1. komponenta

Lecture type	Total
Lectures	45
Exercises	30

* Load is given in academic hour (1 academic hour = 45 minutes)

COURSE AIMS AND OBJECTIVES: The main objective of this course is to bring to the student knowledge of these modern approaches to mathematical modelling and scientific computing, using a wide palette of case study examples. The key of the approach taken in this course is that the focus is to solve a concrete problem using whatever mathematical technique/theory is needed, and not to study particular theories.
Concrete case study examples are used to show how to (i) formulate the problem using the language of mathematics; (ii) analyse the problem and develop solution method; (iii) develop an algorithmic solution and software implementation; (iv) communicate and interpret the results in the context of the original problem.
This format of the course is meant to be a preparation for a professional career of an applied mathematics on the demanding job market. At the same time, it stresses the importance and the power of mathematics and scientific computing and motivates the students for further studying and strengthening of their mathematical skills.

COURSE DESCRIPTION AND SYLLABUS:

1. Models and differential equations in engineering. Classical examples of models in engineering, given by differential equations, analysed in the context of efficient numerical simulations development and model order reduction.
1.1. Case studies examples for mathematical modelling.
1.2. Model order reduction methods POD (Proper Orthogonal Decomposition), PCA (Principal Component Analysis), KLD (Karhunen-Loeve Decomposition), balanced truncation. Development of numerical algorithms using available software packages.
1.3. Numerical simulations. Digital twins. Examples.

2. Data driven modelling. Having available experimental data and an assumed general form of the model, the task is to develop algorithms that derive (learn) equations that generate the experimentally obtained data.
2.1. Selected methods for experimental data driven model development.
2.2. Numerical examples.

3. Randomized algorithms. Many applications inherently contain randomness, and randomness can be artificially introduced to develop more efficient algorithms.
3.1. Monte-Carlo simulations in applications.
3.2. Stochastic optimization with applications.
3.3. Bayesian scientific computing. Data assimilation.

4. Relaxation methods for discrete optimization. NP hard problems in discrete optimization are often solved using relaxation. Concrete examples are used to show how discrete mathematical models yield hard combinatorial problems and how to relax them and solve using standard optimization techniques and available efficient numerical algorithms.
4.1. Case study: spectral cuts in weighted graphs (e.g. min-cut problems for clustering/segmentation).
4.2. Case study: computing meta-stable states of Markov chains.

1. Introduction to Bayesian Scientifi Computing, D. Calvetti, E. Somersalo, Springer, 2007.
2. Numerical Mathematics, A. Quarteroni, R. Sacco, F. Saleri, Springer Texts in Applied mathematics, 2007.
3. Stochastic Simulation and Monte Carlo Methods, C. Graham, D. Talay, Springer Mathematical Foundations of Stochastic Simulation, 2013.
4. Approximation of Large-Scale Dynamical Systems, A. Antoulas, SIAM, 2005.

Mandatory course - Regular study - Computer Science and Mathematics