# MA264 Mathematical Methods and Modelling 3

**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: **

**Synergies:**

**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**:

- Recap: necessary and sufficient conditions for local min/max, convex functions and sets, Jensenâ€™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*