A Mixed Elliptic--Hyperbolic Monge--Ampère Problem: Modeling and a Least-Squares Solver

Abstract: We consider a transport problem motivated by freeform optical design of a parallel-to-far-field reflector. Classical formulations rely on either elliptic or hyperbolic Monge--Ampère equations, leading to globally convex/concave or saddle-shape reflector geometries, respectively. However, the design of smooth reflector arrays may require optical surfaces combining both types of curvature. To address this, we introduce a mixed elliptic--hyperbolic Monge--Ampère system obtained by partitioning the source domain into one elliptic and one hyperbolic subregion, and we develop a least-squares method for its numerical approximation. Numerical examples demonstrate the ability of the proposed framework to compute both isolated and periodic mixed-type transport solutions, yielding smooth reflector configurations.