Will Woolfenden

Geometric Numerical Integration

My final year project was on integration methods for ordinary differential equations which retain qualitative behaviours of the original system. The report first explores symplectic methods, where the flow map is a volume preserving map on areas in the phase portrait. Symplectic methods are very popular in particle physics and celestial problems. The second section is dedicated to positivity preserving methods, with applications to chemical reaction problems. The analysis covers lots of powerful results in the numerical analysis of differential equations.

Phonation

For my third year Mathematics undergraduate project, I investigated phonation - the process of voice production in humans. The vocal cords in the larynx oscillate to produce variations in pressure, leading to complex propagation of sound waves. I considered the vocal cords as opposing masses which move subject to varying pressure in the upper airways. My analysis involved fundamental concepts in linear elasticity and fluid mechanics, and applied numerical methods to formulate results.

Compressed Sensing

A C++ implementation and report in scientific computing, exploring iterative methods for solving systems of linear equations, particularly those with insufficient information for traditional methods. I began by exploring the steepest descent algorithm, used to solve the least squares problem for an overdetermined system. This lay the groundwork to explore normalised iterative hard thresholding, which is a method for instead recovering sparse solutions from underdetermined systems. The analysis involved writing and optimising C++ implementations of these methods.

Conjugate Gradient Method

A C++ implementation and report in scientific computing, exploring the conjugate gradient method for solving a system of linear equations. A particular focus on the efficiency of CG, as opposed to a method like steepest descent. The testing of the implementation explored finite difference approximations to Poisson's equation in two dimensions.