| Private Homepage | https://www.uni-muenster.de/AMM/Jentzen/Mitarbeiter/included.shtml |
| Research Interests | Mathematics for machine learning Numerical approximations for high-dimensional partial differential equations Numerical approximations for stochastic differential equations Deep Learning |
| Selected Publications | •Hairer M, Hutzenthaler M, Jentzen A. Loss of regularity for Kolmogorov equations. Ann. Probab., Vol. 43 (2), 2015, pp 468-527 online •Hutzenthaler M, Jentzen A. On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients. Ann. Probab., Vol. 48 (1), 2020, pp 53-93 online •Hutzenthaler M, Jentzen A, Kloeden PE. Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., Vol. 467 (2130), 2011, pp 1563-1576 online •Fehrman B, Gess B, Jentzen A. Convergence rates for the stochastic gradient descent method for non-convex objective functions. J. Mach. Learn. Res., Vol. 21, 2020, pp Paper No. 136, 48 •Han J, Jentzen A, E W. Solving high-dimensional partial differential equations using deep learning. Proc. Natl. Acad. Sci. USA, Vol. 115 (34), 2018, pp 8505-8510 online •E W, Han J, Jentzen A. Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Commun. Math. Stat., Vol. 5 (4), 2017, pp 349-380 online •Hutzenthaler Martin, Jentzen Arnulf, Kloeden Peter E.. Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. The Annals of Applied Probability, Vol. 22, 2012 •Hutzenthaler M, Jentzen A. Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients. Mem. Amer. Math. Soc., Vol. 236 (1112), 2015, pp v+99 online •Hutzenthaler M, Jentzen A, Kruse T, Nguyen TA, Wurstemberger P. Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations. Proc. A., Vol. 476 (2244), 2020, pp 630-654 online |
| Selected Projects | •Mathematische Theorie zu Tiefem Lernen. Das zentrale Ziel dieses Projektes ist es eine rigorose mathematische Analyse fuer tief lernende Algorithmen zu entwickeln und damit mathematische Aussagen zu beweisen, welche sowohl den Erfolg als auch die Limitierungen von tief lernenden Algorithmen erklaeren. Insbesondere beabsichtigen wir in diesem Projekt (i) eine mathematische Theorie zu hochdimensionalen approximationseigenschaften von tiefen neuronalen Netzwerken zu entwickeln, (ii) geeignete glatte Funktionenfolgen, welche ohne den Fluch der Dimension von tiefen jedoch nicht von flachen neuronalen Netzwerken approximiert werden koennen, anzugeben und (iii) dimensionsunabhaengige Konvergenzraten fuer stochastische Gradientenverfahren mit hoechstens polynomiell in der Dimension wachsenden Fehlerkonstanten zu beweisen.. •Ueberwindung des Fluches der Dimension: Stochastische Approximationsalgorithmen fuer hochdimensionale partielle Differentialgleichungen. Partial differential equations (PDEs) are among the most universal tools used in modeling problems in nature and man-made complex systems. The PDEs appearing in applications are often high dimensional. Such PDEs can typically not be solved explicitly and developing efficient numerical algorithms for high dimensional PDEs is one of the most challenging tasks in applied mathematics. As is well-known, the difficulty lies in the so-called ''curse of dimensionality'' in the sense that the computational effort of standard approximation algorithms grows exponentially in the dimension of the considered PDE. It is the key objective of this research project to overcome this curse of dimensionality and to construct and analyze new approximation algorithms which solve high dimensional PDEs with a computational effffort that grows at most polynomially in both the dimension of the PDE and the reciprocal of the prescribed approximation precision.. •Existenz-, Eindeutigkeit- und Regularitaetseigenschaften von Loesungen von partielle Differentialgleichungen. The goal of this project is to reveal existence, uniqueness, and regularity properties of solutions of partial differential equations (PDEs). In particular, we intend to study existence, uniqueness, and regularity properties of viscosity solutions of degenerate semilinear Kolmogorov PDEs of the parabolic type. We plan to investigate such PDEs by means of probabilistic