## Ralf Hiptmair: Catalogue data in Autumn Semester 2022 |

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 | ||||||||||||||||||||||||||||||||||||||||||||
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401-0663-00L | Numerical Methods for Computer Science | 7 credits | 2V + 2U + 2P | R. Hiptmair | ||||||||||||||||||||||||||||||||||||||||||||

Abstract | The course gives an introduction into fundamental techniques and algorithms of numerical mathematics which play a central role in numerical simulations in science and technology. The course focuses on fundamental ideas and algorithmic aspects of numerical methods. The exercises involve actual implementation of numerical methods in C++. | |||||||||||||||||||||||||||||||||||||||||||||||

Learning objective | * Knowledge of the fundamental algorithms in numerical mathematics * Knowledge of the essential terms in numerical mathematics and the techniques used for the analysis of numerical algorithms * Ability to choose the appropriate numerical method for concrete problems * Ability to interpret numerical results * Ability to implement numerical algorithms afficiently | |||||||||||||||||||||||||||||||||||||||||||||||

Content | First two weeks: A gentle introduction to C++ 1. Computing with Matrices and Vectors 1.1 Fundamentals 1.2 Software and Libraries 1.4 Computational Effort 1.5 Machine Arithmetic and Consequences 2. Direct Methods for (Square) Linear Systems of Equations 2.1 Introduction: Linear Systems of Equations 2.3 Gaussian Elimination 2.6 Exploiting Structure when Solving Linear Systems 2.7 Sparse Linear Systems 3. Direct Methods for Linear Least Squares Problems 3.1 Least Squares Solution Concepts 3.2 Normal Equation Methods 3.3 Orthogonal Transformation Methods 3.3.1 Transformation Idea 3.3.2 Orthogonal/Unitary Matrices 3.3.3 QR-Decomposition 3.3.4 QR-Based Solver for Linear Least Squares Problems 3.4 Singular Value Decomposition 4. Filtering Algorithms 4.1 Filters and Convolutions 4.2 Discrete Fourier Transform (DFT) 4.3 Fast Fourier Transform (FFT) 5. Machine Learning of One-Dimensional Data (Data Interpolation and Data Fitting in 1D) 5.1 Abstract Interpolation (AI) 5.2 Global Polynomial Interpolation 8. Iterative Methods for Non-Linear Systems of Equations 8.1 Introduction 8.2 Iterative Methods 8.3 Fixed-Point Iterations 8.4 Finding Zeros of Scalar Functions 8.5 Newton’s Method in Rn 8.6. Quasi-Newton Method | |||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Lecture materials (PDF documents and codes) will be made available to the participants through the course web page and online repositories. Access information will be communicated in the beginning of the course. | |||||||||||||||||||||||||||||||||||||||||||||||

Literature | U. ASCHER AND C. GREIF, A First Course in Numerical Methods, SIAM, Philadelphia, 2011. A. QUARTERONI, R. SACCO, AND F. SALERI, Numerical mathematics, vol. 37 of Texts in Applied Mathematics, Springer, New York, 2000. W. Dahmen, A. Reusken "Numerik für Ingenieure und Naturwissenschaftler", Springer 2006. W. Gander, M.J. Gander, and F. Kwok "Scientific Computing", Springer 2014. M. Hanke-Bourgeois "Grundlagen der Numerischen Mathematik und des wissenschaftlichen Rechnens", BG Teubner, 2002 P. Deuflhard and A. Hohmann, "Numerische Mathematik I", DeGruyter, 2002 | |||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | The course will be accompanied by programming exercises in C++ relying on the template library EIGEN. Familiarity with C++, object oriented and generic programming is an advantage. Participants of the course are expected to learn C++ by themselves, in case they do not know it already. | |||||||||||||||||||||||||||||||||||||||||||||||

Competencies |
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401-0674-AAL | Numerical Methods for Partial Differential EquationsEnrolment ONLY for MSc students with a decree declaring this course unit as an additional admission requirement. Any other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | 10 credits | 21R | R. Hiptmair | ||||||||||||||||||||||||||||||||||||||||||||

Abstract | Derivation, properties, and implementation of fundamental numerical methods for a few key partial differential equations: convection-diffusion, heat equation, wave equation, conservation laws. Implementation in C++ based on a finite element library. | |||||||||||||||||||||||||||||||||||||||||||||||

