Engineering Tripos Part IIB, 4A3: Turbomachinery, 2021-22
Module Leader
Lecturers
Prof R.J. Miller and Dr L. Xu
Lab Leader
Prof R.J. Miller
Timing and Structure
Michaelmas term. 75% exam / 25% coursework. 12 lectures (including examples classes) + coursework
Prerequisites
3A1 and 3A3 assumed
Aims
The aims of the course are to:
- provide a general understanding of the principles that govern the design of axial flow and radial flow turbomachines.
Objectives
As specific objectives, by the end of the course students should be able to:
- understand the principles underpinning the study of turbomachine aerodynamics.
- know the requirements for different type of turbomachines.
- know the factors which influence the overall design of turbomachine stages and which influence the matching of components.
- know the factors which influence overall design of turbomachines for propulsion and stationary power-plant applications.
- evaluate the performance of turbine and compressor bladerows and stages using mean-line analyses.
- select a design for a given duty.
- present and understand information on stage and machine design.
- describe and understand compressor off-design performance.
- analyse the performance of propulsion systems and stationary power plant.
Content
Applications and Characteristics of Turbomachines (12L, Prof. RJ Miller and Dr LP Xu)
- Stage design and choice of design parameters.
- Specific speed, dynamic scaling and measures of efficiency.
- Analysis of the mean-line flow in compressors and turbines.
- Radial flow turbomachines.
- Characteristics of compressors, pumps and turbines.
- Matching of components: compressors and turbines; nozzles, throttles and diffusers. Compressor off-design problems; stall and its consequences.
- Application of turbomachines: power plant and aircraft propulsion systems.
Coursework
| Coursework | Format |
Due date & marks |
|---|---|---|
|
Cascade Experiment Testing of a turbine cascade in a small wind tunnel to measure the blade surface pressure distribution, loss coefficient and flow exit angle. Time required: About 3 hours in the lab plus 4 hours write up. Learning objectives:
|
Experimental work done in pairs. Individual report. Anonymously marked. |
Reports are due 2 weeks after the date of the experiment. [15/60] |
Booklists
Please refer to the Booklist for Part IIB Courses for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA2
Demonstrate creative and innovative ability in the synthesis of solutions and in formulating designs.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
D1
Wide knowledge and comprehensive understanding of design processes and methodologies and the ability to apply and adapt them in unfamiliar situations.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 01/10/2021 10:41
Engineering Tripos Part IIB, 4A3: Turbomachinery, 2023-24
Module Leader
Lecturers
Prof R.J. Miller and Dr J. Taylor
Lab Leader
Dr J. Taylor
Timing and Structure
Michaelmas term. 75% exam / 25% coursework. 12 lectures (including examples classes) + coursework
Prerequisites
3A1 and 3A3 assumed
Aims
The aims of the course are to:
- provide a general understanding of the principles that govern the design of axial flow and radial flow turbomachines.
Objectives
As specific objectives, by the end of the course students should be able to:
- understand the principles underpinning the study of turbomachine aerodynamics.
- know the requirements for different type of turbomachines.
- know the factors which influence the overall design of turbomachine stages and which influence the matching of components.
- know the factors which influence overall design of turbomachines for propulsion and stationary power-plant applications.
- evaluate the performance of turbine and compressor bladerows and stages using mean-line analyses.
- select a design for a given duty.
- present and understand information on stage and machine design.
- describe and understand compressor off-design performance.
- analyse the performance of propulsion systems and stationary power plant.
Content
Applications and Characteristics of Turbomachines (12L, Prof. RJ Miller and Dr J. Taylor)
- Stage design and choice of design parameters.
- Specific speed, dynamic scaling and measures of efficiency.
- Analysis of the mean-line flow in compressors and turbines.
- Radial flow turbomachines.
- Characteristics of compressors, pumps and turbines.
- Matching of components: compressors and turbines; nozzles, throttles and diffusers. Compressor off-design problems; stall and its consequences.
- Application of turbomachines: power plant and aircraft propulsion systems.
