Učni načrt predmeta

Predmet:
Matematično modeliranje in računske metode v znanosti
Course:
Mathematical Modelling and Scientific Computing
Študijski program in stopnja /
Study programme and level		 Študijska smer /
Study field		 Letnik /
Academic year		 Semester /
Semester
Informacijske in komunikacijske tehnologije, 3. stopnja 1 1
Information and Communication Technologies, 3rd cycle 1 1
Vrsta predmeta / Course type
Izbirni / Elective
Univerzitetna koda predmeta / University course code:
IKT3-928
Predavanja
Lectures		 Seminar
Seminar		 Vaje
Tutorial		 Klinične vaje
work		 Druge oblike
študija		 Samost. delo
Individ. work		 ECTS
15 15 15 105 5

*Navedena porazdelitev ur velja, če je vpisanih vsaj 15 študentov. Drugače se obseg izvedbe kontaktnih ur sorazmerno zmanjša in prenese v samostojno delo. / This distribution of hours is valid if at least 15 students are enrolled. Otherwise the contact hours are linearly reduced and transfered to individual work.

Nosilec predmeta / Course leader:
izr. prof. dr. Gregor Papa
Sodelavci / Lecturers:
dr. Gregor Kosec
Jeziki / Languages:
Predavanja / Lectures:
slovenščina, angleščina / Slovenian, English
Vaje / Tutorial:
Pogoji za vključitev v delo oz. za opravljanje študijskih obveznosti:
Prerequisites:

Zaključen študij druge stopnje naravoslovnih smeri.
Potrebno je znanje matematike, fizike in programiranja.

Completed second-cycle studies in natural sciences. Knowledge of mathematics, physics, and programming is required.

Vsebina:
Content (Syllabus outline):

Matematično modeliranje
• Opredelitev ključnih procesov in količin v modeliranem sistemu
• Opazovanje relacij med vpletenimi procesi in količinami
• Poenostavitvene predpostavke, ki omogočajo obvladovanje kompleksnosti ter reševanje realnih problemov
• Oblikovanje matematičnega okvira, ki najbolje opiše opažanja
Numerične metode
• Sistemi linearnih in nelinearnih enačb
• Interpolacija / metoda najmanjših kvadratov
• Numerično integriranje in odvajanje
• Stohastične metode
• Modeliranje in analiza časovnih vrst
Implementacija in izvedba numeričnih rešitvenih postopkov
• Razumevanje računalniške arhitekture in računalniške aritmetike
• Pregled najpogosteje uporabljenih programskih jezikov in numeričnih knjižnic
• Splošne smernice za implementacijo
• Osnove paralelnega računalništva
Analiza rezultatov in vizualizacija
• Pristopi k vizualizaciji in raziskovanju podatkov
• Statistična analiza
• Validacija in verifikaciji
• Občutljivostna analiza
• Interpretacija rezultatov

Mathematical modelling
• Identification of key processes and quantities in the modelled system
• Observing the relations between involved processes and quantities.
• Simplifying assumptions to manage the complexities of real-world problems and make the problem solvable
• Designing the mathematical framework that best represent the observed relationships
Numerical methods
• Systems of linear and non-linear equations
• Interpolation / least squares
• Numerical integration and differentiation
• Stochastic methods
• Modelling and analysis of time series
Implementation and execution of numerical solution procedures
• Understanding computer architecture and computer arithmetic
• Overview of most common programming Languages and numerical libraries
• General implementation guidelines
• Basics of parallel computing
Result analysis and visualisation
• Visualisation and data exploration approaches
• Statistical analysis
• Validation vs. verification
• Sensitivity analysis
• Interpretation of the results

Temeljna literatura in viri / Readings:

S. Širca, M. Horvat, Computational Methods in Physics, Springer International Publishing DOI: 10.1007/978-3-319-78619-3, 2018

W. H. Press, Numerical recipes 3rd edition: The art of scientific computing. Cambridge university press, 2007

M. T. Heath, Scientific Computing: An Introductory Survey, Revised Second Edition, Society for Industrial and Applied Mathematics, 2018

Berry, Michael W., et al., eds. High-performance scientific computing: algorithms and applications. Springer Science & Business Media, 2012.

