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Analysis of Diffusion Generated Motion for Mean Curvature Flow in Codimension Two: A Gradient-Flow Approach

PDE and Applied Math Seminar

Speaker: Tim Laux, UC Berkeley
Related Webpage: http://math.berkeley.edu/~tim.laux/
Location: 2112 MSB
Start time: Fri, Apr 12 2019, 4:10PM

The Merriman–Bence–Osher (MBO) scheme, also known as diffusion generated motion or thresholding, is an efficient numerical algorithm for computing mean curvature flow (MCF). In this talk, I will briefly discuss the fairly well-understood case of hypersurfaces. Then I will present the first convergence proof of the scheme in codimension two. Our proof is based on a new generalization of the minimizing movements interpretation for hypersurfaces (Esedoglu–Otto ’15) by means of an energy that approximates the Dirichlet energy of the state function. As long as a smooth MCF exists, we establish uniform energy estimates for the approximations away from the smooth solution and prove convergence towards this MCF. The current result which holds in codimension two relies in a very crucial manner on a new sharp monotonicity formula for the thresholding energy. Joint work with Aaron Yip (Purdue).