Coursework
| Coursework | Format |
Due date & marks |
|---|---|---|
|
Cascade Experiment Testing of a turbine cascade in a small wind tunnel to measure the blade surface pressure distribution, loss coefficient and flow exit angle. Time required: About 3 hours in the lab plus 4 hours write up. Learning objectives:
|
Experimental work done in pairs. Individual report. Anonymously marked. |
Reports are due 2 weeks after the date of the experiment. [15/60] |
Booklists
Please refer to the Booklist for Part IIB Courses for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA2
Demonstrate creative and innovative ability in the synthesis of solutions and in formulating designs.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
D1
Wide knowledge and comprehensive understanding of design processes and methodologies and the ability to apply and adapt them in unfamiliar situations.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 30/05/2023 15:24
Engineering Tripos Part IIB, 4A3: Turbomachinery, 2022-23
Module Leader
Lecturers
Prof R.J. Miller and Dr J. Taylor
Lab Leader
Dr J. Taylor
Timing and Structure
Michaelmas term. 75% exam / 25% coursework. 12 lectures (including examples classes) + coursework
Prerequisites
3A1 and 3A3 assumed
Aims
The aims of the course are to:
- provide a general understanding of the principles that govern the design of axial flow and radial flow turbomachines.
Objectives
As specific objectives, by the end of the course students should be able to:
- understand the principles underpinning the study of turbomachine aerodynamics.
- know the requirements for different type of turbomachines.
- know the factors which influence the overall design of turbomachine stages and which influence the matching of components.
- know the factors which influence overall design of turbomachines for propulsion and stationary power-plant applications.
- evaluate the performance of turbine and compressor bladerows and stages using mean-line analyses.
- select a design for a given duty.
- present and understand information on stage and machine design.
- describe and understand compressor off-design performance.
- analyse the performance of propulsion systems and stationary power plant.
Content
Applications and Characteristics of Turbomachines (12L, Prof. RJ Miller and Dr J. Taylor)
- Stage design and choice of design parameters.
- Specific speed, dynamic scaling and measures of efficiency.
- Analysis of the mean-line flow in compressors and turbines.
- Radial flow turbomachines.
- Characteristics of compressors, pumps and turbines.
- Matching of components: compressors and turbines; nozzles, throttles and diffusers. Compressor off-design problems; stall and its consequences.
- Application of turbomachines: power plant and aircraft propulsion systems.
Coursework
| Coursework | Format |
Due date & marks |
|---|---|---|
|
Cascade Experiment Testing of a turbine cascade in a small wind tunnel to measure the blade surface pressure distribution, loss coefficient and flow exit angle. Time required: About 3 hours in the lab plus 4 hours write up. Learning objectives:
|
Experimental work done in pairs. Individual report. Anonymously marked. |
Reports are due 2 weeks after the date of the experiment. [15/60] |
Booklists
Please refer to the Booklist for Part IIB Courses for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA2
Demonstrate creative and innovative ability in the synthesis of solutions and in formulating designs.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
D1
Wide knowledge and comprehensive understanding of design processes and methodologies and the ability to apply and adapt them in unfamiliar situations.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 28/09/2022 14:08
Engineering Tripos Part IIB, 4A3: Turbomachinery, 2018-19
Module Leader
Lecturers
Dr N Atkins and Dr T Hynes
Lab Leader
Dr T Hynes
Timing and Structure
Michaelmas term. 75% exam / 25% coursework. 12 lectures (including examples classes) + coursework
Prerequisites
3A1 and 3A3 assumed
Aims
The aims of the course are to:
- provide a general understanding of the principles that govern the design of axial flow and radial flow turbomachines.
Objectives
As specific objectives, by the end of the course students should be able to:
- understand the principles underpinning the study of turbomachine aerodynamics.
- know the requirements for different type of turbomachines.
- know the factors which influence the overall design of turbomachine stages and which influence the matching of components.
- know the factors which influence overall design of turbomachines for propulsion and stationary power-plant applications.
- evaluate the performance of turbine and compressor bladerows and stages using mean-line analyses.
- select a design for a given duty.
- present and understand information on stage and machine design.
- describe and understand compressor off-design performance.
- analyse the performance of propulsion systems and stationary power plant.
Content
Applications and Characteristics of Turbomachines (12L, Dr N R Atkins and Dr T P Hynes)
- Stage design and choice of design parameters.
- Specific speed, dynamic scaling and measures of efficiency.
- Analysis of the mean-line flow in compressors and turbines.
- Radial flow turbomachines.
- Characteristics of compressors, pumps and turbines.
- Matching of components: compressors and turbines; nozzles, throttles and diffusers. Compressor off-design problems; stall and its consequences.
- Application of turbomachines: power plant and aircraft propulsion systems.
