MA265 2023/24 Content
Aims: The module gives an introduction to the theory of optimisation as well as the fundamentals of approximation theory.
Content:
- Recap: necessary and sufficient conditions for local min/max, convex functions and sets, Jensen’s inequality, level sets.
- Iterative algorithms: gradient descent and line search methods
- Newton's method
- Linear programming with applications in economics and data science
- Constrained optimisation
- Introduction to Neural Networks
- Approximation theory: polynomial approximation, rational approximation, trigonometric approximation
- Discrete Fourier and Cosine Transform with applications in imaging and signal processing
- Introduction to Wavelets
Objectives:
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understand critical points of multivariable functions
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apply various techniques to solve nonlinear optimisation problems and understand their applications, in economics and data science
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use Lagrange multipliers and the Karush–Kuhn–Tucker conditions to solve constrained nonlinear optimisation problems
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understand the basic concepts of approximation theory
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obtain an understanding of different approximation techniques used in the digital sciences
Books:
- Endre Sueli and David F. Mayers, An Introduction to Numerical Analysis, Cambridge University Press, 2003
- S. Boyd. ‘Convex optimization’, Cambridge University Press 2004
- J. D. Powell, ‘Approximation Theory and Methods’, Cambridge University Press, 1981
- N. Trefethen, ‘Approximation Theory and Practice’