Polyhedral reconstruction via Boundary Control method
Dimitra Kyriakopoulou
TL;DR
Extends the Boundary Control method to elliptic Riemannian polyhedra by leveraging Schulze's edge calculus on manifolds with singularities $\mathfrak{M}_k$, establishing a rigorous uniqueness framework for elliptic problems. For hyperbolic polyhedra, it adopts a per-compartment reconstruction strategy and integrates interface/vertex detection criteria across electromagnetic, anisotropic nonlinear, and acoustic media to connect wave phenomena with geometric recovery. The work develops weighted and corner-aware operator theory, including higher-singularity extensions and BC-based coordinatization, model construction, and replication, enabling reconstruction from boundary data in complex domains. By linking physical interface behaviors (transmission/reflection, waveguide coupling, four-wave mixing, and directional couplers) to geometric detection, the paper provides a comprehensive, physics-informed pathway for recovering polyhedral domains with singularities from boundary measurements.
Abstract
We study uniqueness of an elliptic Riemannian polyhedron using the elliptic version for Boundary Control method, which we presented in [1]. We also present interface detection criteria for hyperbolic Riemannian manifolds through introduction of the waveguide notion, the four-wave mixing notion, etc.