Threshold dynamics, also called diffusion generated motion, of
Merriman, Bence, and Osher generates the motion by mean curvature of an
interface by alternating two very simple and computationally efficient
operations: Convolution and thresholding. I will describe new variants that
generate high order geometric motions (such as motion by surface diffusion)
and how to improve the accuracy of the method on uniform grids. Applications
include problems such as inpainting from image processing and the simulation
of grain boundary motion in polycrystalline materials with many grains. The
talk is based on joint works with Steve Ruuth and Richard Tsai and,
separately, Matt Elsey and Peter Smereka. Emphasis will be on grain boundary motion
simulations in 2D and 3D with large number of grains (joint with Elsey and Smereka).
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