Coursework
| Coursework | Format |
Due date & marks |
|---|---|---|
|
Cascade Experiment Testing of a turbine cascade in a small wind tunnel to measure the blade surface pressure distribution, loss coefficient and flow exit angle. Time required: About 3 hours in the lab plus 4 hours write up. Learning objectives:
|
Experimental work done in pairs. Individual report. Anonymously marked. |
Reports are due 2 weeks after the date of the experiment. [15/60] |
Booklists
Please see the Booklist for Group A Courses for references for this module.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA2
Demonstrate creative and innovative ability in the synthesis of solutions and in formulating designs.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
D1
Wide knowledge and comprehensive understanding of design processes and methodologies and the ability to apply and adapt them in unfamiliar situations.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 17/05/2018 13:25
Engineering Tripos Part IIB, 4A3: Turbomachinery, 2019-20
Module Leader
Lecturers
Prof W.N. Dawes and Prof Rob Miller
Lab Leader
Timing and Structure
Michaelmas term. 75% exam / 25% coursework. 12 lectures (including examples classes) + coursework
Prerequisites
3A1 and 3A3 assumed
Aims
The aims of the course are to:
- provide a general understanding of the principles that govern the design of axial flow and radial flow turbomachines.
Objectives
As specific objectives, by the end of the course students should be able to:
- understand the principles underpinning the study of turbomachine aerodynamics.
- know the requirements for different type of turbomachines.
- know the factors which influence the overall design of turbomachine stages and which influence the matching of components.
- know the factors which influence overall design of turbomachines for propulsion and stationary power-plant applications.
- evaluate the performance of turbine and compressor bladerows and stages using mean-line analyses.
- select a design for a given duty.
- present and understand information on stage and machine design.
- describe and understand compressor off-design performance.
- analyse the performance of propulsion systems and stationary power plant.
Content
Applications and Characteristics of Turbomachines (12L, Prof. WN Dawes and Dr LP Xu)
- Stage design and choice of design parameters.
- Specific speed, dynamic scaling and measures of efficiency.
- Analysis of the mean-line flow in compressors and turbines.
- Radial flow turbomachines.
- Characteristics of compressors, pumps and turbines.
- Matching of components: compressors and turbines; nozzles, throttles and diffusers. Compressor off-design problems; stall and its consequences.
- Application of turbomachines: power plant and aircraft propulsion systems.
Coursework
| Coursework | Format |
Due date & marks |
|---|---|---|
|
Cascade Experiment Testing of a turbine cascade in a small wind tunnel to measure the blade surface pressure distribution, loss coefficient and flow exit angle. Time required: About 3 hours in the lab plus 4 hours write up. Learning objectives:
|
Experimental work done in pairs. Individual report. Anonymously marked. |
Reports are due 2 weeks after the date of the experiment. [15/60] |
Booklists
Please see the Booklist for Group A Courses for references for this module.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA2
Demonstrate creative and innovative ability in the synthesis of solutions and in formulating designs.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
D1
Wide knowledge and comprehensive understanding of design processes and methodologies and the ability to apply and adapt them in unfamiliar situations.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 22/10/2019 10:04
Engineering Tripos Part IIA, 3M1: Mathematical Methods, 2021-22
Module Leader
Lecturers
Prof M Girolami, Dr A Kabla and Prof M Gales
Lab Leader
Prof M Girolami
Timing and Structure
Lent term. 16 lectures and coursework.
Aims
The aims of the course are to:
- Teach some mathematical techniques that have wide applicability to many areas of engineering.
Objectives
As specific objectives, by the end of the course students should be able to:
- Find the SVD of a matrix, and understand how this can be used to calculate the rank and pseudo-inverse of the matrix.
- Calculate the least squares solution of a set of linear equations.
- Understand how to apply Principal Component Analysis (PCA) to a problem.
- Apply PCA to reduce the dimensionality of an optimization problem and/or to improve the solution representation.
- 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.
- Implement various sampling schemes to enable parameters of stochastic processes to be estimated.
- Understand the concepts of local and global minima and the conditions for which a global minimum can be obtained.
- Understand the algorithms of the different gradient search methods.
- Solve unconstrained problems using appropriate search methods.
- Solve constrained linear and non-linear optimization problems using appropriately selected techniques.
- Understand how Markov Chain-based algorithms can be used to give reasonable solutions to global optimisation problems.
Content
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. Stochastic (random) processes are important in fields such as signal and image processing, data analysis etc. Optimization methods are routinely used in almost of every branch of engineering, especially in the context of design.
Linear Algebra (4L, Prof G Wells)
- 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
Stochastic Processes (5L, Prof S Godsill)
- 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.
