Lecturer: Marie-Therese Wolfram

Term(s): Term 1

Status for Mathematics students: Core for Maths

Commitment: 30 one hour lectures plus assignments

Assessment: 85% by 2 hour examination, 15% coursework

Formal registration prerequisites: None

Assumed knowledge:

• MA141 Analysis 1
• MA139 Analysis 2
• MA144 Methods for Mathematical Modelling 2
• MA150 Algebra 2
• MA124 Maths by Computer

Useful background:

• ST120 Introduction to Probability

Synergies:

• MA263 Multivariable Analysis
• MA260 Norms Metrics and Topologies
• MA270 Analysis 3

Leads to: The following modules have this module listed as assumed knowledge or useful background:

• MA398 Matrix Analysis and Algorithms
• MA3K1 Mathematics of Machine Learning
• MA3K9 Mathematics of Digital Signal Processing
• MA3H7 Control Theory

Aims: The module gives an introduction to the theory of optimisation as well as the fundamentals of approximation theory.

Content:

1. Recap: necessary and sufficient conditions for local min/max, convex functions and sets, Jensen’s inequality, level sets.
2. Iterative algorithms: gradient descent and line search methods
3. Newton's method
4. Linear programming with applications in economics and data science
5. Constrained optimisation
6. Introduction to Neural Networks
7. Approximation theory: polynomial approximation, rational approximation, trigonometric approximation
8. Discrete Fourier and Cosine Transform with applications in imaging and signal processing
9. Introduction to Wavelets

Objectives:

• understand critical points of multivariable functions

• apply various techniques to solve nonlinear optimisation problems and understand their applications, in economics and data science

• use Lagrange multipliers and the Karush–Kuhn–Tucker conditions to solve constrained nonlinear optimisation problems

• understand the basic concepts of approximation theory

• obtain an understanding of different approximation techniques used in the digital sciences

Books:

• Endre Sueli and David F. Mayers, An Introduction to Numerical Analysis, Cambridge University Press, 2003
• S. Boyd. ‘Convex optimization’, Cambridge University Press 2004
• J. D. Powell, ‘Approximation Theory and Methods’, Cambridge University Press, 1981
• N. Trefethen, ‘Approximation Theory and Practice’

 

Additional Resources