp; Ibragimov, Shokhrukh; Jentzen, Arnulf; Koppensteiner, Sarah 1 2023 A unified existence and uniqueness theorem for stochastic evolution equations. Zbl 1200.60051 Jentzen, A.; Kloeden, P. E. 1 2010 A proof of convergence for stochastic gradient descent in the training of artificial neural networks with ReLU activation for constant target functions. Zbl 07575573 Jentzen, Arnulf; Riekert, Adrian 1 2022 Convergence analysis for gradient flows in the training of artificial neural networks with ReLU activation. Zbl 07595126 Jentzen, Arnulf; Riekert, Adrian 1 2023 A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations. Zbl 07679303 Grohs, Philipp; Hornung, Fabian; Jentzen, Arnulf; von Wurstemberger, Philippe 16 2023 Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations. Zbl 1517.60081 Becker, Sebastian; Gess, Benjamin; Jentzen, Arnulf; Kloeden, Peter E. 10 2023 An overview on deep learning-based approximation methods for partial differential equations. Zbl 07675828 Beck, Christian; Hutzenthaler, Martin; Jentzen, Arnulf; Kuckuck, Benno 9 2023 Space-time error estimates for deep neural network approximations for differential equations. Zbl 07649348 Grohs, Philipp; Hornung, Fabian; Jentzen, Arnulf; Zimmermann, Philipp 4 2023 Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations. Zbl 1515.65025 Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas; Tuan Anh Nguyen 2 2023 Lower bounds for artificial neural network approximations: a proof that shallow neural networks fail to overcome the curse of dimensionality. Zbl 07693253 Grohs, Philipp; Ibragimov, Shokhrukh; Jentzen, Arnulf; Koppensteiner, Sarah 1 2023 Convergence analysis for gradient flows in the training of artificial neural networks with ReLU activation. Zbl 07595126 Jentzen, Arnulf; Riekert, Adrian 1 2023 DNN expression rate analysis of high-dimensional PDEs: application to option pricing. Zbl 1500.35009 Elbrächter, Dennis; Grohs, Philipp; Jentzen, Arnulf; Schwab, Christoph 32 2022 Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning. Zbl 1490.60202 E, Weinan; Han, Jiequn; Jentzen, Arnulf 31 2022 Uniform error estimates for artificial neural network approximations for heat equations. Zbl 1502.65171 Gonon, Lukas; Grohs, Philipp; Jentzen, Arnulf; Kofler, David; Šiška, David 12 2022 Deep neural network approximations for solutions of PDEs based on Monte Carlo algorithms. Zbl 1490.65232 Grohs, Philipp; Jentzen, Arnulf; Salimova, Diyora 11 2022 Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities. Zbl 1501.65084 Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas 10 2022 Full error analysis for the training of deep neural networks. Zbl 1492.65132 Beck, Christian; Jentzen, Arnulf; Kuckuck, Benno 5 2022 A proof of convergence for gradient descent in the training of artificial neural networks for constant target functions. Zbl 1502.65037 Cheridito, Patrick; Jentzen, Arnulf; Riekert, Adrian; Rossmannek, Florian 4 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 2 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 Jentzen, Arnulf; Riekert, Adrian 1 2022 Solving the Kolmogorov PDE by means of deep learning. Zbl 1490.65006 Beck, Christian; Becker, Sebastian; Grohs, Philipp; Jaafari, Nor; Jentzen, Arnulf 38 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 32 2021 Deep splitting method for parabolic PDEs. Zbl 1501.65054 Beck, Christian; Becker, Sebastian; Cheridito, Patrick; Jentzen, Arnulf; Neufeld, Ariel 31 2021 Multilevel Picard iterations for solving smooth semilinear parabolic heat equations. Zbl 1476.65273 E, Weinan; Hutzenthaler, Martin; Jentzen, Arnulf; Kruse, Thomas 22 2021 Solving high-dimensional optimal stopping problems using deep learning. Zbl 1505.91413 Becker, Sebastian; Cheridito, Patrick; Jentzen, Arnulf; Welti, Timo 19 