Numerical Methods for Data Analysis

Credits

Prerequisites

None.

Scientific-disciplinary sector (SSD)

MAT/08 Numerical Analysis.

Examination method

Assessment of a project developed on topics introduced in the course, discussion and oral exam.

Learning
objectives

The course aims to supplement and deepen the knowledge acquired in a first-level numerical analysis course in the fields of numerical linear algebra, optimisation and the FFT and Wavelet transforms. This further study is essentially geared towards solving large-scale problems, especially those inspired by applications requiring ad hoc numerical treatment. With this in mind, laboratory work and the analysis of specific case studies play a central role.

Syllabus

Discrete Fourier Transform and Wavelet Transform (continuous and discrete). Sampling of a function in time and frequency. The Fast Fourier Transform (FFT). Radix-2 and radix-r FFT algorithms. Numerical stability of the FFT. Applications of the FFT: trigonometric interpolation, convolution product, fast algorithms for matrix-vector products with circulant and Toeplitz matrices. Introduction to two-dimensional Fourier and Wavelet transforms. Numerical linear algebra for sparse and large-scale problems. QR factorisation with orthogonal and iterative transformations; Singular Value Decomposition (SVD). Least squares problems: fundamental numerical solution methods and outline of the statistical properties of the solution. Variants: recursive form, generalised problems, constrained problems, nonlinear problems. Regularisation methods for severely ill-conditioned problems (truncated SVD and Tikhonov).

Expected learning
outcomes

By the end of the course, students must demonstrate that they

know and understand the numerical techniques studied, with a clear view of their fields of application;
are able to use the knowledge acquired to solve specific problems, both with software libraries and with purpose-designed and purpose-built codes;
are able to communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
are able to identify the most appropriate methods to analyse and solve a problem relating to the course topics and to interpret the results correctly.

Learning outcomes
to be assessed

Command of the knowledge acquired, clarity of presentation, rigour in the use of language, confidence in using the notions acquired.

Numerical Methods for Data Analysis