Course Description
The lecture is divided into four parts:
Part I focuses on partial differential equations and begins with an overview of common numerical approaches, including finite difference methods, variational methods, moment methods, finite element methods, and the method of lines. The subsequent lectures concentrate in greater detail on three of the most widely used techniques: finite difference methods, variational methods, and the finite element method.
Part II addresses the numerical solution of systems of ordinary differential equations. An overview of commonly used methods is provided, including the Euler method, Taylor series methods, Runge–Kutta methods, collocation methods, multistep methods, and extrapolation techniques. The course then explores collocation and multistep methods in more detail. If time permits, topics such as error control, applications to linear systems of differential equations, and numerical quadrature will also be discussed.
Part III introduces several machine learning techniques for the analysis of large data sets. Following an overview of supervised and unsupervised learning, the course focuses on selected methods such as classification, outlier detection, principal component analysis, and prediction techniques. The accompanying exercises include applications of these methods to astronomical data.
Part IV is devoted to Monte Carlo methods, a class of computational algorithms that rely on repeated random sampling to obtain approximate solutions to a wide range of mathematical and physical problems. The main features of random number generators are introduced, and commonly used sampling techniques are discussed. The course concludes with examples illustrating the use of Monte Carlo simulations in modelling physical processes.
