Analytical methods in mathematical modeling 4010-MAMa

A detailed description of the subject is provided at the level of a given teaching cycle in the Description section.

Term 2026Z:

1. Algorithms – general concepts (2 lectures, 2 exercises):
a. Problem - Model - Algorithm - Implementation
b. Algorithm properties: correctness, complexity, error tolerance, numerical stability
c. Algorithm types: iterative, adaptive, greedy
2. Linear algebra (8 lectures, 8 exercises):
a. Systems of linear equations, Gaussian elimination
b. Matrices, vectors, matrix operations
c. Orthogonality, Euclidean norm, orthogonal projection
d. Matrix factorization, SVD decomposition, matrix conditioning
3. Calculus (6 lectures, 6 exercises):
a. Differential and integral calculus
b. Newton's algorithm
c. Monte-Carlo algorithm
d. Steepest descent algorithm
4. Differential Equations (8 lectures, 8 exercises):
a. Ordinary and partial equations, systems of equations, initial and boundary conditions, etc.
b. Discretization, meshes
c. Galerkin methods
d. Examples of models and their analysis
5. Selected numerical methods
a. Solving ordinary differential equations
b. Solving partial differential equations

The order of issues addressed and the level of detail in their implementation may change slightly.

Course coordinators

Mateusz Dębowski

Type of course

obligatory courses

Mode

Blended learning

Prerequisites (description)

Knowledge of the fundamentals of linear algebra, calculus, and probability theory will facilitate understanding of the subject.

Assessment criteria

1. Completion of the course ends with one common grade for all classes (lecture + laboratory) carried out as part of the course.
2. A sick leave does not exempt students from knowledge of the material. It only entitles them to an individualized form of assessment.
3. Students who have received approval for an individualized course of study are required to contact the course coordinator to determine how to achieve all the learning outcomes assigned to the course. If the above-mentioned outcomes cannot be achieved, the coordinator may refuse to grant credit for the course.
4. Attendance is mandatory. In cases of justified absences, the student is required to contact the course coordinator immediately.
5. Regarding the use of artificial intelligence (AI) tools in the preparation of coursework, the rules set out in the "Regulations on the Use of Artificial Intelligence Tools in the Educational Process for the Computational Engineering Degree Programme" apply.
For this course, the use of AI tools is permitted at Level 1 for the exam, and at Levels 2 and 3 for any project assignments, in accordance with the rules set out in the aforementioned Regulations.
The student must specify the extent to which AI tools were used in preparing the work by submitting a declaration in the prescribed format.
6. Detailed grading criteria are provided at the level of the given academic cycle in the "Notes" section.

Practical placement

Not applicable.

Bibliography

The subject literature has been posted at the level of a given teaching cycle in the Bibliography section.

Term 2026Z:

1. Paweł Bechler, Geometria z algebrą liniową (skrypt) https://www.mimuw.edu.pl/~pbechler/inf_gal/inf_gal_skrypt_2023.1.pdf
2. Włodzimierz Krysicki, Lech Włodarski, Analiza matematyczna w zadaniach część 1, PWN, Warszawa 1977.
3. Andrzej Palczewski, Równania różniczkowe zwyczajne, PWN, Warszawa, 2017
4. Urszula Foryś, Jan Poleszczuk, Modelowanie matematyczne w biologii i medycynie (skrypt) https://mst.mimuw.edu.pl/wyklady/mbm/wyklad.pdf
5. Urszula Foryś, Matematyka w biologii, WNT, Warszawa, 2005
6. Piotr Krzyżanowski, Obliczenia naukowe (skrypt) http://mst.mimuw.edu.pl/wyklady/ona/wyklad.pdf

Notes

Term 2026Z:

Credit for the course is awarded based on:
○ a written exam (verification of learning outcomes W1, W2, W3, W4, W5, W6).

Learning outcomes U1, U2, U3, and U4 are assessed continuously during classes.