H.B. Keller Colloquium

This talk consists of two geometric techniques for computational problems, both motivated by Geometric Measure Theory. They are applied to geometric optimizations and solving partial differential equations (PDEs) on unbounded domains, which are fundamental in shape synthesis and physical simulations. Shape optimization problems often involve non-convex geometric functionals such as surface area. Directly minimizing these energies can lead to stuck at local minima, and initializing a correct surface topology is challenging. Geometric measure theory provides new ways of approaching geometric optimization problems. Classical curves and surfaces are generalized to differential forms representing superpositions of infinitely many curves and surfaces. Under such a representation, the minimal surface problem becomes convex, and standard convex optimization techniques apply. 

The second part of the talk focuses on simulations aided by Kelvin transformation. Many physical simulation problems take place in an unbounded space, requiring solving PDEs on a non-compact domain. Standard numerical approaches rely on coordinate mapping or domain truncation, yielding coordinate singularity or artifacts on the truncation boundary. We describe a general Kelvin transformation technique, which maps the infinite domain to a bounded one without creating singularities. The method is made possible by factoring out an asymptotic of the singularity induced by the coordinate stretching. The resulting transformation of functions can be understood as the natural transformation for fractional densities in geometric measure theory. In the viewpoint of Klein's Erlangen Program, the analysis reveals a "Kelvin Geometry," where objects are functions subject to Kelvin transforms, leaving the PDE of interest invariant. The key to solving the infinite domain problem is to recognize that the boundedness quality of the domain is not a geometrically invariant notion under Kelvin Geometry. Therefore, we can transform the infinite domain problem into a compact one without sacrificing numerical accuracy.