Junior Analysis Seminar – Alexandra Borkowski (KCL)

I will discuss two results on stable finite-time blow-up for semilinear wave equations with quadratic derivative nonlinearities. In one dimension, both the time-derivative and spatial-derivative nonlinearities admit “generalised self-similar” blow-up solutions. I will begin by briefly reviewing a result on the construction and asymptotic stability of such solutions in the time-derivative case.

Following the work of Wang, Abernethy, and Levy (2023), a two-player online game can be used to describe accelerated convex optimisation algorithms. We will begin by developing this viewpoint in continuous time, i.e. by discussing what this game interpretation tells us about accelerated gradient flows. We then ask if a similar framework can be developed for sampling, and construct an analogous two-player game for sampling from probability measures with a log-concave density. In particular, suitable choices of strategies for the two players recover the Underdamped Langevin dynamics. This overall perspective provides a unified way of both understanding the Underdamped Langevin dynamics and obtaining explicit accelerated convergence rates in KL divergence.