## Ralf Hiptmair: Catalogue data in Autumn Semester 2018 |

Name | Prof. Dr. Ralf Hiptmair |

Field | Mathematik |

Address | Seminar für Angewandte Mathematik ETH Zürich, HG G 58.2 Rämistrasse 101 8092 Zürich SWITZERLAND |

Telephone | +41 44 632 34 04 |

Fax | +41 44 632 11 04 |

ralf.hiptmair@sam.math.ethz.ch | |

URL | https://www.math.ethz.ch/sam/the-institute/people/ralf-hiptmair.html |

Department | Mathematics |

Relationship | Full Professor |

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

401-3640-13L | Seminar in Applied Mathematics: Shape Calculus Number of participants limited to 10 | 4 credits | 2S | R. Hiptmair | |

Abstract | Shape calculus studies the dependence of solutions of partial differential equations on deformations of the domain and/or interfaces. It is the foundation of gradient methods for shape optimization. The seminar will rely on several sections of monographs and research papers covering analytical and numerical aspects of shape calculus. | ||||

Objective | * Understanding of concepts like shape derivative, shape gradient, shape Hessian, and adjoint problem. * Ability to derive analytical formulas for shape gradients * Knowledge about numerical methods for the computation of shape gradients. | ||||

Content | Topics: 1. The velocity method and Eulerian shape gradients: Main reference [SZ92, Sect. 2.8–2.11, 2.1, 2.18], covers the “velocity method”, the Hadamard structure theorem and formulas for shape gradients of particular functionals. Several other sections of [SZ92,Ch. 2] provide foundations and auxiliary results and should be browsed, too. 2. Material derivatives and shape derivatives, based on [SZ92, Sect. 2.25–2.32]. 3. Shape calculus with exterior calculus, following [HL13] (without Sections 5 & 6). Based on classical vector analysis the formulas are also derived in [SZ92, Sects 2,19,2.20] and [DZ10, Ch. 9, Sect. 5]. Important background and supplementary information about the shape Hessian can be found in [DZ91, BZ97] and [DZ10, Ch. 9, Sect. 6]. 4. Shape derivatives of solutions of PDEs using exterior calculus [HL17], see also [HL13,Sects. 5 & 6]. From the perspective of classical calculus the topic is partly covered in [SZ92, Sects. 3.1-3.2]. 5. Shape gradients under PDE constraints according to [Pag16, Sect. 2.1] including a presentation of the adjoint method for differentiating constrained functionals [HPUU09, Sect. 1.6]. Related information can be found in [DZ10, Ch. 10, Sect. 2.5] and [SZ92, Sect. 3.3]. 6. Approximation of shape gradients following [HPS14]. Comparison of discrete shape gradients based on volume and boundary formulas, see also [DZ10, Ch. 10, Sect. 2.5]. 7. Optimal shape design based on boundary integral equations following [Epp00b], with some additional information provided in [Epp00a]. 8. Convergence in elliptic shape optimization as discussed in [EHS07]. Relies on results reported in [Epp00b] and [DP00]. Discusses Ritz-Galerkin discretization of optimality conditions for normal displacement parameterization. 9. Shape optimization by pursuing diffeomorphisms according to [HP15], see also [Pag16,Ch. 3] for more details, and [PWF17] for extensions. 10. Distributed shape derivative via averaged adjoint method following [LS16]. | ||||

