Ralf Hiptmair: Catalogue data in Spring 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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| 401-0674-AAL | Numerical Methods for Partial Differential Equations Enrolment 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,among them (convection)-diffusion and heat equations, 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 Second-Order Scalar Elliptic Boundary Value Problems 1.2 Equilibrium Models: Examples 1.3 Sobolev spaces 1.4 Linear Variational Problems 1.5 Equilibrium Models: Boundary Value Problems 1.6 Diffusion Models (Stationary Heat Conduction) 1.7 Boundary Conditions 1.8 Second-Order Elliptic Variational Problems 1.9 Essential and Natural Boundary Conditions 2 Finite Element Methods (FEM) 2.2 Principles of Galerkin Discretization 2.3 Case Study: Linear FEM for Two-Point Boundary Value Problems 2.4 Case Study: Triangular Linear FEM in Two Dimensions 2.5 Building Blocks of General Finite Element Methods 2.6 Lagrangian Finite Element Methods 2.7 Implementation of Finite Element Methods 2.7.1 Mesh Generation and Mesh File Format 2.7.2 Mesh Information and Mesh Data Structures 2.7.2.1 L EHR FEM++ Mesh: Container Layer 2.7.2.2 L EHR FEM++ Mesh: Topology Layer 2.7.2.3 L EHR FEM++ Mesh: Geometry Layer 2.7.3 Vectors and Matrices 2.7.4 Assembly Algorithms 2.7.4.1 Assembly: Localization 2.7.4.2 Assembly: Index Mappings 2.7.4.3 Distribute Assembly Schemes 2.7.4.4 Assembly: Linear Algebra Perspective 2.7.5 Local Computations 2.7.5.1 Analytic Formulas for Entries of Element Matrices 2.7.5.2 Local Quadrature 2.7.6 Treatment of Essential Boundary Conditions 2.8 Parametric Finite Element Methods 3 FEM: Convergence and Accuracy 3.1 Abstract Galerkin Error Estimates 3.2 Empirical (Asymptotic) Convergence of Lagrangian FEM 3.3 A Priori (Asymptotic) Finite Element Error Estimates 3.4 Elliptic Regularity Theory 3.5 Variational Crimes 3.6 FEM: Duality Techniques for Error Estimation 3.7 Discrete Maximum Principle 3.8 Validation and Debugging of Finite Element Codes 4 Beyond FEM: Alternative Discretizations [dropped] 5 Non-Linear Elliptic Boundary Value Problems [dropped] 6 Second-Order Linear Evolution Problems 6.1 Time-Dependent Boundary Value Problems 6.2 Parabolic Initial-Boundary Value Problems 6.3 Linear Wave Equations 7 Convection-Diffusion Problems [dropped] 8 Numerical Methods for Conservation Laws 8.1 Conservation Laws: Examples 8.2 Scalar Conservation Laws in 1D 8.3 Conservative Finite Volume (FV) Discretization 8.4 Timestepping for Finite-Volume Methods 8.5 Higher-Order Conservative Finite-Volume Schemes | ||||||||||||||||||||||||||||||||||||||||||||||||||
| Lecture notes | The lecture will be taught in flipped classroom format: - Video tutorials for all thematic units will be published online. - Tablet notes accompanying the videos will be made available to the audience as PDF. - A comprehensive PDF handout will cover all aspects of the lecture. | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 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. | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-0674-00L | Numerical Methods for Partial Differential Equations Not meant for BSc/MSc students of mathematics. | 10 credits | 2G + 2U + 2P + 4A | 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 | 1.2.1 Elastic Membranes 1.2.2 Electrostatic Fields 1.2.3 Quadratic Minimization Problems 1.3 Sobolev spaces 1.4 Linear Variational Problems 1.5 EquilibriumModels: Boundary Value Problems 1.6 Diffusion Models: Stationary Heat Conduction 1.7 Boundary Conditions 1.8 Second-Order Elliptic Variational Problems 1.9 Essential and Natural Boundary Conditions 2.2 Principles of Galerkin Discretization 2.3 Case Study: Linear FEMfor Two-Point Boundary Value Problems 2.4 Case Study: Triangular Linear FEMin Two Dimensions I 2.4 Case Study: Triangular Linear FEMin Two Dimensions II 2.5 Building Blocks of General Finite Element Methods 2.6 Lagrangian Finite Element Methods 2.7.2 Mesh Information and Mesh Data Structures 2.7.4 Assembly Algorithms 2.7.5 Local Computations 2.7.6 Treatment of Essential Boundary Conditions 2.8 Parametric Finite Element Methods I 2.8 Parametric Finite Element Methods II 3.1 Abstract Galerkin Error Estimates 3.2 Empirical (Asymptotic) Convergence of Lagrangian FEM 3.3 A Priori (Asymptotic) Finite Element Error Estimates I 3.3 A Priori (Asymptotic) Finite Element Error Estimates II 3.3 A Priori (Asymptotic) Finite Element Error Estimates III 3.4 Elliptic Regularity Theory 3.5 