MA265 Methods of Mathematical Modelling 3

Lecturer: Marie-Therese Wo lfram

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:

Useful background:

Synergies:

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

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

Content:

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