ENGINEERING TRIPOS PART IIB – 2012/2013 (module not running this year)

Module 5R1 - Stochastic Processes and Optimization Methods

 AIMS

This module aims to teach some mathematical techniques that have wide applicability to many areas of engineering. Stochastic (random) processes are important in fields such as signal and image processing, data analysis etc. Stochastic (“random” search) optimization methods are widely used as “methods of last resort” to solve difficult, real-world optimization problems.

 LECTURE SYLLABUS

 Stochastic Optimization Methods (6L, Dr G.T. Parks)

• Common Issues: performance measures, archiving
• Monte Carlo Sampling: basic concepts, solution representation and generation, sampling biasing, convergence criteria, treatment of constraints, algorithm performance
• Simulated Annealing: basic concepts, solution representation and generation, the annealing schedule, enhancements and modifications, convergence criteria, treatment of constraints, algorithm performance
• Genetic Algorithms: basic concepts, solution representation, selection, crossover, mutation, covergence criteria, treatment of constraints, algorithm performance
• Tabu Search: basic concepts, solution representation, local search, intensification, diversification, convergence criteria, treatment of constraints, algorithm performance
• Multiobjective Optimization: archiving, multiobjective simulated annealing, multiobjective genetic algorithms
• Case Study: multiobjective optimization of pressurised water reactor reload cores

 Stochastic Processes (7L, Prof W.J. Fitzgerald)

• Theory of Markovian Stochastic Processes: elements of statistical mechanics, Brownian motion and the Langevin equation, basic theory of stochastic processes, the Chapman-Kolmogorov and Fokker-Planck equations, solutions of the equations of Brownian motion, transition between potential wells, the Ehrenfest urn, the master equation
• Theory of Non-Markovian Stochastic Processes: relevant variables and non-Markovian processes, projection method and the adiabatic elimination procedure, the linear chain, adiabatic elimination and multiplicative noise, fluctuating potential barriers, the reduced model theory
• Extreme Value Theory - with applications

COURSEWORK

• Writing of a Biased Monte Carlo Sampling (BMCS) algorithm (in C++, FORTRAN or MATLAB)
• Investigation of the performance of the BMCS algorithm on Keane’s Bump Function
• Implementation of Simulated Annealing (SA) or Genetic Algorithm (GA) (self-written or modification of publicly available software)
• Investigation of the performance of SA or GA on Keane’s Bump Function
• Report detailing work done and outcomes of investigations

Approx. 14 hours work

Stochastic Processes

• Investigation of Markov Chain Monte Carlo (MCMC) methods
• Implementation of Importance Sampling
• Comparison of MCMC and other sampling schemes

Approx. 13 hours work

OBJECTIVES

On completion of the module students should:

• Understand the strengths and weaknesses of stochastic optimization methods;
• Understand the algorithms of the different search methods;
• Appreciate how the control parameters affect the performance of the search methods;
• Know how to measure and compare the performance of stochastic optimization methods;
• Be able to select an appropriate optimization method for a specific problem;
• Be able to apply the different search methods to both constrained and unconstrained optimization problems.

Stochastic Processes

• 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.

REFERENCES

Please see the Booklist for Group R Courses for references for this module.

Last updated: August 2011