Optimization (7L, Prof M Girolami)
- 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)
- Constrained Optimization: Lagrange and Karush-Kuhn-Tucker (KKT) multipliers; penalty and barrier functions
- Global optimisation: Simulated Annealing
Coursework
Exploring Principal Component Analysis for dimensional reduction and data representation.
There is no Full Technical Report (FTR) associated with this module.
Booklists
Please refer to the Booklist for Part IIA Courses for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
E4
Understanding of and ability to apply a systems approach to engineering problems.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
P8
Ability to apply engineering techniques taking account of a range of commercial and industrial constraints.
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
Last modified: 20/05/2021 07:40
Engineering Tripos Part IIA, 3M1: Mathematical Methods, 2024-25
Module Leader
Lecturers
Prof M Girolami, Prof G Wells and Dr H Ge
Lab Leader
Prof M Girolami
Timing and Structure
Lent term. 16 lectures and coursework.
Aims
The aims of the course are to:
- Teach some mathematical techniques that have wide applicability to many areas of engineering.
Objectives
As specific objectives, by the end of the course students should be able to:
- Find the SVD of a matrix, and understand how this can be used to calculate the rank and pseudo-inverse of the matrix.
- Calculate the least squares solution of a set of linear equations.
- Understand how to apply Principal Component Analysis (PCA) to a problem.
- Apply PCA to reduce the dimensionality of an optimization problem and/or to improve the solution representation.
- 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.
- Implement various sampling schemes to enable parameters of stochastic processes to be estimated.
- Understand the concepts of local and global minima and the conditions for which a global minimum can be obtained.
- Understand the algorithms of the different gradient search methods.
- Solve unconstrained problems using appropriate search methods.
- Solve constrained linear and non-linear optimization problems using appropriately selected techniques.
- Understand how Markov Chain-based algorithms can be used to give reasonable solutions to global optimisation problems.
Content
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. Stochastic (random) processes are important in fields such as signal and image processing, data analysis etc. Optimization methods are routinely used in almost of every branch of engineering, especially in the context of design.
Linear Algebra (5L, Prof G Wells)
- 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
Stochastic Processes (5L, Dr H Ge)
- 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.
Optimization (6L, Prof M Girolami)
- 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)
- Constrained Optimization: Lagrange and Karush-Kuhn-Tucker (KKT) multipliers; penalty and barrier functions
Coursework
Exploring Principal Component Analysis for dimensional reduction and data representation.
There is no Full Technical Report (FTR) associated with this module.
Booklists
Please refer to the Booklist for Part IIA Courses for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
E4
Understanding of and ability to apply a systems approach to engineering problems.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
P8
Ability to apply engineering techniques taking account of a range of commercial and industrial constraints.
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
Last modified: 22/01/2025 13:56
Engineering Tripos Part IIA, 3M1: Mathematical Methods, 2023-24
Module Leader
Lecturers
Prof M Girolami, Prof G Wells and Prof M Gales
Lab Leader
Prof M Girolami
Timing and Structure
Lent term. 16 lectures and coursework.
Aims
The aims of the course are to:
- Teach some mathematical techniques that have wide applicability to many areas of engineering.
Objectives
As specific objectives, by the end of the course students should be able to:
- Find the SVD of a matrix, and understand how this can be used to calculate the rank and pseudo-inverse of the matrix.
- Calculate the least squares solution of a set of linear equations.
- Understand how to apply Principal Component Analysis (PCA) to a problem.
- Apply PCA to reduce the dimensionality of an optimization problem and/or to improve the solution representation.
- 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.
- Implement various sampling schemes to enable parameters of stochastic processes to be estimated.
- Understand the concepts of local and global minima and the conditions for which a global minimum can be obtained.
- Understand the algorithms of the different gradient search methods.
- Solve unconstrained problems using appropriate search methods.
- Solve constrained linear and non-linear optimization problems using appropriately selected techniques.
- Understand how Markov Chain-based algorithms can be used to give reasonable solutions to global optimisation problems.
Content
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. Stochastic (random) processes are important in fields such as signal and image processing, data analysis etc. Optimization methods are routinely used in almost of every branch of engineering, especially in the context of design.
Linear Algebra (5L, Prof G Wells)
- 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
Stochastic Processes (5L, Prof M Gales)
- 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.
Optimization (6L, Prof M Girolami)
- 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)
- Constrained Optimization: Lagrange and Karush-Kuhn-Tucker (KKT) multipliers; penalty and barrier functions
Coursework
Exploring Principal Component Analysis for dimensional reduction and data representation.
