Philipp Grohs: Catalogue data in Spring Semester 2016 |
| Name | Dr. Philipp Grohs |
| URL | http://www.sam.math.ethz.ch/~pgrohs |
| Department | Mathematics |
| Relationship | Lecturer |
| Number | Title | ECTS | Hours | Lecturers | |
|---|---|---|---|---|---|
| 401-2654-00L | Numerical Analysis II | 6 credits | 3V + 2U | P. Grohs | |
| Abstract | The central topic of this course is the numerical treatment of ordinary differential equations. It focuses on the derivation, analysis, efficient implementation, and practical application of single step methods and pay particular attention to structure preservation. | ||||
| Learning objective | The course aims to impart knowledge about important numerical methods for the solution of ordinary differential equations. This includes familiarity with their main ideas, awareness of their advantages and limitations, and techniques for investigating stability and convergence. Further, students should know about structural properties of ordinary diferential equations and how to use them as guideline for the selection of numerical integration schemes. They should also acquire the skills to implement numerical integrators in MATLAB and test them in numerical experiments. | ||||
| Content | 1 Einleitung 1.1 Anfangswertprobleme (AWP) 1.2 Beispiele und Grundbegriffe 1.2.1 Okologie 1.2.2 Chemische Reaktionskinetik 1.2.3 Physiologie 1.2.4 Mechanik 1.3 Theorie 1.3.1 Existenz und Eindeutigkeit von Loesungen 1.3.2 Lineare AWPe 1.3.3 Sensitivitaet 1.3.3.1 Grundbegriffe 1.3.3.2 Unser Problem: das Anfangswertproblem 1.3.3.3 Wohlgestelltheit 1.3.3.4 Asymptotische Kondition 1.3.3.5 Schlecht konditionierte AWPe 1.4 Polygonzugverfahren 1.4.1 Das explizite Euler-Verfahren 1.4.2 Das implizite Euler-Verfahren 1.4.3 Implizite Mittelpunktsregel 1.4.4 Stoermer-Verlet-Verfahren 2 Einschrittverfahren 2.1 Grundlagen 2.1.1 Abstrakte Einschrittverfahren 2.1.2 Konsistenz 2.1.3 Konvergenz 2.1.4 Das Aequivalenzprinzip 2.1.5 Reversibilitaet 2.2 Kollokationsverfahren 2.2.1 Konstruktion 2.2.2 Konvergenz von Kollokationsverfahren 2.3 Runge-Kutta-Verfahren 2.3.1 Konstruktion 2.3.2 Konvergenz 2.4 Extrapolationsverfahren 2.4.1 Der Kombinationstrick 2.4.2 Extrapolationsidee 2.4.3 Extrapolation von Einschrittverfahren 2.4.4 Lokale Extrapolations-Einschrittverfahren 2.4.5 Ordnungssteuerung 2.4.6 Extrapolation reversibler Einschrittverfahren 2.5 Splittingverfahren 2.6 Schrittweitensteuerung 3 Stabilitaet 3.1 Modellproblemanalyse 3.2 Vererbung asymptotischer Stabilitaet 3.3 Nichtexpansivitaet 3.4 Gleichmaessige Stabilitaet 3.5 Steifheit 3.6 Linear-implizite Runge-Kutta-Verfahren 3.7 Exponentielle Integratoren 3.8 Differentiell-Algebraische Anfangswertprobleme 3.8.1 Grundbegriffe 3.8.2 Runge-Kutta-Verfahren fuer Index-1-DAEs 3.8.3 DAEs mit hoeherem Index 4 Strukturerhaltende numerische Integration 4.1 Polynomiale Invarianten 4.2 Volumenerhaltung 4.3 Verallgemeinerte Reversibilitaet 4.4 Symplektizitaet 4.4.1 Symplektische Evolutionen Hamiltonscher Differentialgleichungen 4.4.2 Symplektische Integratoren 4.4.3 Rueckwaertsanalyse 4.4.4 Modifizierte Gleichungen: Fehleranalyse 4.4.5 Strukturerhaltende modifizierte Gleichungen 4.5 Methoden fuer oszillatorische Differentialgleichungen | ||||
| Lecture notes | Lecture slides including supplements will be provided electronically. | ||||
| Literature | Note: Extra reading is not considered important for understanding the course subjects. Deuflhard and Bornemann: Numerische Mathematik II - Integration gewöhnlicher Differentialgleichungen, Walter de Gruyter & Co., 1994. Hairer and Wanner: Solving ordinary differential equations II - Stiff and differential-algebraic problems, Springer-Verlag, 1996. Hairer, Lubich and Wanner: Geometric numerical integration - Structure-preserving algorithms for ordinary differential equations}, Springer-Verlag, Berlin, 2002. L. Gruene, O. Junge "Gewoehnliche Differentialgleichungen", Vieweg+Teubner, 2009. Hairer, Norsett and Wanner: Solving ordinary differential equations I - Nonstiff problems, Springer-Verlag, Berlin, 1993. Walter: Gewöhnliche Differentialgleichungen - Eine Einführung, Springer-Verlag, Berlin, 1972. Walter: Ordinary differential equations, Springer-Verlag, New York, 1998. | ||||
| Prerequisites / Notice | Homework problems involve MATLAB implementation of numerical algorithms. | ||||
| 401-5650-00L | Zurich Colloquium in Applied and Computational Mathematics  | 0 credits | 2K | R. Abgrall, H. Ammari, P. Grohs, R. Hiptmair, A. Jentzen, S. Mishra, S. Sauter, C. Schwab | |
| Abstract | Research colloquium | ||||
| Learning objective | |||||
| 406-0141-AAL | Linear Algebra and Numerical Analysis 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. | 5 credits | 11R | P. Grohs | |
| Abstract | Introduction to Linear Algebra and Numerical Analysis for Engineers. The contents of the course are covered in the book "Introduction to Linear Algebra" by Gilbert Strang (SIAM, 2003). MATLAB is used as a tool to formulate and implement numerical algorithms. | ||||
| Learning objective | To acquire basic knowledge of Linear Algebra and of a few fundamental numerical techniques. The course is meant to hone analytic and algorithmic skills. | ||||
| Content | 1. Vectors and vector spaces 2. Solving linear systems of equations (Gaussian elimination) 3. Orthogonality 4. Determinants 5. Eigenvalues and eigenvectors 6. Linear transformations 7. Numerical linear algebra in MATLAB 8. (Piecewise) polynomial interpolation 9. Splines | ||||
| Literature | G. Strang, "Introduction to linear algebra", Third edition, 2003, ISBN 0-9614088-9-8, http://math.mit.edu/linearalgebra/ T. Sauer. "Numerical analysis", Addison-Wesley 2006 | ||||