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Geometric Approaches to Computational Problems

Date: Wednesday, June 27, 2018 16:00 - 17:00
Speaker: Albert Chern (TU Berlin)
Location: Mondi Seminar Room 2, Central Building
Series: CS Talk Series
Host: Chris Wojtan

Abstract:

Numerical algorithms for computational problems are usually developed based on modeling technique and analytical approximation. In addition to the classical methods, taking differential geometry into account can significantly advance long lasting computational challenges, and reveal intriguing relation between distant subjects. In this talk, I will present three examples from my recent work: Perfectly matched layers, vortex detections, and the isometric embedding problem.Perfectly Matched Layer (PML) is a widely adopted non-reflecting boundary treatment for wave simulations. The main challenge has been reducing numerical reflections from a discretized PML. We present a new discrete PML that produces no numerical reflection at all. The reflectionless discrete PML is discovered through a straightforward derivation using Discrete Complex Analysis in discrete differential geometry.The next two topics are the vortex detection problem for incompressible fluids and the isometric embedding problem in geometry processing. Perhaps surprisingly, these two seemingly unrelated problems boil down to the same optimization problem over hypersphere-valued functions. By minimizing a GinzburgLandau energy our algorithms can construct codimension-2 level-set functions for vortices fluid, as well as realizations of surfaces given only metric data.
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