I. Kuščer, A. Kodre, Matematika v fiziki in tehniki, DMFA-založništvo, 2016, [COBISS.SI-ID:287827456]

Cilji in kompetence:
Objectives and competences:

Cilj predmeta je razviti temeljno razumevanje računskih metod v znanosti, začenši s prenosom problemov iz realnega sveta v matematični okvir, kot so sistemi (parcialnih) diferencialnih enačb, diskretni modeli ali statistični modeli. Matematičnih modelov, ki opisujejo kompleksne probleme realnega sveta, v večini primerov ni mogoče rešiti analitično. Zato bomo pri predmetu obravnavali ustrezne numerične metode, kot so numerična integracija kompleksnih parcialnih diferencialnih enačb, reševanje nelinearnih problemov, Monte Carlo simulacije, itd. Te tehnike bomo implementirali z uporabo ustreznega programskega jezika in jih izvedli na različnih računalniških arhitekturah. Na koncu bomo obravnavali analizo podatkov in osnove vizualizacije, ki jo potrebujemo za interpretacijo rezultatov.

The objective of this course is to develop a foundational understanding of scientific computing, starting with the translation of real-world problems into a mathematical framework, such as systems of (partial) differential equations, discrete models, or statistical models. In most cases, mathematical models that describe complex real-world problems cannot be solved in a closed form. Therefore, the course will cover appropriate numerical methods, such as the numerical integration of complex partial differential equations, solving non-linear problems, Monte Carlo simulation, etc. These techniques will be implemented using a suitable programming language and executed on different computer architectures. Finally, we will cover the data analysis and basic visualization needed for the interpretation of results.

Predvideni študijski rezultati:
Intendeded learning outcomes:

Študenti, ki bodo uspešno opravili ta predmet, se bodo seznanili z razvojem in uporabo matematičnih modelov, numeričnih metod in algoritmov, ki se uporabljajo za izvajanje simulacij in analiz znanstvenih in inženirskih problemov.

Students who successfully complete this course will gain insight into the development and application of mathematical models, numerical methods, and algorithms used to perform simulations and analyses of scientific and engineering problems.

Metode poučevanja in učenja:
Learning and teaching methods:

Predavanja, konzultacije, individualno delo

Lectures, consultations, individual work

Načini ocenjevanja:
Delež v % / Weight in %
Assesment:
Seminar
50 %
Seminar
Zagovor seminarja
50 %
Oral defense of seminar work
Reference nosilca / Lecturer's references:
1. M. Jančič, G. Kosec; Strong form mesh-free hp-adaptive solution of linear elasticity problem, Engineering with computers, vol. 40, 2024 [DOI: 10.1007/s00366-023-01843-6][COBISS: 153678339]
2. M. Depolli, M. Žebre, U. Stepišnik, G. Kosec; Simulation of a former ice field with Parallel Ice Sheet Model : Snežnik study case, Climate of the past, vol. 20, 2024 [DOI: 10.5194/cp-20-1471-2024]
3. F. Strniša, T. Tinta, G. Herndl, G. Kosec; Dynamic population modeling of Bacterioplankton community response to gelatinous marine zooplankton bloom collapse and its impact on marine nutrient balance, Progress in oceanography, 2024 [DOI: 10.1016/j.pocean.2024.103312][COBISS: 202508291]
4. U. Duh, V. Shankar, G. Kosec; Discretization of non-uniform rational B-spline (NURBS) models for meshless isogeometric analysis, Journal of scientific computing, 2024 [DOI: 10.1007/s10915-024-02597-z][COBISS: 201006595]
5. M. Jančič, J. Slak, G. Kosec; Monomial augmentation guidelines for RBF-FD from accuracy versus computational time perspective, Journal of scientific computing, vol. 87, 2021 [DOI: 10.1007/s10915-020-01401-y][COBISS: 52715011]