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Numerical PDEs and Applied Topology

Anil N. Hirani
University of Illinois at Urbana-Champaign


Exterior calculus generalizes vector calculus to manifolds. For numerical solutions of PDEs on meshes this language has been discretized as finite element exterior calculus and discrete exterior calculus. I'll first give a very brief introduction to these discretizations. Tools from geometry and topology, such as Hodge theory, and basic ideas from cohomology and homology will be seen to be an integral part of these discretizations.

A specific example I'll describe will be the computation of harmonic forms. This is a crucial first step in a finite element solution of even the most basic elliptic PDE like Poisson's equation. I'll show how the availability of a homology basis allows one to find a basis for discrete harmonic forms using least squares. When viewed appropriately, the concepts, language, and software for these PDE discretizations can be easily used to solve some interesting problems in data analysis. A slight generalization also leads to some problems in computational topology. Specifically, this involves moving from 2-norms to 1-norms. In some sense, this is an example of how work in numerical PDEs can lead to a very combinatorial and classical problem in computational topology.

BRIEF BIO: Anil Hirani is an applied mathematician and computer scientist on the faculty in the CS department at University of Illinois at Urbana-Champaign. Before Illinois he was at NASA's Jet Propulsion Lab, Sony, and Sun Microsystems. He has Master's degrees in CS from Stanford and Caltech and a Ph.D from Caltech in CS with minors in Mathematics, and in Control and Dynamical Systems.

Tuesday, October 16, 2012
11:00AM AP&M 2402