## Arnulf Jentzen: Catalogue data in Autumn Semester 2018 |

Name | Dr. Arnulf Jentzen |

Field | Applied Mathematics |

URL | http://www.sam.math.ethz.ch/~jentzena |

Department | Mathematics |

Relationship | Assistant Professor |

Number | Title | ECTS | Hours | Lecturers | |
---|---|---|---|---|---|

401-4357-68L | On Deep Artificial Neural Networks and Partial Differential Equations | 4 credits | 2G | A. Jentzen | |

Abstract | In this lecture we rigorously analyse approximation capacities of deep artificial neural networks and prove that deep artificial neural networks do overcome the curse of dimensionality in the numerical approximation of solutions of partial differential equations (PDEs). | ||||

Objective | The aim of this course is to teach the students a decent knowledge on deep artificial neural networks and their approximation capacities. | ||||

Content | In recent years deep artificial neural networks (DNNs) have very successfully been used in numerical simulations for a series of computational problems ranging from computer vision, image classification, speech recognition, and natural language processing to computational advertisement. Such numerical simulations indicate that deep artificial neural networks seem to admit the fundamental power to overcome the curse of dimensionality when approximating the high-dimensional functions appearing in the above named applications. In this lecture we rigorously analyse approximation capacities of deep artificial neural networks and prove that deep artificial neural networks do overcome the curse of dimensionality in the numerical approximation of solutions of partial differential equations (PDEs). In particular, this course includes (i) a rigorous mathematical introduction to artificial neural networks, (ii) an introduction to some partial differential equations, and (iii) results on approximation capacities of deep artificial neural networks. | ||||

Lecture notes | Lecture notes will be available as a PDF file. | ||||

Literature | Related literature: * Arnulf Jentzen, Diyora Salimova, and Timo Welti, A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients. arXiv:1809.07321 (2018), 48 pages. Available online at [https://arxiv.org/abs/1809.07321]. * Philipp Grohs, Fabian Hornung, Arnulf Jentzen, and Philippe von Wurstemberger, A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations. arXiv:1809.02362 (2018), 124 pages. Available online at [https://arxiv.org/abs/1809.02362]. * Andrew R. Barron, Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inform. Theory 39 (1993), no. 3, 930--945. | ||||

Prerequisites / Notice | Prerequisites: Analysis I and II, Elementary Probability Theory, and Measure Theory | ||||

401-4657-00L | Numerical Analysis of Stochastic Ordinary Differential Equations Alternative course title: "Computational Methods for Quantitative Finance: Monte Carlo and Sampling Methods" | 6 credits | 3V + 1U | A. Jentzen, L. Yaroslavtseva | |

Abstract | Course on numerical approximations of stochastic ordinary differential equations driven by Wiener processes. These equations have several applications, for example in financial option valuation. This course also contains an introduction to random number generation and Monte Carlo methods for random variables. | ||||

Objective | The aim of this course is to enable the students to carry out simulations and their mathematical convergence analysis for stochastic models originating from applications such as mathematical finance. For this the course teaches a decent knowledge of the different numerical methods, their underlying ideas, convergence properties and implementation issues. | ||||

Content | Generation of random numbers Monte Carlo methods for the numerical integration of random variables Stochastic processes and Brownian motion Stochastic ordinary differential equations (SODEs) Numerical approximations of SODEs Applications to computational finance: Option valuation | ||||

Lecture notes | Lecture notes are available as a PDF file: see Learning materials. | ||||

Literature | P. Glassermann: Monte Carlo Methods in Financial Engineering. Springer-Verlag, New York, 2004. P. E. Kloeden and E. Platen: Numerical Solution of Stochastic Differential Equations. Springer-Verlag, Berlin, 1992. | ||||

Prerequisites / Notice | Prerequisites: Mandatory: Probability and measure theory, basic numerical analysis and basics of MATLAB programming. a) mandatory courses: Elementary Probability, Probability Theory I. b) recommended courses: Stochastic Processes. Start of lectures: Wednesday, September 19, 2018. Date of the End-of-Semester examination: Wednesday, December 19, 2018, 13:00-15:00; students must arrive before 12:30 at ETH HG E 19. Room for the End-of-Semester examination: ETH HG E 19. Exam inspection: Tuesday, February 26, 2019, 12:00-13:00 at HG D 7.2. Please bring your legi. | ||||

401-5650-00L | Zurich Colloquium in Applied and Computational Mathematics | 0 credits | 2K | R. Abgrall, R. Alaifari, H. Ammari, R. Hiptmair, A. Jentzen, C. Jerez Hanckes, S. Mishra, S. Sauter, C. Schwab | |

Abstract | Research colloquium | ||||

Objective |