Covariant tomography of fields

TL;DR

This paper introduces covariant tomography, a geometric method for solving inverse boundary problems that reconstructs fields from boundary data using a tower algorithm, applicable to Maxwell equations and validated in low dimensions.

Contribution

It develops a covariant framework with a tower algorithm to reduce complex IBVPs to first-order systems, establishing solvability criteria for higher-order equations.

Findings

Reconstruction of currents and gauge potentials from boundary data.

The tower algorithm reduces higher-order systems to coupled first-order equations.

Validation through electromagnetic potential reconstruction in three-dimensional space.

Abstract

This paper develops 'covariant tomography', a local framework for solving Inverse Boundary Value Problems (IBVP) for parallel transport equation on star-shaped domains. By integrating geometric decomposition with specific interior extensions - radial, heat equation, or harmonic - the method reconstructs currents and gauge potentials from boundary data. The choice of extension directly dictates the regularity of the recovered interior fields. A primary contribution is the 'tower' algorithm, which reduces higher-order systems, such as Maxwell equations, to a sequence of coupled first-order equations. We establish a formal solvability criterion (Theorem \ref{Th_TowerTheorem}), proving that higher-order IBVPs are solvable if and only if this tower is sequentially solvable. The framework is validated through low-dimensional examples and electromagnetic potential reconstruction in $\mathbb{R}^{3}$.