MA3J8 Approximation Theory and Applications

Introductory description

This module will provide students with a foundation in approximation theory, driven by its applications in scientific computing and data science.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Part 1: univariate approximation

• spline approximation of smooth functions in 1D
• polynomial and trigonometric approximation of analytic functions in 1D
• linear best approximation

Part 2: Multi-variate approximation: details will depend on the progress through Part 1 and available time, but the idea of Part 2 is to cover a few selected examples of high-dimensional approximation theory, for example a sub-set of the following:
wavelets and nonlinear approximation,
applications to image processing, such as image compression,
compressed sensing and best n-term approximation,
ridge functions and neural networks.

Throughout the lecture each topic will cover (1) approximation rates, (2) algorithms, and (3) examples, typically implemented in Python. Any programming aspects of the module will not be examinable.

Learning outcomes

By the end of the module, students should be able to:

• demonstrate understanding of key concepts, theorems and calculations of univariate approximation theory.
• demonstrate understanding of a selection of the basic concepts, theorems and calculations of multivariate approximation theory.
• demonstrate understanding of basic algorithms and examples used in approximation theory.

Indicative reading list

Approximation Theory and Methods, M. J. D. Powell
Approximation Theory and Approximation Practice, N. Trefethen
A course in approximation theory, E.W.Cheney and W.A.Light
Nonlinear approximation, R. DeVore (Acta Numerica)

Subject specific skills

Formalisation of function approximation, connection to applications in modelling, numerical analysis and machine learning. Specific topics:

• Abstract approximation in Hilbert and Banach spaces
• Connect approximation rates and function regularity, in particular analyticity
• Approximation by trigonometric and algebraic polynomials
• Multi-variate approximation
• Approximation via Least Squares Fitting
• High-dimensional problems
• Practical implementation of algorithms
• Introduction to wavelets and applications to image processing.

Transferable skills

Analytical thinking and problem solving: Numerical models and methods of the kind that will be studied are crucial in many areas of science, engineering, and machine learning. The course trains students to connect rigorous mathematical ideas with concrete numerical algorithms, programming and scientific computing. E.g. it will train students to better select suitable computational tools for approximation and inverse problems with are ubiquitous across many disciplines.