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 10 2021 Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients. Zbl 1482.65143 Jentzen, Arnulf; Kurniawan, Ryan 9 2021 Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values. Zbl 1473.35670 Andersson, Adam; Jentzen, Arnulf; Kurniawan, Ryan 9 2021 On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear parabolic partial differential equations. Zbl 1490.60200 Beck, Christian; Hutzenthaler, Martin; Jentzen, Arnulf 9 2021 Strong error analysis for stochastic gradient descent optimization algorithms. Zbl 1460.65071 Jentzen, Arnulf; Kuckuck, Benno; Neufeld, Ariel; von Wurstemberger, Philippe 8 2021 Non-convergence of stochastic gradient descent in the training of deep neural networks. Zbl 1494.65044 Cheridito, Patrick; Jentzen, Arnulf; Rossmannek, Florian 7 2021 On existence and uniqueness properties for solutions of stochastic fixed point equations. Zbl 1462.60096 Beck, Christian; Gonon, Lukas; Hutzenthaler, Martin; Jentzen, Arnulf 4 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 3 2021 On the strong regularity of degenerate additive noise driven stochastic differential equations with respect to their initial values. Zbl 1471.34116 Jentzen, Arnulf; Kuckuck, Benno; Müller-Gronbach, Thomas; Yaroslavtseva, Larisa 2 2021 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 60 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 52 2020 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 51 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 39 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 24 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 21 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 17 2020 Convergence rates for the stochastic gradient descent method for non-convex objective functions. Zbl 1520.68143 Fehrman, Benjamin; Gess, Benjamin; Jentzen, Arnulf 13 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 12 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 8 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. 3 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 106 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 64 2019 Deep optimal stopping. Zbl 1495.60029 Becker, Sebastian; Cheridito, Patrick; Jentzen, Arnulf 59 2019 A mild Itô formula for SPDEs. Zbl 1423.35472 Da Prato, Giuseppe; Jentzen, Arnulf; Röckner, Michael 32 2019 Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg-Landau equations. Zbl 1403.60052 Becker, Sebastian; Jentzen, Arnulf 24 2019 Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients. Zbl 1477.65020 Conus, Daniel; Jentzen, Arnulf; Kurniawan, Ryan 23 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 12 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 10 2019 Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations. Zbl 1403.60059 Jentzen, Arnulf; Salimova, Diyora; Welti, Timo 10 2019 Regularity properties for solutions of infinite dimensional Kolmogorov equations in Hilbert spaces. Zbl 1431.60051 Andersson, Adam; Hefter, Mario; Jentzen, Arnulf; Kurniawan, Ryan 5 2019 Solving high-dimensional partial differential equations using deep learning. Zbl 1416.35137 Han, Jiequn; Jentzen, Arnulf; E, Weinan 390 2018 Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations. Zbl 1432.65011 Hutzenthaler, Martin; Jentzen, Arnulf; Wang, Xiaojie 26 2018 Exponential moments for numerical approximations of stochastic partial differential equations. Zbl 1416.60072 Jentzen, Arnulf; Pušnik, Primož 11 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 9 2018 On the mild Itô formula in Banach spaces. Zbl 1402.60067 Cox, Sonja; Jentzen, Arnulf; Kurniawan, Ryan; Pušnik, Primož 4 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 244 2017 On the differentiability of solutions of stochastic