Legacy display Course
This is an archived course. The content might be broken.
Vorlesung im Wintersemester 2015/16, Lecture in winter semester 2015/16:
Wissenschaftliches Rechnen I, Scientific Computing I (V3E1)
Prof. Dr. Marc Alexander Schweitzer
Quick Links
Aktuelles
- Wegen der Panorama of Mathematics Tagung zu Trends und Entwicklugen in der Mathematik entfällt die Vorlesung am 22.10.
- Die Vorlesung wir auf Englisch gelesen!
- Beide Übungen nun im größeren Raum 6.020
- Lösungen der Programmieraufgaben bitte per email an die Tutoren
- Bitte auf lesbare Abgaben, insbesonders DEUTLICH LESBARE Namen achten
- Aufgrund einer anderen Veranstaltung findet die Übung am Freitag, 2015-11-27 einmalig wieder im Raum 5.002 statt
- Übung am Dies Academicus, 2015-12-02, findet statt
- Vorlesung 2015-12-22 findet statt
- Keine Übungen 2015-12-21 bis 2016-01-10
Inhalt und Ziele der Vorlesung
Die Mathematik stellt eine wichtige Grundlage für viele Anwendungsbereiche des täglichen Lebens dar. Ingenieure, Logistikexperten und Ökonomen profitieren in gleicher Weise von mathematischen Methoden und Modellen. Jedoch kann nur ein Bruchteil der auftretenden Probleme analytisch gelöst werden, der Großteil ist mit Papier und Bleistift nicht zu bewältigen. Aus diesem Grund nutzt man zur Umsetzung der immer komplexer werdenden Verfahren den Computer als effizientes Hilfsmittel.
Der Hörer dieser Einführungsvorlesung lernt grundlegende Konzepte, Algorithmen und Methoden des wissenschaftlichen Rechnens kennen.
Er soll am Ende in der Lage sein, mit den erlernten Kenntnissen selbständig Methoden zu entwickeln, zu analysieren und umzusetzen, mit denen anwendungsorientierte Probleme effizient und genau gelöst werden können.
Die Auswahl der Inhalte orientiert sich dabei am
Modulhandbuch
für den Bachelorstudiengang Mathematik.
Da genügend Bedarf angemeldet wurde, wird die Vorlesung in Englisch zu lesen.
Themen
- Mathematische Modellierung: first principles, Erhaltungsgrößen
- Skalenaspekte: Entdimensionalisierung, Homogenisierung
- Klassifikation von partiellen Differentialgleichungen
- Diskretisierung: Finite Differenzen, Finite Elemente, Adaptivität, Fehlerschätzer
Literatur
- Eck, Garcke, Knabner: Mathematische Modellierung
- Braess, Finite Elemente: Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie
- Hackbusch, Theorie und Numerik elliptischer Differentialgleichungen, auch als preprint erhältlich
- Adams, Fournier, Sobolev spaces
News
- Because of Panorama of Mathematics there will be no lecture on Thursday, 2015-10-22.
- The lecture will be held in English!
- Both tutorials will now take place 0830-1000 in the larger room 6.020
- Please send in your solutions to programming exercises via mail to your tutorial's teaching assistant.
- Please take care of legible submission, in particular names.
- Because of another event, the tutorial on Friday, 2015-11-27 takes place in room 5.002
- Tutorial on Dies Academicus, 2015-12-02, will take place
- Lecture 2015-12-22 will be held
- No tutorials 2015-12-21 until 2016-01-10
Content and Goals of the lecture
Mathematics is fundamental for many fields of application. Engineering, logistics, economics among many others are fields thriving because of the use of mathematical models and methods. Unfortunately, only a minor fraction of the mathematical problems found in these fields can be solved analytically. Most of those problems are not tangible with just theory, paper and pencil.
The attendees of this introductory lecture learn basic notions, algorithms and methods of scientific computing.
After attending, the listeners should be able to use the notions learned in the lecture to develop, analyse and implement methods allowing the efficient solution of application oriented problems.
The selection of topics is based on the
module handbook
for the Bachelor programme in mathematics.
Since sufficient demand has been met, the lecture will be held in English.
Topics
- Mathematical Modeling: first principles, conservation equations
- Scaling: nondimensionalization, homogenization
- Classification of partial differential equations
- discretization: finite diffrences, finite elements, adaptivity, error erstimators
Literature
- Braess, Finite Elements: Theory, fast solvers, and applications in solid mechanics
- Hackbusch, Elliptic Differential Equations: Theory and numerical treatment
- Adams, Fournier, Sobolev spaces
Prerequisites
Prerequisites for this lecture are the topics and exercises of the preceding lecturesAlgorithmische Mathematik I (V1G5), Algorithmische Mathematik II (V1G6), V2E1 Einführung die Grundlagen der Numerik (V2E1)
These prerequisite topics include:
- Calculus, multidimensional differentiation and integration and Taylor expansion
- Elementary combinatorics and probability theory
- Structured programming, in particular C, Python
- Polynomial Interpolation
- Least Squares Approximation
- Conditioning of problems, Stability of algorithms
- Inner products, Orthogonality, Hilbert Spaces, Best approximation in Hilbert spaces
- Orthogonalization, Gram-Schmidt, Orthogonal Polynomials
- Numerical Integration, Quadrature
- Solution of systems of linear equations: Gauss elimination, LU decompostion, Cholesky decomposition, QR decomposition
- Solution of nonlinear equations: bisection, Newton
- Classical iterative methods for systems of linear equations: Jacobi, Gauss-Seidel, Richardson, SOR
- Krylov subspaces, Krylov subspace methods: Gradient descent, CG, PCG, MINRES, GMRES
- Preconditioning of iterative methods: PCG
- Eigenvalues, Eigenvectors, Bounds for the spectrum, Power iteration, Lanczos, Arnoldi, QR algorithm
Lecture times
| Dates: | Tuesday | 1000 – 1200 |
| Thursday | 0800 – 1000 | |
| Location: | Wegelerstraße 10 - Zeichensaal | |
BASIS
Tutorials
Registration for tutorials and homework assignments in the first lecture Tuesday, 2015-10-20. Please, be present to be able to find alternative timeslots if needed.The contact person for tutorials and homework assignments is Sa Wu.
