MA26410 Mathematical Methods and Modelling 3
Introductory description
Some modern technology became possible due to computing power and mathematical methods. Examples include search algorithms, data compression, artificial intelligence, etc. This module will develop explicit mathematical methods used in some of these applications.
Module aims
The module gives an introduction to the theory and practice of optimisation as well as the fundamentals of approximation theory.
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.
 Recap: necessary and sufficient conditions for local min/max, convex functions and sets, Jensen’s inequality, level sets.
 Differentiable unconstrained problems: existence of local minimisers, Hessian, classification of critical points.
 Convex linear and quadratic problems: linear programming, least squares and regression, steepest descent, singular value decomposition, stochastic gradient descent, Gibbs inequalities, entropy minimisation, applications (line search and optimisation in 1D, linear and logistic regression, backward propagation for neural network training, support vector machines).
 Constrained optimisation: Lagrange multipliers, Karush–Kuhn–Tucker conditions.
 The discrete Fourier transform: Parseval’s theorem, FFT in applications (convolutions in PDEs, signal processing, audio and video compression).
 Approximation properties of complete orthonormal systems: Hilbert spaces, Fourier basis and classical orthogonal polynomials.
 Linear least squares: interpolation by algebraic polynomials (Chebyshev polynomials, Lagrange polynomials), interpolation by trigonometric polynomials (Fourier series).
Learning outcomes
By the end of the module, students should be able to:
 understand critical points of multivariable functions
 apply various techniques to solve nonlinear optimisation problems and understand their applications, including to machine learning
 use Lagrange multipliers and the Karush–Kuhn–Tucker conditions to solve constrained nonlinear optimisation problems
 have a working knowledge of discrete Fourier transformation
 understand the basic concepts of Hilbert space theory and its relation to approximation by orthogonal functions, including approximation of functions by algebraic and trigonometric polynomials
Indicative reading list
S. Boyd. ‘Convex optimization’, Cambridge University Press 2004
Y. Nesterov, ‘Lectures on Convex Optimization’, Springer, 2018
J. D. Powell, ‘Approximation Theory and Methods’, Cambridge University Press, 1981
N. Trefethen, ‘Approximation Theory and Practice’
Subject specific skills
The module gives an introduction to the theory and practice of optimisation as well as the fundamentals of approximation theory. The students will learn applications of these techniques to a range of modern technologies.
Transferable skills
The algorithmic techniques taught have widespread "real world" applications. Examples include ranking in search engines, linear programming and optimisation, signal analysis and machine learning.
Study time
Type  Required 

Lectures  20 sessions of 1 hour (20%) 
Online learning (independent)  9 sessions of 1 hour (9%) 
Private study  13 hours (13%) 
Assessment  58 hours (58%) 
Total  100 hours 
Private study description
Working on assignments, going over lecture notes, text books, exam revision.
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Assessment group D
Weighting  Study time  

Assignments  15%  20 hours 
Examination  85%  38 hours 

Assessment group R
Weighting  Study time  

Inperson Examination  Resit  100% 
Feedback on assessment
Marked homework (both assessed and formative) is returned and discussed in smaller classes. Exam feedback is given.
Courses
This module is Core option list A for:

UMAAGV17 Undergraduate Mathematics and Philosophy
 Year 2 of GV17 Mathematics and Philosophy
 Year 2 of GV17 Mathematics and Philosophy
 Year 2 of GV17 Mathematics and Philosophy
 Year 2 of UMAAGV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations
This module is Option list A for:

UCSAG4G1 Undergraduate Discrete Mathematics
 Year 2 of G4G1 Discrete Mathematics
 Year 2 of G4G1 Discrete Mathematics
 Year 2 of UCSAG4G3 Undergraduate Discrete Mathematics

UPXAGF13 Undergraduate Mathematics and Physics (BSc)
 Year 2 of GF13 Mathematics and Physics
 Year 2 of GF13 Mathematics and Physics

UPXAFG31 Undergraduate Mathematics and Physics (MMathPhys)
 Year 2 of FG31 Mathematics and Physics (MMathPhys)
 Year 2 of FG31 Mathematics and Physics (MMathPhys)

USTAGG14 Undergraduate Mathematics and Statistics (BSc)
 Year 2 of GG14 Mathematics and Statistics
 Year 2 of GG14 Mathematics and Statistics
This module is Option list B for:

USTAY602 Undergraduate Mathematics,Operational Research,Statistics and Economics
 Year 2 of Y602 Mathematics,Operational Research,Stats,Economics
 Year 2 of Y602 Mathematics,Operational Research,Stats,Economics