Thresholding Algorithms for Curvature Driven Flows

Curvature driven flows arise in a number of models from materials science, physics, and biology. PDE's can be used to model these flows, but they tend to be non-linear and somewhat complicated. In the early 90's, Bence-Merriman-Osher (BMO) developed a very simple "thresholding algorithm" for producing mean curvature motion for a hypersurface. Their ideas have been extended over the years to mean curvature motion for filaments in space, and fourth order flows such as Willmore flow and Surface Diffusion. In this talk I will discuss the thresholding algorithms for mean curvature and Willmore flows. In particular I will detail a new proof of the convergence of the original BMO algorithm to mean curvature flow, which does not use the maximum principle. This new technique seems promising for proving the convergence of the algorithm for filament motion, which involves vector valued functions.