There is no Full Technical Report (FTR) associated with this module.
Booklists
Please refer to the Booklist for Part IIA Courses for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
E4
Understanding of and ability to apply a systems approach to engineering problems.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
P8
Ability to apply engineering techniques taking account of a range of commercial and industrial constraints.
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
Last modified: 31/12/2023 12:49
Engineering Tripos Part IIA, 3M1: Mathematical Methods, 2017-18
Module Leader
Lecturers
Prof G Csanyi, Dr G Wells and Prof M Gales
Lab Leader
Prof G Csanyi
Timing and Structure
Lent term. 16 lectures and coursework.
Aims
The aims of the course are to:
- Teach some mathematical techniques that have wide applicability to many areas of engineering.
Objectives
As specific objectives, by the end of the course students should be able to:
- Find the SVD of a matrix, and understand how this can be used to calculate the rank and pseudo-inverse of the matrix.
- Calculate the least squares solution of a set of linear equations.
- Understand how to apply Principal Component Analysis (PCA) to a problem.
- 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.
- Implement various sampling schemes to enable parameters of stochastic processes to be estimated.
- Understand the concepts of local and global minima and the conditions for which a global minimum can be obtained.
- Understand the algorithms of the different gradient search methods.
- Solve unconstrained problems using appropriate search methods.
- Solve constrained linear and non-linear optimization problems using appropriately selected techniques.
- Apply PCA to reduce the dimensionality of an optimization problem and/or to improve the solution representation.
Content
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. Stochastic (random) processes are important in fields such as signal and image processing, data analysis etc. Optimization methods are routinely used in almost of every branch of engineering, especially in the context of design.
Linear Algebra (4L, Dr G Wells)
- 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
Stochastic Processes (4L, Prof M Gales)
- 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.
Optimization (8L, Prof G Csanyi)
- 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)
- Constrained Optimization: Lagrange and Kuhn-Tucker multipliers; penalty and barrier functions
Coursework
Exploring Principal Component Analysis for dimensional reduction and data representation.
There is no Full Technical Report (FTR) associated with this module.
Booklists
Please see the Booklist for Part IIA Courses for references for this module.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
E4
Understanding of and ability to apply a systems approach to engineering problems.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
P8
Ability to apply engineering techniques taking account of a range of commercial and industrial constraints.
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
Last modified: 17/02/2018 14:00
Engineering Tripos Part IIA, 3M1: Mathematical Methods, 2022-23
Module Leader
Lecturers
Prof M Girolami and Prof M Gales
Lab Leader
Prof M Girolami
Timing and Structure
Lent term. 16 lectures and coursework.
Aims
The aims of the course are to:
- Teach some mathematical techniques that have wide applicability to many areas of engineering.
Objectives
As specific objectives, by the end of the course students should be able to:
- Find the SVD of a matrix, and understand how this can be used to calculate the rank and pseudo-inverse of the matrix.
- Calculate the least squares solution of a set of linear equations.
- Understand how to apply Principal Component Analysis (PCA) to a problem.
- Apply PCA to reduce the dimensionality of an optimization problem and/or to improve the solution representation.
- 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.
- Implement various sampling schemes to enable parameters of stochastic processes to be estimated.
- Understand the concepts of local and global minima and the conditions for which a global minimum can be obtained.
- Understand the algorithms of the different gradient search methods.
- Solve unconstrained problems using appropriate search methods.
- Solve constrained linear and non-linear optimization problems using appropriately selected techniques.
- Understand how Markov Chain-based algorithms can be used to give reasonable solutions to global optimisation problems.
Content
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. Stochastic (random) processes are important in fields such as signal and image processing, data analysis etc. Optimization methods are routinely used in almost of every branch of engineering, especially in the context of design.
Linear Algebra (4L, Prof G Wells)
- 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
Stochastic Processes (5L, Prof S Godsill)
- 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.
Optimization (7L, Prof M Girolami)
- 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)
- Constrained Optimization: Lagrange and Karush-Kuhn-Tucker (KKT) multipliers; penalty and barrier functions
- Global optimisation: Simulated Annealing
Coursework
Exploring Principal Component Analysis for dimensional reduction and data representation.
There is no Full Technical Report (FTR) associated with this module.
Booklists
Please refer to the Booklist for Part IIA Courses for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
E4
Understanding of and ability to apply a systems approach to engineering problems.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
P8
Ability to apply engineering techniques taking account of a range of commercial and industrial constraints.
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
Last modified: 24/05/2022 12:53
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