Learning objective | Main skills to be acquired in this course: * Ability to implement fundamental numerical methods for the solution of partial differential equations efficiently. * Ability to modify and adapt numerical algorithms guided by awareness of their mathematical foundations. * Ability to select and assess numerical methods in light of the predictions of theory * Ability to identify features of a PDE (= partial differential equation) based model that are relevant for the selection and performance of a numerical algorithm. * Ability to understand research publications on theoretical and practical aspects of numerical methods for partial differential equations. * Skills in the efficient implementation of finite element methods on unstructured meshes. This course is neither a course on the mathematical foundations and numerical analysis of methods nor an course that merely teaches recipes and how to apply software packages. | |||||||||||||||||||||||||||||||||||||||||||||||

Content | 1 Case Study: A Two-point Boundary Value Problem [optional] 1.1 Introduction 1.2 A model problem 1.3 Variational approach 1.4 Simplified model 1.5 Discretization 1.5.1 Galerkin discretization 1.5.2 Collocation [optional] 1.5.3 Finite differences 1.6 Convergence 2 Second-order Scalar Elliptic Boundary Value Problems 2.1 Equilibrium models 2.1.1 Taut membrane 2.1.2 Electrostatic fields 2.1.3 Quadratic minimization problems 2.2 Sobolev spaces 2.3 Variational formulations 2.4 Equilibrium models: Boundary value problems 3 Finite Element Methods (FEM) 3.1 Galerkin discretization 3.2 Case study: Triangular linear FEM in two dimensions 3.3 Building blocks of general FEM 3.4 Lagrangian FEM 3.4.1 Simplicial Lagrangian FEM 3.4.2 Tensor-product Lagrangian FEM 3.5 Implementation of FEM in C++ 3.5.1 Mesh file format (Gmsh) 3.5.2 Mesh data structures (DUNE) 3.5.3 Assembly 3.5.4 Local computations and quadrature 3.5.5 Incorporation of essential boundary conditions 3.6 Parametric finite elements 3.6.1 Affine equivalence 3.6.2 Example: Quadrilaterial Lagrangian finite elements 3.6.3 Transformation techniques 3.6.4 Boundary approximation 3.7 Linearization [optional] 4 Finite Differences (FD) and Finite Volume Methods (FV) [optional] 4.1 Finite differences 4.2 Finite volume methods (FVM) 5 Convergence and Accuracy 5.1 Galerkin error estimates 5.2 Empirical Convergence of FEM 5.3 Finite element error estimates 5.4 Elliptic regularity theory 5.5 Variational crimes 5.6 Duality techniques [optional] 5.7 Discrete maximum principle [optional] 6 2nd-Order Linear Evolution Problems 6.1 Parabolic initial-boundary value problems 6.1.1 Heat equation 6.1.2 Spatial variational formulation 6.1.3 Method of lines 6.1.4 Timestepping 6.1.5 Convergence 6.2 Wave equations [optional] 6.2.1 Vibrating membrane 6.2.2 Wave propagation 6.2.3 Method of lines 6.2.4 Timestepping 6.2.5 CFL-condition 7 Convection-Diffusion Problems [optional] 7.1 Heat conduction in a fluid 7.1.1 Modelling fluid flow 7.1.2 Heat convection and diffusion 7.1.3 Incompressible fluids 7.1.4 Transient heat conduction 7.2 Stationary convection-diffusion problems 7.2.1 Singular perturbation 7.2.2 Upwinding 7.3 Transient convection-diffusion BVP 7.3.1 Method of lines 7.3.2 Transport equation 7.3.3 Lagrangian split-step method 7.3.4 Semi-Lagrangian method 8 Numerical Methods for Conservation Laws 8.1 Conservation laws: Examples 8.2 Scalar conservation laws in 1D 8.3 Conservative finite volume discretization 8.3.1 Semi-discrete conservation form 8.3.2 Discrete conservation property 8.3.3 Numerical flux functions 8.3.4 Montone schemes 8.4 Timestepping 8.4.1 Linear stability 8.4.2 CFL-condition 8.4.3 Convergence 8.5 Higher order conservative schemes [optional] 8.5.1 Slope limiting 8.5.2 MUSCL scheme 8.6. FV-schemes for systems of conservation laws [optional] "optional" indicates that the corresponding topic might be skipped depending on the progress of the course. | |||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | The lecture will be taught in flipped classroom format: - Video tutorials for all thematic units will be published online. - Solution of homework problems will partly be covered by video tutorials. - Lecture documents and tablet notes accompanying the videos will be made available to the audience as PDF. | |||||||||||||||||||||||||||||||||||||||||||||||

