The Optimisation and Numerical Analysis seminar is held regularly during termtime.
Wednesday 15 October 2025, 15:00-16:00
Arts, Room 103
Wednesday 22 October 2025, 14:00-15:00
Arts, Lecture Room 4
Fluid mechanics and magnetohydrodynamics often involve intricate differential and topological structures, such as vorticity and magnetic field knots, which are critical to the underlying physics. Numerical discretization errors can break these structures, leading to wrong solutions.
In this talk, we present two examples in topological (magneto)hydrodynamics: relaxation and dynamo. Relaxation addresses the evolution of magnetic fields from given initial conditions in plasma physics, focusing on the existence and properties of stationary states. Open questions, including the Parker hypothesis, highlight the role of magnetic field line topology, particularly knots, in constraining relaxation processes. Conversely, the dynamo problem examines the exponential growth of magnetic fields.
We emphasise the importance of structure-preserving numerical methods, specifically those that conserve helicity and topology. Using finite element de Rham complexes within the framework of finite element exterior calculus, we derive schemes that precisely preserve these structures, ensuring robust and physically meaningful simulations.
Wednesday 5 November 2025, 14:00-15:00
Watson Building, B16
In this talk, I will introduce tropical geometry - a variant of algebraic geometry which provides a geometric lens through which to view non-smooth optimisation problems, and that has become increasingly studied in applications such as computational biology, economics, and computer science. We will review various types of convexity which arise in tropical problems, and we propose a new gradient descent method for solving tropical optimisation problems. Theoretical results establish global solvability for tropically quasi-convex problems, while numerical experiments demonstrate the method's superior performance over classical descent for tropical optimisation problems which exhibit tropical quasi-convexity but not classical convexity. Notably, tropical gradient descent seamlessly integrates into advanced optimisation methods, such as Adam, offering improved overall performance.