representations of the Feynman-Kac type. We also intend to study the connections of such PDEs to optimal control problems.. •Regularitaetseigeschaften und approximationen fuer stochastische gewoehnliche und partielle Differentialgleichungen mit nicht global Lipschitz-stetigen Nichtlinearitaeten. A number of stochastic ordinary and partial differential equations from the literature (such as, for example, the Heston and the 3/2-model from financial engineering, (overdamped) Langevin-type equations from molecular dynamics, stochastic spatially extended FitzHugh-Nagumo systems from neurobiology, stochastic Navier-Stokes equations, Cahn-Hilliard-Cook equations) contain non-globally Lipschitz continuous nonlinearities in their drift or diffusion coefficients. A central aim of this project is to investigate regularity properties with respect to the initial values of such stochastic differential equations in a systematic way. A further goal of this project is to analyze the regularity of solutions of the deterministic Kolmogorov partial dfferential equations associated to such stochastic differential equations. Another aim of this project is to analyze weak and strong convergence and convergence rates of numerical approximations for such stochastic differential equations.. |
| Project membership Mathematics Münster | C: Models and Approximations C1: Evolution and asymptotics C4: Geometry-based modelling, approximation, and reduction |
| Current Publications | • Cox S, Hutzenthaler M, Jentzen A, Neerven J, Welti T Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions. IMA J. Numer. Anal. Vol. 41 (1), 2021, pp 493-548 online • Jentzen A, Kuckuck B, Neufeld A, Wurstemberger P Strong error analysis for stochastic gradient descent optimization algorithms. IMA J. Numer. Anal. Vol. 41 (1), 2021, pp 455-492 online • Andersson A, Jentzen A, Kurniawan R Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values. J. Math. Anal. Appl. Vol. 495 (1), 2021, pp 124558, 33 online • Cheridito Patrick, Jentzen Arnulf, Rossmannek Florian Efficient approximation of high-dimensional functions with neural networks. IEEE Transactions on Neural Networks and Learning Systems Vol. 0, 2021 online • Beck C, Hornung F, Hutzenthaler M, Jentzen A, Kruse T Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations. J. Numer. Math. Vol. 28 (4), 2020, pp 197-222 online • Hutzenthaler M, Jentzen A On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients. Ann. Probab. Vol. 48 (1), 2020, pp 53-93 online • Berner J, Grohs P, Jentzen A 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. SIAM J. Math. Data Sci. Vol. 2 (3), 2020, pp 631-657 online • Jentzen A, Pušnik P Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities. IMA J. Numer. Anal. Vol. 40 (2), 2020, pp 1005-1050 online • Fehrman B, Gess B, Jentzen A Convergence rates for the stochastic gradient descent method for non-convex objective functions. J. Mach. Learn. Res. Vol. 21, 2020, pp Paper No. 136, 48 online |
| Current Projects | • Dynamical systems and irregular gradient flows online • Mathematical Theory for Deep Learning online • Existence, uniqueness, and regularity properties of solutions of partial differential equations online • Regularity properties and approximations for stochastic ordinary and partial differential equations with non-globally Lipschitz continuous nonlinearities online • Overcoming the curse of dimensionality: stochastic algorithms for high-dimensional partial differential equations online • EXC 2044 - C1: Evolution and asymptotics online • EXC 2044 - C3: Interacting particle systems and phase transitions online | ajentzen@uni-muenster.de |
| Phone | +49 251 83-33793 |
| FAX | +49 251 83-32729 |
| Room | 120.005 |
| Secretary | Sekretariat des Instituts für Analysis und Numerik Frau Claudia Giesbert Telefon +49 251 83-33792 Fax +49 251 83-32729 Zimmer 120.002 |
| Address | Prof. Dr. Arnulf Jentzen Angewandte Mathematik Münster: Institut für Analysis und Numerik Fachbereich Mathematik und Informatik der Universität Münster Orléans-Ring 10 48149 Münster |
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