Anil Hirani

Ph.D. California Institute of Technology, 2003

Research Statement

Discretization of exterior calculus (DEC): vector calculus generalizes to nonlinear manifolds as exterior calculus and DEC attempts to build a complete mathematical and computational framework in which theorems from the smooth world have discrete analogs.

Numerical solution of partial differential equations (PDEs): formulation using differential forms and solution using DEC, for PDE vector problems for which traditional vector finite element methods are unstable, e.g. Poisson's equation in first order form or flow in porous media.

Computational algebraic topology: computation of topological invariants of simplicial complexes using DEC and computational Hodge theory, with applications e.g. in mesh topology, machine learning, or sensor networks coverage analysis.

Optimized triangulations: creation of optimized triangulations for numerical PDEs and computational topology, e.g. well-centered triangulations, in which each simplex contains its circumcenter.

Computational astrodynamics: adaptive representation and computation of gravitational field of small irregular bodies like asteroids and comets for fast trajectory propagation; applications of dynamical systems to low thrust space mission trajectory design.