## Christoph Schwab: Catalogue data in Autumn Semester 2020 |

Name | Prof. Dr. Christoph Schwab |

Field | Mathematik |

Address | Seminar für Angewandte Mathematik ETH Zürich, HG G 57.1 Rämistrasse 101 8092 Zürich SWITZERLAND |

Telephone | +41 44 632 35 95 |

Fax | +41 44 632 10 85 |

christoph.schwab@sam.math.ethz.ch | |

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

Department | Mathematics |

Relationship | Full Professor |

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

401-3650-68L | Numerical Analysis Seminar: Mathematics of Deep Neural Network Approximation Number of participants limited to 6. Consent of Instructor needed. | 4 credits | 2S | C. Schwab | |

Abstract | The seminar will review recent _mathematical results_ on approximation power of deep neural networks (DNNs). The focus will be on mathematical proof techniques to obtain approximation rate estimates (in terms of neural network size and connectivity) on various classes of input data including, in particular, selected types of PDE solutions. | ||||

Objective | |||||

Content | Presentation of the Seminar: Deep Neural Networks (DNNs) have recently attracted substantial interest and attention due to outperforming the best established techniques in a number of tasks (Chess, Go, Shogi, autonomous driving, language translation, image classification, etc.). In big data analysis, DNNs achieved remarkable performance in computer vision, speech recognition and natural language processing. In many cases, these successes have been achieved by heuristic implementations combined with massive compute power and training data. For a (bird's eye) view, see https://arxiv.org/abs/1901.05639 and, more mathematical and closer to the seminar theme, https://arxiv.org/abs/1901.02220 The seminar will review recent _mathematical results_ on approximation power of deep neural networks (DNNs). The focus will be on mathematical proof techniques to obtain approximation rate estimates (in terms of neural network size and connectivity) on various classes of input data including, in particular, selected types of PDE solutions. Mathematical results support that DNNs can equalize or outperform the best mathematical results known to date. Particular cases comprise: high-dimensional parametric maps, analytic and holomorphic maps, maps containing multi-scale features which arise as solution classes from PDEs, classes of maps which are invariant under group actions. Format of the Seminar: The seminar format will be oral student presentations, combined with written report. Student presentations will be based on a recent research paper selected in two meetings at the start of the semester. Grading of the Seminar: Passing grade will require a) 1hr oral presentation with Q/A from the seminar group and b) typed seminar report (``Ausarbeitung'') of several key aspects of the paper under review. Each seminar topic will allow expansion to a semester or a master thesis in the MSc MATH or MSc Applied MATH. Disclaimer: The seminar will _not_ address recent developments in DNN software, eg. TENSORFLOW, and algorithmic training heuristics, or programming techniques for DNN training in various specific applications. | ||||

401-3651-00L | Numerical Analysis for Elliptic and Parabolic Partial Differential Equations Course audience at ETH: 3rd year ETH BSc Mathematics and MSc Mathematics and MSc Applied Mathematics students. Other ETH-students are advised to attend the course "Numerical Methods for Partial Differential Equations" (401-0674-00L) in the CSE curriculum during the spring semester. | 10 credits | 4V + 1U | C. Schwab | |

Abstract | This course gives a comprehensive introduction into the numerical treatment of linear and nonlinear elliptic boundary value problems, related eigenvalue problems and linear, parabolic evolution problems. Emphasis is on theory and the foundations of numerical methods. Practical exercises include MATLAB implementations of finite element methods. | ||||

Objective | Participants of the course should become familiar with * concepts underlying the discretization of elliptic and parabolic boundary value problems * analytical techniques for investigating the convergence of numerical methods for the approximate solution of boundary value problems * methods for the efficient solution of discrete boundary value problems * implementational aspects of the finite element method | ||||

Content | The course will address the mathematical analysis of numerical solution methods for linear and nonlinear elliptic and parabolic partial differential equations. Functional analytic and algebraic (De Rham complex) tools will be provided. Primal, mixed and nonstandard (discontinuous Galerkin, Virtual, Trefftz) discretizations will be analyzed. Particular attention will be placed on developing mathematical foundations (Regularity, Approximation theory) for a-priori convergence rate analysis. A-posteriori error analysis and mathematical proofs of adaptivity and optimality will be covered. Implementations for model problems in MATLAB and python will illustrate the theory. A selection of the following topics will be covered: * Elliptic boundary value problems * Galerkin discretization of linear variational problems * The primal finite element method * Mixed finite element methods * Discontinuous Galerkin Methods * Boundary element methods * Spectral methods * Adaptive finite element schemes * Singularly perturbed problems * Sparse grids * Galerkin discretization of elliptic eigenproblems * Non-linear elliptic boundary value problems * Discretization of parabolic initial boundary value problems | ||||

Literature | SUPPLEMENTARY Literature (core material will be in lecture notes) Brenner, Susanne C.; Scott, L. Ridgway The mathematical theory of finite element methods. Third edition. Texts in Applied Mathematics, 15. Springer, New York, 2008. xviii+397 pp. A. Ern and J.L. Guermond: Theory and Practice of Finite Element Methods, Springer Applied Mathematical Sciences Vol. 159, Springer, 1st Ed. 2004, 2nd Ed. 2015. R. Verfürth: A Posteriori Error Estimation Techniques for Finite Element Methods, Oxford University Press, 2013 Additional Literature: D. Braess: Finite Elements, THIRD Ed., Cambridge Univ. Press, (2007). (Also available in German.) Brezis, Haim Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. xiv+599 pp. D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, vol. 69 SMAI Mathématiques et Applications, Springer, 2012 [DOI: 10.1007/978-3-642-22980-0] V. Thomee: Galerkin Finite Element Methods for Parabolic Problems, SECOND Ed., Springer Verlag (2006). | ||||

Prerequisites / Notice | Practical exercises based on MATLAB Former title of the course unit: Numerical Methods for Elliptic and Parabolic Partial Differential Equations | ||||

401-5650-00L | Zurich Colloquium in Applied and Computational Mathematics | 0 credits | 1K | R. Abgrall, R. Alaifari, H. Ammari, R. Hiptmair, S. Mishra, S. Sauter, C. Schwab | |

Abstract | Research colloquium | ||||

Objective |