Lecturer: Martin Lotz

Term(s): Term 1

THIS MODULE IS NOT AVAILABLE TO MATHS (G100/G103) STUDENTS

Commitment: 30 one-hour lectures plus assignments

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

This module will be examined in the first week of Term 3

Formal registration prerequisites: None

Assumed knowledge:

• MA140 Mathematical Analysis 1 or MA142 Calculus 1
• MA152 Mathematical Analysis 2 or MA143 Calculus 2
• MA145 Mathematical Methods and Modelling 2 or MA133 Differential Equations
• MA149 Linear Algebra or MA148 Vectors and Matrices

Useful background:

• ST120 Introduction to Probability or ST118 Probability 1
  

Synergies: 

• MA263 Multivariable Analysis
• MA222 Metric Spaces
• MA271 Mathematical 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