evolution equations with respect to their initial values. Zbl 1373.34092 Andersson, Adam; Jentzen, Arnulf; Kurniawan, Ryan; Welti, Timo 7 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 5 2017 On stochastic differential equations with arbitrary slow convergence rates for strong approximation. Zbl 1355.65012 Jentzen, Arnulf; Müller-Gronbach, Thomas; Yaroslavtseva, Larisa 19 2016 Renormalized powers of Ornstein-Uhlenbeck processes and well-posedness of stochastic Ginzburg-Landau equations. Zbl 1342.60094 E, Weinan; Jentzen, Arnulf; Shen, Hao 12 2016 An exponential Wagner-Platen type scheme for SPDEs. Zbl 1345.60068 Becker, Sebastian; Jentzen, Arnulf; Kloeden, Peter E. 12 2016 Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients. Zbl 1330.60084 Hutzenthaler, Martin; Jentzen, Arnulf 114 2015 Loss of regularity for Kolmogorov equations. Zbl 1322.35083 Hairer, Martin; Hutzenthaler, Martin; Jentzen, Arnulf 44 2015 A Milstein scheme for SPDEs. Zbl 1318.60072 Jentzen, Arnulf; Röckner, Michael 36 2015 Galerkin approximations for the stochastic Burgers equation. Zbl 1267.60071 Blömker, Dirk; Jentzen, Arnulf 47 2013 Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations. Zbl 1283.60098 Hutzenthaler, Martin; Jentzen, Arnulf; Kloeden, Peter E. 35 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. 202 2012 Regularity analysis for stochastic partial differential equations with nonlinear multiplicative trace class noise. Zbl 1241.60030 Jentzen, Arnulf; Röckner, Michael 41 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. 226 2011 Taylor approximations for stochastic partial differential equations. Zbl 1240.35001 Jentzen, Arnulf; Kloeden, Peter 74 2011 Efficient simulation of nonlinear parabolic SPDEs with additive noise. Zbl 1223.60050 Jentzen, Arnulf; Kloeden, Peter; Winkel, Georg 36 2011 Convergence of the stochastic Euler scheme for locally Lipschitz coefficients. Zbl 1253.65006 Hutzenthaler, Martin; Jentzen, Arnulf 26 2011 Higher order pathwise numerical approximations of SPDEs with additive noise. Zbl 1228.65015 Jentzen, Arnulf 19 2011 Taylor expansions of solutions of stochastic partial differential equations with additive noise. Zbl 1220.35202 Jentzen, Arnulf; Kloeden, Peter 35 2010 Taylor expansions of solutions of stochastic partial differential equations. Zbl 1201.60064 Jentzen, Arnulf 13 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 5 2010 A unified existence and uniqueness theorem for stochastic evolution equations. Zbl 1200.60051 Jentzen, A.; Kloeden, P. E. 1 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. 78 2009 The numerical approximation of stochastic partial differential equations. Zbl 1205.60130 Jentzen, A.; Kloeden, P. E. 59 2009 Pathwise numerical approximations of SPDEs with additive noise under non-global Lipschitz coefficients. Zbl 1176.60051 Jentzen, Arnulf 40 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. 39 2009 A random Euler scheme for Carathéodory differential equations. Zbl 1160.65003 Jentzen, A.; Neuenkirch, A. 25 2009 Pathwise Taylor schemes for random ordinary differential equations. Zbl 1162.65305 Jentzen, Arnulf; Kloeden, Peter E. 14 2009 Pathwise convergence of numerical schemes for random and stochastic differential equations. Zbl 1179.65009 Jentzen, A.; Kloeden, P. E.; Neuenkirch, A. 2 2009 Pathwise convergent higher order numerical schemes for random ordinary differential equations. Zbl 1140.65015 Kloeden, Peter E.; Jentzen, Arnulf 19 2007 all cited Publications top 5 cited Publications all top 5 Cited by 1,963 Authors 66 Jentzen, Arnulf 27 Mao, Xuerong 26 Hutzenthaler, Martin 24 Kloeden, Peter Eris 24 Wang, Xiaojie 16 Liu, Wei 15 E, Weinan 15 Gan, Siqing 15 Hong, Jialin 13 Karniadakis, George Em 13 Reisinger, Christoph 13 Schwab, Christoph 12 