I will present two recent projects related to front propagation & optimal control.
The first of these (joint with A. Oberman and R. Takei) deals with 2-scale and 3-scale computations in geometric optics. We propose a new & efficient method to homogenize first-order Hamilton–Jacobi PDEs. Unlike the prior cell-problem methods, our algorithm is based on homogenizing the related geodesic distance function. We illustrate by computing the effective velocity profiles for a number of periodic and "random" composite materials.
The second project (joint with A. Kumar) deals with multiple criteria for optimality (e.g., fastest versus shortest trajectories) and optimality under integral constraints. We show that an augmented PDE on a higher-dimensional domain describes all Pareto-optimal trajectories. Our numerical method uses the causality of this PDE to approximate its discontinuous viscosity solution efficiently. The method is illustrated by problems in robotic navigation (e.g., minimizing the path length and exposure to an enemy observer simultaneously).