ENGINEERING TRIPOS PART IIA –
Module 3M1 – Mathematical methods
16 Lectures + coursework
This module aims to teach some mathematical techniques that have wide applicability to many areas of engineering. Linear Algebra provides important mathematical tools that are not only essential to solve many technical and computational problems, but also help in obtaining a deeper understanding of many areas of engineering. Optimization methods are routinely used in almost of every branch of engineering, especially in the context of design. Stochastic (random) processes are important in fields such as signal and image processing, data analysis etc.
1. Linear Algebra (4L)
- Revision of IB material
- Matrix norms, condition numbers, conditions for convergence of iterative schemes
- Positive definite matrices
- Singular Value Decomposition (SVD), pseudo-inverse of a matrix and least squares solutions of Ax = b
- Principal Component Analysis
- Markov matrices and applications
2. Optimization (8L)
- Introduction: Formulation of optimization problems; conditions for local and global minimum in one, two and multi-dimensional problems
- Unconstrained Optimization: gradient search methods (Steepest Descent, Newton’s Method, Conjugate Gradient Method)
- Linear programming (Simplex Method)
- Network optimization
- Constrained Optimization: Lagrange and Kuhn-Tucker multipliers; penalty and barrier functions
- Application of Principal Component Analysis to optimization problems
3. Stochastic Processes (4L)
- Definition of a stochastic process, Markov assumption (with examples), the Chapman-Kolmogorov (CK) equation, conversion of a particular CK integral equation into a differential equation (for the case of Brownian motion)
- The general Fokker-Planck equation with particular examples (Brownian motion, Ornstein-Uhlenbeck process)
- Introduction to sampling Gibbs sampler, Metropolis Hastings, Importance sampling with applications
By the end of the course students should:
- Be able to find the SVD of a matrix, and to understand how this can be used to calculate the rank and pseudo inverse of the matrix;
- Be able to calculate the least squares solution of a set of linear equations;
- Understand how to apply PCA to a problem;
- Be able to represent linear iterative schemes using linear algebra and understand what influences the rate of convergence.
Understand the definitions and application areas of Stochastic Processes;
Understand the principle of Markov Chains;
Be able to implement various sampling schemes to enable parameters of stochastic processes to be estimated.
- Understand the concepts of local and global minimum and the conditions for which a global minimum can be obtained;
- Understand the algorithms of the different gradient search methods;
- Be able to solve unconstrained problems using appropriate search methods;
- Be able to select an appropriate optimization method for a specific problem;
- Be able to solve a constrained linear and non-linear problem using an appropriately selected technique;
- Be able to solve network optimization problems;
- Be able to apply PCA to reduce the dimensionality of an optimization problem and/or to improve the solution representation.
Please see the Booklist for Part IIA Courses for references for this module.