Kovács, Mihály 12 Kruse, Thomas 12 Larsson, Stig 12 Tambue, Antoine 11 Lang, Annika 11 Liu, Mingzhu 11 Przybyłowicz, Paweł 11 Song, Minghui 11 Szpruch, Lukasz 10 Becker, Sebastian 10 Han, Jiequn 10 Hu, Junhao 10 Kruse, Raphael 10 Lord, Gabriel James 10 Mishra, Siddhartha 10 Ngo, Hoang-Long 10 Petersen, Philipp Christian 10 Redmann, Martin 10 Sabanis, Sotirios 10 Ying, Lexing 10 Yuan, Chenggui 9 Andersson, Adam 9 Beck, Christian 9 Bréhier, Charles-Edouard 9 Cui, Jianbo 9 Grohs, Philipp 9 Guo, Qian 9 Kumar, Chaman 9 Lu, Jianfeng 9 Warin, Xavier 9 Yang, Haizhao 8 Cao, Yanzhao 8 Gao, Shuaibin 8 Li, Xiaoyue 8 Neuenkirch, Andreas 8 Qi, Ruisheng 8 Yan, Zhenya 8 Yang, Hongfu 8 Zhang, Qimin 8 Zhao, Weidong 7 Blömker, Dirk 7 Kamrani, Minoo 7 Kuznetsov, Dmitriĭ Feliksovich 7 Liu, Zhihui 7 Mukam, Jean Daniel 7 Perdikaris, Paris G. 7 Wu, Fuke 6 Bayer, Christian 6 Cheridito, Patrick 6 Duan, Jinqiao 6 Fan, Yuwei 6 Jiao, Yuling 6 Kutyniok, Gitta 6 Lindner, Felix 6 Mirzaee, Farshid 6 Müller-Gronbach, Thomas 6 Nguyen, Tuan-Anh 6 Oosterlee, Cornelis Willebrordus 6 Petersson, Andreas 6 Schneider, Reinhold 6 Šiška, David 6 Taguchi, Dai 6 Yaroslavtseva, Larisa 6 Zhang, Wei 5 Asai, Yusuke 5 Bachouch, Achref 5 Bao, Feng 5 Cao, Wanrong 5 Darbon, Jerome 5 Dareiotis, Konstantinos Anastasios 5 Del Moral, Pierre 5 dos Reis, Gonçalo 5 Gerencsér, Máté 5 Gonon, Lukas 5 Gu, Yiqi 5 Han, Xiaoying 5 Hu, Ruimeng 5 Huang, Chengming 5 Huang, Jianguo 5 Kühn, Christian 5 Kurniawan, Ryan 5 Li, Yangrong 5 Lu, Xiliang 5 Luong, Duc-Trong 5 Nguwi, Jiang Yu 5 Oh, Tadahiro 5 Pham, Huyên 5 Privault, Nicolas ...and 1,863 more Authors all top 5 Cited in 245 Serials 100 Journal of Computational Physics 76 Journal of Computational and Applied Mathematics 42 Computer Methods in Applied Mechanics and Engineering 42 Applied Mathematics and Computation 39 SIAM Journal on Numerical Analysis 39 Journal of Scientific Computing 31 Applied Numerical Mathematics 31 SIAM Journal on Scientific Computing 31 Communications in Computational Physics 27 Stochastic and Partial Differential Equations. Analysis and Computations 25 Journal of Mathematical Analysis and Applications 24 Discrete and Continuous Dynamical Systems. Series B 23 BIT 22 The Annals of Applied Probability 21 Stochastic Processes and their Applications 18 Numerical Algorithms 18 International Journal of Computer Mathematics 17 Computers & Mathematics with Applications 16 Communications in Nonlinear Science and Numerical Simulation 16 SIAM Journal on Financial Mathematics 14 Mathematics of Computation 14 Stochastic Analysis and Applications 13 Quantitative Finance 12 Journal of Complexity 12 Advances in Computational Mathematics 11 Multiscale Modeling & Simulation 10 Journal of Differential Equations 10 Monte Carlo Methods and Applications 10 Electronic Journal of Probability 10 SN Partial Differential Equations and Applications 9 Mathematics and Computers in Simulation 9 Numerische Mathematik 9 Stochastics and Dynamics 9 Analysis and Applications (Singapore) 8 SIAM Journal on Control and Optimization 8 Physica D 8 Foundations of Computational Mathematics 8 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 7 Mathematical Methods in the Applied Sciences 7 Chaos, Solitons and Fractals 7 Applied Mathematics and Optimization 7 Calcolo 7 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 7 Computational and Applied Mathematics 7 Applied Mathematical Finance 7 Mathematical Finance 7 Methodology and Computing in Applied Probability 7 Differentsial’nye Uravneniya i Protsessy Upravleniya 7 Advances in Difference Equations 7 East Asian Journal on Applied Mathematics 7 Research in the Mathematical Sciences 6 Journal of Computational Mathematics 6 Discrete and Continuous Dynamical Systems. Series S 6 Numerical Mathematics: Theory, Methods