# MA3J8 Approximation Theory and Applications

**Lecturer: **Professor Christoph Ortner

**Term(s): **2

**Status for Mathematics students: **List A

**Commitment: **30 lectures

**Assessment: **100% Exam

**Prerequisites: **There are no formal prerequisites beyond the core module MA222 Metric Spaces, but any programming module and any of the following modules would be useful complements: MA228 Numerical Analysis, MA250 Introduction to Partial Differential Equations, MA3G7 Functional Analysis I, MA3G1 Theory of Partial Differential Equations, MA3H0 Numerical Analysis and PDE.

**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, e.g., 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, including e.g. global polynomials, trigonometric polynomials, splines, radial basis functions, ridge functions as well as methods to construct the approximations, e.g., interpolation, least-squares, Gaussian process, neural networks.

Outline Syllabus:

- Review of background, in particular function spaces, regularity of functions

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 whether feasible)

- multi-variate approximation by tensor products (low dimension)

Part 2: Multi-variate approximation: cover a selection 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.

**Learning Outcomes:**

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

- Demonstrate understanding of a selection of the basic 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:**

*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)