Mathematical Modeling of Complex Systems
Mathematical modelling is a distinctly generic and interdisciplinary branch of science which is applied in almost all branches of natural, technical and, last but not least, e-science. The purpose of this course is to present selected topics related to the issues of the synthesis of models for complex dynamic processes, their simulations and calibration. The presentation will include basic techniques and ideas, available modelling and simulation tools and examples of their practical use.
The student will be able to design models based on basic physical principles and design models from data.
Basic steps of the model synthesis process, basics of nonlinear dynamics.
2) Complex dynamics:
Models of complex and self-organizing systems; determinism, predictability and causality in (complex) dynamic systems; stochastic processes, Fokker-Planck equation; synchronization.
3) Analysis of complex dynamic systems:
Spectral methods (Fourier and vawelet analysis), Lyapunov exponent, correlation dimension.
4) Advanced simulation:
Basics of numerical integration. Simulation of differential-algebraic equations. Simulation of models with distributed parameters; method of final elements, offline methods (with examples from ecology, heat conduction, Black-Scholes financial model).
Simulation of stochastic systems (Monte Carlo approaches, Markov chains). Simulation tools: Matlab, Simulink, Femlab.
5) Data driven modelling of complex dynamic systems:
Basics of linear regression and instrumental variables method. Nonparametric model identification (neural networks, Gaussian processes). Bayesian approach to the identification of complex dynamic systems. Applications.
Student must complete first-cycle study programmes in natural sciences, technical disciplines or computer science.
Literature and references