Computational Problems from a Geometric Perspective

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Albert Chern

Albert Chern
Post-Doctoral Researcher, Technical University of Berlin
Monday, February 11, 2019 @ 11:00am-12:00pm
Room 1242, CSE building

 

Abstract:

Simulating physical reality in computational sciences and computer graphics rely on mathematical modelings and numerical approximation theories.  In differential geometry—a seemingly distant subject from computational mathematics—one explores the abstraction of calculus on curved or abstract spaces.  It turns out that by approaching some long-lasting computational challenges through a geometric perspective, one may discover surprising insights or even simple solutions.  In this talk, I will give three examples from my recent research work.  These include wave simulations, fluid simulations and surface reconstruction problems.

In simulating propagating waves, what is highly demanded is a boundary treatment which does not produce artificial reflections.  Since the 1970’s there have been theories for designing such boundaries, yet all of their existing discretizations produce numerical reflections.  Using the discrete complex analysis, a research area in geometry processing and discrete differential geometry, I will present the first boundary treatment that produces no reflection at all.  

In fluid simulations, one main challenge is to capture the intricate vortex dynamics, which are often lost in a finite computational resolution.  Through a change of variable the governing equations for the classical fluids become a much simpler Schrödinger equation known in quantum mechanics.  Schrödinger-based fluid solver is also more robust in reproducing vortex dynamics.  The bridge between the two equations is discovered through a geometrization of the level set functions for vortex lines.  I will also elucidate the possible new understanding about fluid dynamics based on this framework.

Constructing surfaces from their intrinsic metric is an important process for visualizations.  However a direct optimization to this problem usually leads to a tangled surface with unwanted singularities.  By understanding the topology of the group of 3D rotations and quaternions, one can construct a surface with all singularities resolved. 

 

Bio:

Albert Chern is a postdoctoral researcher in mathematics at Technical University of Berlin.  He received his bachelor and master degree in mathematics at National Taiwan University, and received his PhD in applied and computational mathematics at Caltech.  Albert’s research focuses lie in geometry processing, physical simulation, and discrete differential geometry in computer graphics.  He also has a broad interest in mathematical physics, differential geometry, algebraic topology, and numerical analysis.

Faculty Host: Ravi Ramamoorthi