Content:

The Module will provide students with a foundation in approximation theory, driven by its applications in scientific computing and data science.

In approximation theory a function that is difficult or impossible to evaluate directly, e.g., an unknown constitutive law or the solution of a PDE, is to be approximated as efficiently as possible from a more elementary class of functions, the approximation space. The module will explore different choices of approximation spaces and how they can be effective in different applications chosen from typical scientific computing and data science, including e.g. global polynomials, trigonometric polynomials, splines, radial basis functions, ridge functions (neural networks) as well as methods to construct the approximations, e.g., interpolation, least-squares, Gaussian process.

Outline Syllabus:

Part 1: univariate approximation

- spline approximation of smooth functions in 1D
- polynomial and trigonometric approximation of analytic functions in 1D
- linear best approximation
- best n-term approximation (to be decided)
- multi-variate approximation by tensor products in $\mathbb{R}^d$, curse of dimensionality

Part 2: Multi-variate approximation: details will depend on the progress through Part 1 and available time, but the idea of Part 2 is to cover a few selected examples of high-dimensional approximation theory, for example a sub-set of the following:

- mixed regularity, splines and sparse grids, Smolyak algorithm
- radial basis functions and Gaussian processes
- ridge functions and neural networks
- compressed sensing and best n-term approximation

Throughout the lecture each topic will cover (1) approximation rates, (2) algorithms, and (3) examples, typically implemented in Julia or Python. Any programming aspects of the module will not be examinable.

Learning Outcomes:

By the end of the module students should be able to:

• Demonstrate understanding of key concepts, theorems and calculations of univariate approximation theory.
  
• Demonstrate understanding of a selection of the basic concepts, theorems and calculations of multivariate approximation theory.
• Demonstrate understanding of basic algorithms and examples used in approximation theory.

Books:

I plan to develop lecture notes, possibly a mix of traditional and online notebooks, but they will only become available as we progress through the module.

Approximation Theory and Methods, M. J. D. Powell
Approximation Theory and Approximation Practice, N. Trefethen
A course in approximation theory, E.W.Cheney and W.A.Light
Nonlinear approximation, R. DeVore (Acta Numerica)