Mathematical modelling

2022/2023
Programme:
Practical Mathematics
Year:
3 year
Semester:
first or second
Kind:
optional
Group:
I1
ECTS:
5
Language:
slovenian
Hours per week – 1. or 2. semester:
Lectures
2
Seminar
0
Tutorial
1
Lab
1
Content (Syllabus outline)

Using software in optimization:
optimization on scalar fields, constrained optimization, discrete catenary problem
Selected problems from calculus of variations:
continuous catenary problem, brachistochrone problem, minimal time in brachistochrone problem
Statistical simulations:
pseudo random numbers generation, simulation of complex processes, simulation of simple games
Numerical solution of some simple problems concerning partial differential equations:
the problem of thin membrane, the problem of
soup bubble
Fundamentals of computer aided geometric design:
construction of parametric curves and surfaces, approximation of circular arcs, application of parametric curves in robotics and design

Readings

C. Moler: Numerical Computing with Matlab, SIAM, 2004
E. Zakrajšek: Matematično modeliranje, DMFA založništvo, 2004
D. E. Knuth: The Art of Computer Programming, vol. 2, Addison-Wesley, 1997
G. Farin: Curves and Surfaces for Computer Aided Geometric Design: a practical guide, Academic press, 1993
W. M. Tikhomirov: Stories about Maxima and Minima, AMS, 1991

Objectives and competences

Students will acquire knowledge on numerical optimization. In particular, they will learn how to solve constrained optimization problems. They will apply mathematical theory to solve some classical problems of modeling and calculus of variations. They will learn about pseudorandom generators and statistical simulations. Furthermore, some basic problems concerning simple partial differential equations will be solved. Finally, some algorithms for construction of parametric curves and surfaces with application in robotics and design will be presented.

Intended learning outcomes

Knowledge and understanding: Knowledge and understanding of basic algorithms for numerical optimization and their application to standard problems of mathematical modeling. Understanding generation of pseudorandom numbers with application in statistical simulation. Understanding of basic algorithm for numerical solution of partial differential equations. Ability of construction parametric curves and surfaces with applications in robotics and design.
Application: Mathematical modeling can be used in almost any field of natural sciences.
It is more and more useful also in economy and social sciences. Mathematical models are indispensable tool for solving several problems arising in modern society.
Reflection: Integrating theoretical and practical procedures for solving practical problems.
Transferable skills: Selection of a stable algorithm to solve the particular problem, which arises in practice. Knowledge is transmitted to virtually all sciences: natural sciences, technical sciences, computer sciences, economy, etc.

Learning and teaching methods

Lectures, exercises, homeworks, consultations

Assessment

Type (written exam, project, homeworks):
two obligatory homeworks required to apply for an oral exam,
written exam
project defence
Grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Emil Žagar:
JAKLIČ, Gašper, KANDUČ, Tadej, PRAPROTNIK, Selena, ŽAGAR, Emil. Energy minimizing mountain ascent. Journal of optimization theory and applications, ISSN 0022-3239, 2012, vol. 155, is. 2, str. 680-693. ,. [COBISS-SI-ID 4382935]
JAKLIČ, Gašper, ŽAGAR, Emil. Curvature variation minimizing cubic Hermite interpolants. Applied mathematics and computation, ISSN 0096-3003. [Print ed.], 2011, vol. 218, iss. 7, str. 3918-3924. [COBISS-SI-ID 16049241]
JAKLIČ, Gašper, ŽAGAR, Emil. Planar cubic G [sup] 1 interpolatory splines with small strain energy. Journal of Computational and Applied Mathematics, ISSN 0377-0427. [Print ed.], 2011, vol. 235, iss. 8, str. 2758-2765. [COBISS-SI-ID 15770969]