Literature | Chapters of the following books provide supplementary reading (detailed references in course material): * D. Braess: Finite Elemente, Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie, Springer 2007 (available online). * S. Brenner and R. Scott. Mathematical theory of finite element methods, Springer 2008 (available online). * A. Ern and J.-L. Guermond. Theory and Practice of Finite Elements, volume 159 of Applied Mathematical Sciences. Springer, New York, 2004. * Ch. Großmann and H.-G. Roos: Numerical Treatment of Partial Differential Equations, Springer 2007. * W. Hackbusch. Elliptic Differential Equations. Theory and Numerical Treatment, volume 18 of Springer Series in Computational Mathematics. Springer, Berlin, 1992. * P. Knabner and L. Angermann. Numerical Methods for Elliptic and Parabolic Partial Differential Equations, volume 44 of Texts in Applied Mathematics. Springer, Heidelberg, 2003. * S. Larsson and V. Thomée. Partial Differential Equations with Numerical Methods, volume 45 of Texts in Applied Mathematics. Springer, Heidelberg, 2003. * R. LeVeque. Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, UK, 2002. However, study of supplementary literature is not important for for following the course. | |||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | Mastery of basic calculus and linear algebra is taken for granted. Familiarity with fundamental numerical methods (solution methods for linear systems of equations, interpolation, approximation, numerical quadrature, numerical integration of ODEs) is essential. Important: Coding skills and experience in C++ are essential. Homework assignments involve substantial coding, partly based on a C++ finite element library. The written examination will be computer based and will comprise coding tasks. | |||||||||||||||||||||||||||||||||||||||||||||||

401-2673-AAL | Numerical Methods for CSEEnrolment ONLY for MSc students with a decree declaring this course unit as an additional admission requirement. Any other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | 9 credits | 19R | R. Hiptmair | ||||||||||||||||||||||||||||||||||||||||||||

Abstract | he course gives an introduction into fundamental techniques and algorithms of numerical mathematics which play a central role in numerical simulations in science and technology. The course focuses on fundamental ideas and algorithmic aspects of numerical methods. The exercises involve actual implementation of numerical methods in C++. | |||||||||||||||||||||||||||||||||||||||||||||||

Learning objective | * Knowledge of the fundamental algorithms in numerical mathematics * Knowledge of the essential terms in numerical mathematics and the techniques used for the analysis of numerical algorithms * Ability to choose the appropriate numerical method for concrete problems * Ability to interpret numerical results * Ability to implement numerical algorithms afficiently | |||||||||||||||||||||||||||||||||||||||||||||||

Content | * Direct Methods for linear systems of equations * Least Squares Techniques * Data Interpolation and Fitting * Filtering Algorithms * Approximation of Functions * Numerical Quadrature * Iterative Methods for non-linear systems of equations | |||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Lecture materials (PDF documents and codes) will be made available to participants. | |||||||||||||||||||||||||||||||||||||||||||||||

Literature | U. ASCHER AND C. GREIF, A First Course in Numerical Methods, SIAM, Philadelphia, 2011. A. QUARTERONI, R. SACCO, AND F. SALERI, Numerical mathematics, vol. 37 of Texts in Applied Mathematics, Springer, New York, 2000. W. Dahmen, A. Reusken "Numerik für Ingenieure und Naturwissenschaftler", Springer 2006. M. Hanke-Bourgeois "Grundlagen der Numerischen Mathematik und des wissenschaftlichen Rechnens", BG Teubner, 2002 P. Deuflhard and A. Hohmann, "Numerische Mathematik I", DeGruyter, 2002 | |||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | Solid knowledge about fundamental concepts and technques from linear algebra & calculus as taught in the first year of science and engineering curricula. The course will be accompanied by programming exercises in C++ relying on the template library EIGEN. Familiarity with C++, object oriented and generic programming is an advantage. Participants of the course are expected to learn C++ by themselves. | |||||||||||||||||||||||||||||||||||||||||||||||

401-2673-00L | Numerical Methods for CSE | 9 credits | 2V + 2U + 4P | R. Hiptmair | ||||||||||||||||||||||||||||||||||||||||||||

Abstract | The course gives an introduction into fundamental techniques and algorithms of numerical mathematics which play a central role in numerical simulations in science and technology. The course focuses on fundamental ideas and algorithmic aspects of numerical methods. The exercises involve actual implementation of numerical methods in C++. | |||||||||||||||||||||||||||||||||||||||||||||||

Learning objective | * Knowledge of the fundamental algorithms in numerical mathematics * Knowledge of the essential terms in numerical mathematics and the techniques used for the analysis of numerical algorithms * Ability to choose the appropriate numerical method for concrete problems * Ability to interpret numerical results * Ability to implement numerical algorithms afficiently | |||||||||||||||||||||||||||||||||||||||||||||||