Timeslots
| Number | Day | Time | Room | |
|---|---|---|---|---|
| 1 | Wednesday | 0830 – 1000 | 6.020, Wegelerstr. 6 | Sa Wu |
| 2 | Friday | 1215 – 1345 | 6.020, Wegelerstr. 6 | Denis Düsseldorf |
Homework Assignments
Worksheets with homework assignments are distributed and put on the website Tuesdays. Please, submission your homework assignments Tuesdays right before the lecture on week after handout with special exceptions for i.a. holidays.Programming exercises will be based mainly in Python/NumPy/matplotlib. Please send in your solutions to programming exercises via mail to your tutorial's teaching assistant.
This combination is a very useful for quick implementation. Algorithms can be put into code fast. Plots can be produced with little effort. Much of what is needed for the lecture can be found in the following examples.
Some Documentation and Tutorials can be found at the following links.- The Python Tutorial, Version 2.7.8
- matplotlib
One easy way to obtain all necessary Python packages is Anaconda.
More installation alternatives, suggestions and instructions can be found on the websites for NumPy and matplotlib.
Model solutions are base on Python in version 2.7.8. For editing and writing Python code, any good editor will do. We recommend Vim or Notepad++.
Worksheets
| Number | Worksheet | Due | Remarks and errata |
|---|---|---|---|
| 0 | blatt00.pdf | Warm up sheet, not graded, no submission. German, English version will follow shortly | |
| exercise_sheet_00.pdf | english version | ||
| typo in programming exercise 2b), should be \epsilon = 2^{-k}, corrected online | |||
| 1 | exercice_sheet_01.pdf | Tuedsday, 2015-11-03 | From a student remark to 4b): yes, the resulting equation is of second order, but with \tau replaced with t |
| 2 | exercice_sheet_02.pdf | Tuedsday, 2015-11-10 | 7a) should be just smooth, i.e. arbitrarily often differentiable instead of analytic |
| 3 | exercice_sheet_03.pdf | Tuedsday, 2015-11-17 | |
| 4 | exercice_sheet_04.pdf | Tuedsday, 2015-11-24 | Exercise 13, Friedrich consistency only for constant \frac{\Delta t}{\Delta x}, sign error in Lax-Wendroff, corrected now |
| Programming exercise 5, b=1, T=1 | |||
| Exercise 15, translation typo, u_O = u_E | |||
| 5 | exercice_sheet_05.pdf | Tuedsday, 2015-12-01 | Exercise 17b) \pi/2 missing |
| Exercise 18, new hint | |||
| 6 | exercice_sheet_06.pdf | Tuedsday, 2015-12-08 | Exercise 21b) \Omega connected is redundant here |
| 7 | exercice_sheet_07.pdf | Tuedsday, 2015-12-15 | |
| 8 | exercice_sheet_08.pdf | Tuedsday, 2015-01-12 | |
| 9 | exercice_sheet_09.pdf | Tuedsday, 2015-01-19 | |
| 10 | exercice_sheet_10.pdf | Tuedsday, 2015-01-26 | |
| 11 | exercice_sheet_11.pdf | Tuedsday, 2015-02-02 | |
| Exercise 41, property $F_i(x,y)=a+bx+cy+dxy$ for each component of $F=(F_0,F_1)$, bilinear should be affine | |||
| 12 | exercice_sheet_12.pdf | Some exercises as revision aid. No submission or grading | |
Exams
- Oral exams, length 30 minutes in room 6.007
- Please do not forget signing up for the exam via BASIS
- Exam slots available
- Monday 2016-02-15, 0900-1130, 1300-1630
- Tuesday, 2016-02-23, 0900-1130
- Wednesday, 2016-02-24, 0900-1130, 1300-1630
- Thursday, 2016-02-25, 0900-1130, 1300-1630
- Friday, 2016-02-26, 0900-1130, 1300-1630
- Registration for exam slots, first come first served via lists
- Thursday, 2016-02-04, end of lecture and 1300-1600 in office We. 04, 0.002
- Friday, 2016-02-05, 1300-1600 in office We. 04, 0.002
- Monday, 2016-02-08, 1300-1600 in office We. 04, 0.002
- Tuesday, 2016-02-09, beginning of lecture
- Missing choice of timeslot: Lukas Dreyer, Gabriel Aguirre. Please contact us as soon as possible.