and Applications 6 Communications in Theoretical Physics 5 Statistics & Probability Letters 5 Computational Mechanics 5 European Journal of Applied Mathematics 5 Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 5 Engineering Analysis with Boundary Elements 5 Finance and Stochastics 5 Abstract and Applied Analysis 5 Communications in Mathematics and Statistics 5 Proceedings of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences 5 SIAM Journal on Mathematics of Data Science 4 Inverse Problems 4 Journal of the Franklin Institute 4 Memoirs of the American Mathematical Society 4 Systems & Control Letters 4 Constructive Approximation 4 MCSS. Mathematics of Control, Signals, and Systems 4 Neural Networks 4 Japan Journal of Industrial and Applied Mathematics 4 Journal of Nonlinear Science 4 SIAM Journal on Applied Dynamical Systems 4 International Journal of Numerical Analysis and Modeling 4 Mathematical Biosciences and Engineering 4 Oberwolfach Reports 4 Advances in Applied Mathematics and Mechanics 4 Journal of Computational Dynamics 4 SIAM/ASA Journal on Uncertainty Quantification 3 Journal of Statistical Physics 3 Nonlinearity 3 Physics Letters. A 3 ZAMP. Zeitschrift für angewandte Mathematik und Physik 3 The Annals of Probability 3 Journal of Optimization Theory and Applications 3 Chinese Annals of Mathematics. Series B 3 Journal of Theoretical Probability 3 SIAM Review 3 Potential Analysis 3 Discrete and Continuous Dynamical Systems 3 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 3 Nonlinear Dynamics 3 Vietnam Journal of Mathematics 3 Infinite Dimensional Analysis, Quantum Probability and Related Topics 3 Journal of Evolution Equations 3 Computational Methods in Applied Mathematics 3 Journal of Numerical Mathematics 3 Journal of Applied Mathematics and Computing ...and 145 more Serials all top 5 Cited in 43 Fields 919 Numerical analysis (65-XX) 757 Probability theory and stochastic processes (60-XX) 383 Partial differential equations (35-XX) 363 Computer science (68-XX) 127 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 116 Ordinary differential equations (34-XX) 103 Systems theory; control (93-XX) 67 Biology and other natural sciences (92-XX) 65 Fluid mechanics (76-XX) 64 Calculus of variations and optimal control; optimization (49-XX) 58 Statistics (62-XX) 53 Approximations and expansions (41-XX) 52 Dynamical systems and ergodic theory (37-XX) 40 Operations research, mathematical programming (90-XX) 36 Statistical mechanics, structure of matter (82-XX) 30 Mechanics of deformable solids (74-XX) 25 Integral equations (45-XX) 23 Operator theory (47-XX) 21 Functional analysis (46-XX) 17 Quantum theory (81-XX) 11 Linear and multilinear algebra; matrix theory (15-XX) 11 Harmonic analysis on Euclidean spaces (42-XX) 10 Real functions (26-XX) 6 Special functions (33-XX) 6 Classical thermodynamics, heat transfer (80-XX) 5 General and overarching topics; collections (00-XX) 5 Difference and functional equations (39-XX) 5 Geophysics (86-XX) 5 Information and communication theory, circuits (94-XX) 4 Optics, electromagnetic theory (78-XX) 3 Global analysis, analysis on manifolds (58-XX) 2 Number theory (11-XX) 2 Functions of a complex variable (30-XX) 2 Integral transforms, operational calculus (44-XX) 2 Astronomy and astrophysics (85-XX) 1 History and biography (01-XX) 1 Mathematical logic and foundations (03-XX) 1 Commutative algebra (13-XX) 1 Category theory; homological algebra (18-XX) 1 Measure and integration (28-XX) 1 Convex and discrete geometry (52-XX) 1 General topology (54-XX) 1 Mechanics of particles and systems (70-XX) Citations by Year Wikidata Timeline The data are displayed as stored in Wikidata under a Creative Commons CC0 License. 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