Course description
The lecture will cover four parts:
Part I will concentrate on partial differential equations and will
begin with an overview of popular approaches, such as finite difference
methods, variational methods, moment methods, finite element methods and
method of lines. In the following lectures we will focus more
extensively on three of the most widely used methods, namely finite
difference, variational and the finite element.
Part II will focus on the numerical solution of ordinary differential
equation systems. An overview of commonly used methods will be given
such as the Eulerian Approach, Taylor series approaches, Runge-Kutta,
collocation, multi-step and extrapolation methods. In the following we
will focus in some more details on collocation and multistep methods. If
time permits error control and applications to linear differential
equation systems and numerical quadrature will be discussed, as well.
Part III will introduce several machine learning techniques that enable
us to analyze large amounts of data. After an overview of unsupervised
and supervised learning, we will focus on selected techniques, such as
classification, outlier detection, principal component analysis, and
predictions. The exercises will contain applications of the techniques
to astronomical data.
The fourth part will focus on Monte Carlo methods, a class of
computational algorithms that rely on repeated random sampling to
provide approximate solutions to a variety of mathematical and physical
problems. Main features of random number generators will be introduced
and most common sampling techniques will be discussed. Then some example
of how Monte Carlo simulations are used for the modelling of physical
processes will be given.
