Lecture Basics in Applied Mathematics (WS 2024/2025)

Professors: Prof. Dr. Moritz Diehl, Prof. Dr. Patrick Dondl, Prof. Dr. Angelika Rohde

Assistant: Ben Deitmar (organizational), Coffi Hounkpe (coffi.hounkpe@mathematik.uni-freiburg.de), Thomas Müller (mueller.studiert@posteo.de), Leo Simpson (leo.bill.simpson@gmail.com)

Lectures: Tue & Thu, 8-10 a.m. Location: HS 2 Albertstr. 23b

Tutorial seesions: Thu. 10-12 am. Location: Ernst-Zermelo-Str. 1 SR 218

ETCS: 12 Points

Language: English

Contents

This course provides an introduction into the basic concepts, notions, definitions and results in probability theory, numerics and optimization, accompanied with programming projects in Python. Besides deepen mathematical skills in principle, the course lays the foundation of further classes in these three areas.

The first 5 weeks (14th Oct. - 15th Nov.) will be on probability theory with lectures held by Angelika Rohde and assistance by Thomas Müller.

The next 5 weeks (18th Nov. - 20. Dez.) will be on numerics with lectures held by Patrick Dondl and assistance by Coffi Hounkpe.

The final 5 weeks (7th Jan. - 7th Feb.) will be on optimization with lectures held by Moritz Diehl and assistance by Leo Simpson.

Lecture Notes

The lecture notes for the stochastics part by Angelika Rohde can be found here.

The lecture notes for the numerics part by Patrick Dondl will be posted here when available. The (German) source for the lecture notes is Numerik 3x9 by Sören Bartels (available for students in PDF format on the website of the university library). A recommended English textbook could be Numerical Mathematics by Alfio Quarteroni, Riccardo Sacco and Fausto Saleri.

Contents 

Week 1:

Matrix-factorizations, i.e., LU & Cholesky.
Some introduction to basic problems of numerics, i.e., conditioning, stability, complexity (on the example of matrix factorization).

Week 2:
Gauß-elimination & pivoting
Least squares problems (incl. QR decomposition)

Week 3:
Iterative methods for linear systems (Jacobi and Gauss-Seidel methods)
Polynomial interpolation

Week 4:
(Fast) Fourier transform
Numerical quadrature

Week 5:
Root finding, Newton-Raphson,
Conjugate gradients

Exercises

Weekly exercises to the lecture will be posted here. 

Each of the three parts of the lecture will have its own exercises and reaching 50% of available points in the exercises of each of the three parts will be necessary for passing the course. Specifically, you will need 50% of the points from exercises 1-5, also 50% of the points from exercises 6-10 and again 50% of the points from exercises 11-14.

Stochastic Replacement Exercise:

Anyone who could not be in Freiburg during the stochastic part of the lecture, will have the opportunity to get credit for it, if they get more than 50% of the points in this exercise sheet and are able to explain two (randomly selected) solutions they handed in. The exercise is due by the 07. of January 2025. Please send your solutions to the assistant Ben Deitmar (ben.deitmar@stochastik.uni-freiburg.de).

Programming exercises

The programming part of this course is made up of:

- a weekly programming exercise appended to the regular exercise sheets (graded together with the exercise sheets)

- a small programming project during the last week of the semester under Moritz Diehl (graded separately)

Programming project

The programming project will take place during the first week after the end of classes, i.e. between the 10.02.2025 and the 14.02.2025. The room 218 in Ernst-Zermelo-Str. 1 is booked as a study space for you during this time. You can work in groups of 2-3 people.

To find a topic for your project, please talk to an assistant of one of the three parts of the lecture before the 03.02.2025. Finding your own topic for the project is encouraged, but it must be confirmed by an assistant.

At the end of the week (on the 14.02.2025) you should hold a short talk about your project, where you explain your chosen problem and how you implemented your solution. A week later you are to hand in a short report of around 5 pages on your project.

Tutorial sessions

The exercises of the previous week will be reviewed in (mandatory) weekly tutorial session lead by Julian Wiedermann.

You are allowed to miss two tutorial sessions during the semester. Any further absence without a valid excuse may result in having to repeat the course.

Time: Thu. 10-12 am.

Location: Ernst-Zermelo-Str. 1 SR 218