At the intersection of differential equations and optimization: inverse problems, path planning and Krylov subspaces

Abstract

Four problems at the intersection of optimization and partial differential equations are presented. First, a problem in remote sensing of the marine atmospheric boundary layer is discussed. A method that exploits the low-rank structure of the electromagnetic field is used to infer the refractive index profile of the lower atmosphere. The second problem is concerned with 3D X-ray imaging of large objects at nanometer scale resolution. A massively parallel optimization method is used to perform the reconstruction from measurements of an object outside of the depth of focus. The third problem presents a path planning problem where an evader is choosing his trajectory to hinder the surveillance of an observer. An algorithm to compute optimal strategies using ideas from convex optimization, game theory and optimal control is described. The final chapter presents a practical framework to apply Krylov subspace methods to differential operators.