Variational Crimes 3.6.1 Linear Output Functionals 3.6.2 Case Study: Computation of Boundary Fluxes with FEM 3.6.3 Lagrangian FEM: L2-Estimates 3.7 Discrete Maximum Principle 3.8 Validation and Debugging of Finite Element Codes 4.1 Finite Difference Methods (FDM) 4.2 Finite Volume Methods (FVM) 4.3 Spectral Galerkin Methods 4.4 Collocation Methods 6.1 Initial-Value Problems (IVPs) for Ordinary Differential Equations (ODEs) 6.2 Introduction: Polygonal Approximation Methods 6.3.2 (Asymptotic) Convergence of Single-Step Methods 6.3 General Single-Step Methods 6.4 Explicit Runge-Kutta Single-Step Methods (RKSSMs) 6.5 Adaptive Stepsize Control 7.1 Model Problem Analysis 7.2 Stiff Initial-Value Problems 7.3 Implicit Runge-Kutta Single-Step Methods 7.4 Semi-Implicit Runge-Kutta Methods 7.5 Splitting Methods 9.2.1 Heat Equation 9.2.2 Heat Equation: Spatial Variational Formulation 9.2.3 Stability of Parabolic Evolution Problems 9.2.4 Spatial Semi-Discretization: Method of Lines 9.2.7 Timestepping for Method-of-Lines ODE 9.2.8 Fully Discrete Method of Lines: Convergence 9.3.1 Models for Vibrating Membrane 9.3.2 Wave Propagation 9.3.3 Method of Lines for Wave Propagation 9.3.4 Timestepping for Semi-Discrete Wave Equations 9.3.5 The Courant-Friedrichs-Levy (CFL) Condition 10.1.1 Modeling Fluid Flow 10.1.2 Heat Convection and Diffusion 10.1.3 Incompressible Fluids 10.1.4 Time-Dependent (Transient) Heat Flow in a Fluid 10.2.1 Singular Perturbation 10.2.2 Upwinding 10.2.2.1 Upwind Quadrature 10.2.2.2 Streamline Diffusion 10.3.1 Method of Lines 10.3.2 Transport Equation 10.3.3 Lagrangian Split-Step Method 10.3.4 Semi-Lagrangian Method | ||||||||||||||||||||||||||||||||||||||||||||||||||
| Lecture notes | The lecture will be taught in flipped classroom format: - Video tutorials for all thematic units will be published online. - Tablet notes accompanying the videos will be made available to the audience as PDF. - A comprehensive lecture document will cover all aspects of the course. | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 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. | ||||||||||||||||||||||||||||||||||||||||||||||||||
Competencies |
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| 401-2673-AAL | Numerical Methods for CSE Enrolment 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 | Introduction into fundamental techniques and algorithms of numerical mathematics which play a central role in numerical simulations in science and technology. | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 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 in C++ | ||||||||||||||||||||||||||||||||||||||||||||||||||
| Content | 1. Computing with Matrices and Vectors 2. Direct Methods for Linear Systems of Equations 3. Direct Methods for Linear Least Squares Problems 4. Filtering Algorithms 5. Data Interpolation and Data Fitting in 1D 6. Approximation of Functions in 1D 7. Numerical Quadrature 8. Iterative Methods for Non-linear Systems of Equations 12. Numerical Integration - Single Step Methods 13. Single Step Methods for Stiff Initial Value Problems | ||||||||||||||||||||||||||||||||||||||||||||||||||
| Lecture notes | https://people.math.ethz.ch/~grsam/HS16/NumCSE/NumCSE16.pdf | ||||||||||||||||||||||||||||||||||||||||||||||||||
| Literature | 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 U. Ascher and C. Greif "A first course in Numerical Methods" | ||||||||||||||||||||||||||||||||||||||||||||||||||
| Prerequisites / Notice | Examination will be conducted at the computer and will involve coding in C++/Eigen. A course covering the material is taught in English every autumn term (course unit 401-0663-00L). Course documents, exercises and examinations are available online. | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 401-3667-22L | Case Studies Seminar (Spring Semester 2022) | 3 credits | 2S | V. C. Gradinaru, R. Hiptmair, R. Käppeli, 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. | ||||||||||||||||||||||||||||||||||||||||||||||||||
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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. | ||||||||||||||||||||||||||||||||||||||||||||||||||
| Prerequisites / Notice | The talks ar in presence only (no zoom)! Student talks are in parallel sessions in the two rooms, the invited talks take place in the larger lecture hall. 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 | ||||||||||||||||||||||||||||||||||||||||||||||||||
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