Content | * Computing with Matrices and Vectors * Direct Methods for linear systems of equations * Least Squares Techniques * Data Interpolation and Fitting * Iterative Methods for non-linear systems of equations * Filtering Algorithms * Approximation of Functions * Numerical Quadrature | |||||||||||||||||||||||||||||||||||||||||||||||

Lecture notes | Lecture materials (PDF documents and codes) will be made available to the participants through the course web page, whose address will be announced in the beginning of the course. | |||||||||||||||||||||||||||||||||||||||||||||||

Literature | U. ASCHER AND C. GREIF, A First Course in Numerical Methods, SIAM, Philadelphia, 2011. A. QUARTERONI, R. SACCO, AND F. SALERI, Numerical mathematics, vol. 37 of Texts in Applied Mathematics, Springer, New York, 2000. W. Dahmen, A. Reusken "Numerik für Ingenieure und Naturwissenschaftler", Springer 2006. W. Gander, M.J. Gander, and F. Kwok "Scientific Computing", Springer 2014. M. Hanke-Bourgeois "Grundlagen der Numerischen Mathematik und des wissenschaftlichen Rechnens", BG Teubner, 2002 P. Deuflhard and A. Hohmann, "Numerische Mathematik I", DeGruyter, 2002 | |||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | The course will be accompanied by programming exercises in C++ relying on the template library EIGEN. Knowledge of C++ is taken for granted. | |||||||||||||||||||||||||||||||||||||||||||||||

Competencies |
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401-3650-72L | Rational Approximation and Interpolation Does not take place this semester. | 4 credits | 2S | R. Hiptmair | ||||||||||||||||||||||||||||||||||||||||||||

Abstract | The seminar covers theory and algorithms for rational interpolation based on classical and modern literature. The various topics have to be presented by groups of students. | |||||||||||||||||||||||||||||||||||||||||||||||

Learning objective | Participants of the seminar should acquire familiarity with the theoretical properties of approximation by means of rational functions as well as knowledge about algorithms used for computing approximating or interpolating rational functions. | |||||||||||||||||||||||||||||||||||||||||||||||

Content | The simplest and most widely used function system for approximation in computational mathematics are polynomials. They are ideally suited for smooth (analytic) functions. However, in many application we encounter functions with kinks and other kinds of singularities. In this case approximation by rational functions, that is, quotients of polynomials, may be vastly superior. This is why rational approximation and interpolation is receiving increased attention for the construction of surrogate models in model order reduction. This seminar will study a number of research papers dealing with both theoretical and algorithmic aspects of rational approximation and interpolation. | |||||||||||||||||||||||||||||||||||||||||||||||

Literature | Will be announced in due course | |||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | Good skills in analysis are required as well as basic familiarity with numerical methods for interpolation and approximation with polynomials. Preparatory meeting: Mon, Sep 19, 2022, 18:00 on ZOOM, Meeting ID: 698 4220 0325, Password: RAP HS22 Every presentation has to be done jointly by a group of 2-3 students with presenters selected at random. Every participant will have to present on 2-3 occasions. See https://people.math.ethz.ch/~hiptmair/Seminars/RAP_22/SeminarRAP_HS22.pdf for more information. | |||||||||||||||||||||||||||||||||||||||||||||||

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401-3667-72L | Case Studies Seminar (Autumn Semester 2022) | 3 credits | 2S | V. C. Gradinaru, R. Hiptmair | ||||||||||||||||||||||||||||||||||||||||||||

Abstract | 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. Students have to register their presentations online on https://rw.ethz.ch/the-programme/case-studies.html by the first week of the teaching period. | |||||||||||||||||||||||||||||||||||||||||||||||

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Content | 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 (containing articles from, e.g., Nature, Science, Scientific American, etc.). If the underlying paper comprises more than 15 pages, two or three consecutive case studies presentations delivered by different students can be based on it. Consistency in layout, style, and contents of those presentations is expected. Students have to register their presentations online on https://rw.ethz.ch/the-programme/case-studies.html by the first week of the teaching period. | |||||||||||||||||||||||||||||||||||||||||||||||

Prerequisites / Notice | The talks might be given via Zoom; talks in presence should be also streamed in Zoom. 75% attendance and a short presentation on a published paper out of a list or on some own project are mandatory. Students have to register their presentations online until the second Wednesday of the semester on https://rw.ethz.ch/the-programme/case-studies.html The student talks will be grouped by subject, so we'll decide the actual dates of the individual talks. 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. | |||||||||||||||||||||||||||||||||||||||||||||||

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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 | |||||||||||||||||||||||||||||||||||||||||||||||

Learning objective |