Mathematical Modelling
Section outline
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Pri Matematičnem modeliranju študenti (relativno) realne probleme rešujejo s pomočjo matematičnih modelov. Študenti se spoznajo s timskim delom in lahko sami, skupaj s kolegi v ekipi (in ob pomoči učiteljske ekipe) izpeljejo projekt od začetka do konca.
Predmet je razdeljen na tri glavna poglavja:
- Linearni modeli: sistemi linearnih enačb, posplošeni inverzi matrik
- Nelinearni modeli: vektorske funkcije vektorske spremenljivke, sistemi nelienarnih enačb, krivulje in ploskve
- Krivulje in ploskve
- Dinamični modeli: diferencialne enačbe in dinamični sistemi
During the course of Mathematical Modelling, the students will be solving (relatively) real-life problems with the help of mathematical models. Students will be introduced to teamwork and will be given the opportunity to carry out a complete, relatively complex project from the beginning to the very end together with their colleagues (and with the help of the teaching team).
The course consists of three main topics:
- Linear models: systems of linear equations, generalized inverses of matrices
- Nonlinear models: vector functions of a vector variable, nonlinear systems, curves and surfaces
- Curves and surfaces
- Dynamic models: differential equations and dynamical systems
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Lectures: Introduction, a linear matematical model, some basics of linear algebra.
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Lectures: Generalized matrix pseudoinverses and their use in solving linear systems, the Moore-Penrose pseudoinverse.
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Lecture: SVD and MP inverse computation, underdetermined systems with many and overdetermined systems with no solutions.
Extra lecture (instead of the next week's one): Vector and matrix norm, PCA, introduction into nonlinear models (see the updated lecture notes) and this pdf. -
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No lectures this week.
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Lectures: Vector functions. Derivative/Jacobian matrix. Linear approximation. Tangent method. Newton method. Gauss-Newton method.
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Modified 18/03/20, 07:24
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Lecture: Rate of convergence of Newton's method. Newton's optimization. Gradient descent. Broyden's method.
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Podroben opis reševanja 1. naloge iz tedna 5 in 1. naloge iz tedna 6.
Uploaded 26/03/18, 14:55
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Lectures: Curves - different parametrizations, arc length. (Pages up to 115 in the notes)
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Lectures: Curves - different parametrizations, polar coordinates, arc length, area bounded by the curve, natural parametrization, curvature of parametric curves, plotting plane curves.
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Lectures: Parametrization of surfaces. Surface of revolution. Tangent plane. Differential equations: introduction.
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Lectures: Differential equations - Applications, separation of variables, first order linear ODE, variation of constants,
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Lectures: Orthogonal trajectories. Homogeneous DE, direction field, Euler's method, Runge-Kutta methods. DOPRI5.
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Lectures: Transforming an ODE of higher order into a system of first order ODE. Systems of differential equations. Autonomous linear systems. Numerical methods for systems.
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Lectures: Phase portraits, linearization of a nonlinear system, autonomous linear DE of higher order.
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Lectures: Wronskian determinant, nonhomogeneous second order DEs.
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