Literature | References: [BZ97] Dorin Bucur and Jean-Paul Zolsio. Anatomy of the shape hessian via lie brackets. Annali di Matematica Pura ed Applicata, 173:127–143, 1997. 10.1007/BF01783465. [DP00] Marc Dambrine and Michel Pierre. About stability of equilibrium shapes. M2AN Math. Model. Numer. Anal., 34(4):811–834, 2000. [DZ91] Michel C. Delfour and Jean-Paul Zolésio. Velocity method and Lagrangian formulation for the computation of the shape Hessian. SIAM J. Control Optim., 29(6):1414–1442, 1991. [DZ10] M.C. Delfour and J.-P. Zolésio. Shapes and Geometries, volume 22 of Advances in Design and Control. SIAM, Philadelphia, 2nd edition, 2010. [EHS07] Karsten Eppler, Helmut Harbrecht, and Reinhold Schneider. On convergence in elliptic shape optimization. SIAM J. Control Optim., 46(1):61–83 2007. [Epp00a] Karsten Eppler. Boundary integral representations of second derivatives in shape optimization. Discuss. Math. Differ. Incl. Control Optim., 20(1):63–78, 2000. German-Polish Conference on Optimization—Methods and Applications (Żagań, 1999). [Epp00b] Karsten Eppler. Optimal shape design for elliptic equations via BIE-methods. Int. J. Appl. Math. Comput. Sci., 10(3):487–516, 2000. [HL13] Ralf Hiptmair and Jingzhi Li. Shape derivatives in differential forms I: an intrinsic perspective. Ann. Mat. Pura Appl. (4), 192(6):1077–1098, 2013. [HL17] R. Hiptmair and J.-Z. Li. Shape derivatives in differential forms II: Application to scattering problems. Report 2017-24, SAM, ETH Zürich, 2017. To appear in Inverse Problems. [HP15] Ralf Hiptmair and Alberto Paganini. Shape optimization by pursuing diffeomorphisms. Comput. Methods Appl. Math., 15(3):291–305, 2015. [HPS14] R. Hiptmair, A. Paganini, and S. Sargheini. Comparison of approximate shape gradients. BIT Numerical Mathematics, 55:459–485, 2014. [HPUU09] M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich. Optimization with PDE constraints, volume 23 of Mathematical Modelling: Theory and Applications. Springer, New York, 2009. [LS16] Antoine Laurain and Kevin Sturm. Distributed shape derivative via averaged adjoint method and applications. ESAIM Math. Model. Numer. Anal., 50(4):1241–1267,2016. [Pag16] A. Paganini. Numerical shape optimization with finite elements. Eth dissertation 23212, ETH Zurich, 2016. [PWF17] A. Paganini, F. Wechsung, and P.E. Farell. Higher-order moving mesh methods for pde-constrained shape optimization. Preprint arXiv:1706.03117 [math.NA], arXiv, 2017. [SZ92] J. Sokolowski and J.-P. Zolesio. Introduction to shape optimization, volume 16 of Springer Series in Computational Mathematics. Springer, Berlin, 1992. | ||||

Prerequisites / Notice | Knowledge of analysis and functional analysis; knowledge of PDEs is an advantage and so is some familiarity with numerical methods for PDEs | ||||

401-3667-68L | Case Studies Seminar (Autumn Semester 2018) | 3 credits | 2S | V. C. Gradinaru, R. Hiptmair, K. Nipp, M. Reiher | |

Abstract | In the CSE Case Studies Seminar invited speakers from ETH, from other universities as well as from industry give a talk on an applied topic. Beside of attending the scientific talks students are asked to give short presentations (10 minutes) on a published paper out of a list. | ||||

Objective | |||||

Prerequisites / Notice | 75% attendance and a short presentation on a published paper out of a list or on some own project are mandatory. Students that realize that they will not fulfill this criteria have to contact the teaching staff or de-register before the end of semester from the Seminar if they want to avoid a "Fail" in their documents. Later de-registrations will not be considered. | ||||

401-4503-68L | Reading Course: Reduced Basis Methods | 4 credits | 2G | R. Hiptmair | |

Abstract | Reduced Basis Methods (RBM) allow the efficient repeated numerical soluton of parameter depedent differential equations, which arise, e.g., in PDE-constrained optimization, optimal control, inverse problems, and uncertainty quantification. This course introduces the mathematical foundations of RBM and discusses algorithmic and implementation aspects. | ||||

Objective | * Knowledge about the main principles underlying RBMs * Familiarity with algorithms for the construction of reduced bases * Knowledge about the role of and techniques for a posteriori error estimation. * Familiarity with some applications of RBMs. | ||||

