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## ECM2704 - Numerics and Optimisation (2015)

MODULE TITLE | Numerics and Optimisation | CREDIT VALUE | 15 |
---|---|---|---|

MODULE CODE | ECM2704 | MODULE CONVENER | Dr Bob Beare (Coordinator) |

DURATION: TERM | 1 | 2 | 3 |
---|---|---|---|

DURATION: WEEKS | 11 weeks | 0 | 0 |

Number of Students Taking Module (anticipated) | 145 |
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When designing any kind of system, the relationship between the system data and the actual physical performance of the system can be very complicated. For this reason, engineers use numerical optimisation to deliver a a system structure. The process of identifiying an objective, variables and constraints on system design is known as modeling. Next, engineers apply an optimisation algorithm. Since there are different kinds of algorithms, this course will acquaint you with some of the standard approaches to finding a solution to your question.

An example of this in everyday business would be a chemical company whose objective is to deliver a specific product from its three factories to a dozen retail outlets. The variables would include plant capacity, weekly demand, and shipping costs. The constraints might be distance, product stability, market timing, or seasonal demand fluctuations.

Building on the Matlab skills you acquired in ECM1704, you will study nonlinear equations, systems of linear equations, time-stepping of ordinary differential equations, and the finding of minima or functions of many variables.

Prerequisite module: ECM1701, ECM1705 or NSC1002 (Natural Science Students) or equivalent

This module explores the use of computers to solve mathematical problems by means of numerical approximation. The techniques discussed form the basis of the numerical simulation and computer modelling of problems in science and business. Topics to be covered include solving nonlinear equations, solving systems of linear equations, time-stepping of ordinary differential equations, and finding minima of functions of many variables. IMPORTANT: the module builds on the Matlab programming from the first year Scientific Computing module (ECM1704): for students who have taken this module, there will be revision of Matlab in the first week. However, the Scientific Computing module is not a prerequisite, so, for students who have not taken it, there will be an intensive introduction to Matlab in the first week computer classes.

On successful completion of this module, **you should be able to:**

**Module Specific Skills and Knowledge:**

1 demonstrate a working knowledge of the theory and practical implementation of basic numerical methods;

2 explore applications and ideas underpinning more advanced methods that are developed in third/fourth stage modules and project work;

**Discipline Specific Skills and Knowledge:**

3 show knowledge of the subject material of the module, which will provide the basis for future study of numerical analysis and its application to all areas of science and business: for example, meteorology, control theory, operations research;

4 understand computation as a natural method for tackling such problems;

**Personal and Key Skills**

5 demonstrate theoretical and practical mathematical skills, including programming.

6. formulate and solve problems.

7. communicate computer results and mathematical derivations effectively.

8. work in teams and use a variety of sources to write reports.

- revision/introduction to Matlab as a tool for numerical programming;

- what numerical analysis is: examples and applications;

- nonlinear scalar equations;

- Newton-Raphson;

- bisection method;

- iterative techniques;

- systems of equations;

- applications in Matlab;

- systems of linear equations;

- Gaussian elimination and the LU decomposition of matrices;

- Gauss-Seidel and Jacobi iteration;

- matrix eigenvalue problems and iterative schemes;

- power method, shifted and inverse power method;

- introduction to matrix norms and condition numbers;

- illustrations in Matlab;

- numerics for ODEs;

- forwards and backwards Euler;

- implicit schemes;

- Runge-Kutta schemes;

- multi-step methods;

- Adams-Bashforth techniques;

- local/global truncation errors;

- stability analysis;

- Matlab-based simulations;

- what optimisation is: examples and applications;

- revision of Hessians;

quadratic forms, eigenvalues and convexity;

- optimisation in one dimension;

- bracketing, golden section search;

- optimisation in two dimensions;

- one at a time method;

- steepest descents method, Newton-Raphson method;

- conjugate gradients; Matlab simulations;

- search-based methods;

- informal discussion of methods such as genetic algorithms, simulated annealing, and swarm-based optimisation;

- computer simulations.

Scheduled Learning & Teaching Activities | 44.00 | Guided Independent Study | 106.00 | Placement / Study Abroad | 0.00 |
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Category | Hours of study time | Description |

Scheduled learning and teaching activities | 24 | Lectures |

Scheduled learning and teaching activities | 2 | Tests |

Scheduled learning and teaching activities | 3 | Feedback |

Scheduled learning and teaching activities | 4 | Discussion sesssions |

Scheduled learning and teaching activities | 11 | Tutorials |

Guided independent study | 106 | Lecture and assessment preparation; wider reading |

Form of Assessment | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|

Not applicable | |||

Coursework | 20 | Written Exams | 80 | Practical Exams |
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Form of Assessment | % of Credit | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|---|

Written exam – closed book | 80 | 2 hours | 1, 3, 5, 6, 7 | Available on request |

Coursework – based on questions submitted for formative assessment | 20 | 2 problem sheets | 1, 2, 3, 4, 5, 6, 7, 8 | Written comments on returned coursework, customized marksheet |

Original Form of Assessment | Form of Re-assessment | ILOs Re-assessed | Time Scale for Re-reassessment |
---|---|---|---|

All above | Written exam (100%) | All | August Ref/Def period |

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk

Reading list for this module:

Type | Author | Title | Edition | Publisher | Year | ISBN | Search |
---|---|---|---|---|---|---|---|

Set | Gerald C.F. & Wheatley P.O. | Applied Numerical Analysis | 7th | Anderson-Wesley | 2004 | 978-8131717400 | [Library] |

Set | Kharab A. & Guenther R.B. | An Introduction To Numerical Methods: a MATLAB Approach | Chapman & Hall | 2012 | 978-1439868997 | [Library] | |

Set | Adby, P.R. & Dempster, M.A.H | Introduction to Optimization Methods | Chapman & Hall | 1974 | 0-412-11040-7 | [Library] | |

Set | Press, W.H., Flannery, B.P., Teukolsky, S.A. & Vetterling, W.T | Numerical Recipes: the Art of Scientific Computing | 3rd edition | Cambridge University Press | 2007 | 13: 9780521880688 | [Library] |

Extended | Iserles A. | A first course in numerical analysis of differential equations | Cambridge University Press | 1996 | 000-0-521-55376-8 | [Library] | |

Extended | Yang, X-S | Introduction to Computational Mathematics | World Scientific | 2008 | 13-978-981-281-81 | [Library] |

CREDIT VALUE | 15 | ECTS VALUE | 7.5 |
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PRE-REQUISITE MODULES | ECM1701, ECM1705 |
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CO-REQUISITE MODULES |

NQF LEVEL (FHEQ) | 5 | AVAILABLE AS DISTANCE LEARNING | No |
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ORIGIN DATE | Friday 09 January 2015 | LAST REVISION DATE | Friday 06 November 2015 |

KEY WORDS SEARCH | Numerical analysis; differential equations; optimisation; minimisation; matrices; Gaussian elimination; MATLAB. |
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