MA93415 Numerical Algorithms and Optimisation
Introductory description
This module provides students with knowledge (and practice) of important numerical optimisation concepts at the intersection between mathematics and scientific computing. Algorithmic structures, data structures, numerical method construction and performance assessment will form key parts of the module, with applications and use cases concentrated on topics in linear algebra, signal processing and optimisation.
Module aims
Numerical Algorithms and Optimisation teaches students the theory and implementation of a set of computational algorithms that provide the fundamental toolkit for advanced data analysis, simulation and optimisation. The syllabus will be drawn from the following list of topics: algorithmic structures (iteration, recursion, memorization) and computational complexity, data structures (linked lists, stacks and queues, binary indexed trees), sorting and search algorithms, Fast Fourier Transform, automatic differentiation, linear systems and the Conjugate Gradient algorithm, Singular Value Decomposition, convex and nonconvex optimisation, constrained optimisation, linear programming, Dijkstra's algorithm and dynamic programming, discreteevent simulation.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
 algorithmic structures (iteration, recursion, memorization) and computational complexity
 data structures (linked lists, stacks and queues, binary indexed trees)
 sorting and search algorithms
 Fast Fourier Transform and its applications
 Topics in numerical linear algebra: solving linear systems, conjugate gradient algorithm, singular value decomposition
 unconstrained continuous optimisation: multivariate minimisation, NelderMead algorithm, automatic differentiation, gradient descent
 constrained continuous optimisation: method of Lagrange multipliers, linear programming
 discrete optimisation: Dijkstra's algorithm, dynamic programming, combinatorial optimisation
Learning outcomes
By the end of the module, students should be able to:
 Demonstrate a deep fundamental understanding of the most important computational algorithms used for advanced data analysis, mathematical modelling and optimisation of complex systems.
 Understand algorithmic structures like iteration, recursion and memorization and be able to apply them.
 Perform mathematical manipulation and implement computational problemsolving techniques.
 Identify the most appropriate approach for computational solution of a mathematical problem and understand problems that can arise such as numerical error, poor conditioning or instability.
 Design and apply both discrete and continuous approaches depending on the requirements of a particular problem.
 Develop efficient code for the computational solution and analysis and optimisation problems, select appropriate data structures and algorithms, present and visualise algorithm outputs and results of analyses in a clear and informative way.
 Distinguish the computational complexity of an algorithm and the practical constraints it imposes on problem solving.
Indicative reading list
Lloyd N. Trefethen and David Bau, Numerical Linear Algebra, 3rd Edition, SIAM, 1997.
William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. Numerical recipes: The art of scientific computing, 3rd Edition, Cambridge University Press, New York, NY, USA, 2007.
David Kincaid and Ward Cheney. Numerical analysis : mathematics of scientific computing, 3rd Edition, American Mathematical Society, 2009.
Research articles in the field will complement the textbooks above.
Interdisciplinary
The module sits naturally between mathematics, computer science and elements from related disciplines such as
physics, biology or image processing.
Subject specific skills
See learning outcomes: data structure recognition and usage, discrete mathematical analysis, algorithm construction, performance analysis, advanced linear algebra, optimisation techniques.
Transferable skills
Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence. This will be grounded in enhanced software development knowledge (including transferable ‘software carpentry’ skills such as version control and continuous integration testing) and deployed for a wide variety of problems in interdisciplinary contexts.
Study time
Type  Required 

Lectures  30 sessions of 1 hour (20%) 
Supervised practical classes  10 sessions of 1 hour (7%) 
Private study  48 hours (32%) 
Assessment  62 hours (41%) 
Total  150 hours 
Private study description
The students will engage with the material via exercises and assignments, which are constructed as an embodiment of theoretical material learned during the lectured components of the module. Lecture and practical class preparation (20h), engaging with online material (16h) and work on formative elements (12h, be they pen and paper based or programming) is anticipated to fill up the rest of the independent learning time. Checkpoints will be provided along the way, and time allowance for software installation and early steps, as well as exam revision is also considered.
Costs
No further costs have been identified for this module.
You must pass all assessment components to pass the module.
Assessment group D1
Weighting  Study time  

Assessed coursework  60%  50 hours 
Assignments with feedback provided within 2 weeks from submission, so as to enable feedforward into the following assessment and ultimately study. These will combine theoretical work (derivation, proofs, algorithmic formulation), implementation, data visualisation and interpretation. Discussions about the wider implications of the material will also be encouraged. 

Oral examination  40%  12 hours 
Feedback on assessment
Written feedback on assignments (annotated workbooks) plus informal/general oral feedback during classwork sessions.
Written feedback from examiners on the oral examination.
Courses
This module is Core for:
 Year 1 of TMAAG1PF Postgraduate Taught Mathematics of Systems
This module is Optional for:
 Year 1 of TMAAG1P9 Postgraduate Taught Interdisciplinary Mathematics
 Year 1 of TMAAG1P0 Postgraduate Taught Mathematics

TPXAF345 Postgraduate Taught Modelling of Heterogeneous Systems (PGDip)
 Year 1 of F345 Modelling of Heterogeneous Systems (PGDip)
 Year 2 of F345 Modelling of Heterogeneous Systems (PGDip)