Literature | Main reference: Hesthaven, Jan S.; Rozza, Gianluigi; Stamm, Benjamin, Certified reduced basis methods for parametrized partial differential equations. SpringerBriefs in Mathematics, 2016 Supplementary reference: Quarteroni, Alfio; Manzoni, Andrea; Negri, Federico, Reduced basis methods for partial differential equations. An introduction. Unitext 92, Springer, Cham, 2016. | ||||

Prerequisites / Notice | This is a reading course, which will closely follow the book by J. Hesthaven, G. Rozza and B. Stamm. Participants are expected to study particular sections of the book every week, which will then be discussed during the course sessions. | ||||

401-4671-00L | Advanced Numerical Methods for CSE | 9 credits | 4V + 2U + 1P | R. Hiptmair, C. Jerez Hanckes | |

Abstract | This course discusses modern numerical methods involving complex algorithms and intricate data structures that render an efficient implementation non-trivial. The focus will be on boundary element methods, hierarchical matrix techniques, convolution quadrature, and algebraic multigrid methods. | ||||

Objective | - Appreciation of the interplay of functional analysis, advanced calculus, numerical linear algebra, and sophisticated data structures in modern computer simulation technology. - Knowledge about the main ideas and mathematical foundations underlying boundary element methods, hierarchical matrix techniques, convolution quadrature, and reduced basis methods. - Familiarity with the algorithmic challenges arising with these methods and the main ways on how to tackle them. - Knowledge about the algorithms' complexity and suitable data structures. - Ability to understand details of given implementations. - Skills concerning the implementation of algorithms and data structures in C++. | ||||

Content | 1 Boundary Element Methods (BEM) 1.1 Elliptic Model Boundary Value Problem: Electrostatics . . . . . . . . 1.2 Boundary Representation Formulas . . . . . . . . . . . . . . . . . . 1.3 Boundary Integral Equations (BIEs) . . . . . . . . . . . . . . . . . . 1.4 Boundary Element Methods in Two Dimensions . . . . . . . . . . . . . . . . . . . 1.5 Boundary Element Methods on Closed Surfaces . . . . . . . . . . . . . . . . . . . 1.6 BEM: Various Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Local Low-Rank Compression of Non-Local Operators 2.1 Examples: Non-Local Operators . . . . . . . . . . . . . . . . . . . . . 2.2 Approximation of Kernel Collocation Matrices . . . . . . . . . . . . . . . 2.3 Clustering Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Hierarchical Matrices . . . . . . . . . . . . . . . . . . . 3 Convolution Quadrature 3.1 Basic Concepts and Tools 3.2 Convolution Equations: Examples . . . . . . . . . . . . . . 3.3 Implicit-Euler Convolution Quadrature . . . . . . . . . . . . 3.5 Runge-Kutta Convolution Quadrature . . . . . . . . . . . . 3.6 Fast Oblivious Convolution Quadrature . . . . . . . . 4 Algebraic Multigrid Methods | ||||

Lecture notes | Lecture material will be created during the course and will be made available online and in chapters. | ||||

Literature | S. Sauter and Ch. Schwab, Boundary Element Methods, Springer 2010 O. Steinbach, Numerical approximation methods for elliptic boundary value problems, Springer 2008 M. Bebendorf, Hierarchical matrices: A means to efficiently solve elliptic boundary value problems, Springer 2008 W. Hackbusch, Hierarchical Matrices, Springer 2015 S. Boerm, Efficient Numerical Methods for Non-Local Operators: H2-Matrix Compression, Algorithms and Analysis, EMS 2010 S. Boerm, Numerical Methods for Non-Local Operators, Lecture Notes Univ. Kiel 2017 M. Hassell and F.-J. Sayas, Convolution Quadrature for Wave Simulations J.-C. Xu and L. Zikatanov, Algebraic Multirgrid Methods, Acta Numerica, 2017 Ch. Wagner, Introduction to Algebraic Multigrid, Lecture notes IWR Heidelberg, 1999, https://perso.uclouvain.be/alphonse.magnus/num2/amg.pdf | ||||

Prerequisites / Notice | - Familiarity with basic numerical methods (as taught in the course "Numerical Methods for CSE"). - Knowledge about the finite element method for elliptic partial differential equations (as covered in the course "Numerical Methods for Partial Differential Equations"). | ||||

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 |