The Calderon Problem Revisited: Reconstruction With Resonant Perturbations

TL;DR

This paper introduces a novel method for reconstructing potentials in the Calderón problem by using resonant perturbations with small particles, enabling linearization and explicit reconstruction via complex geometrical optics solutions.

Contribution

It demonstrates how resonant perturbations at specific frequencies can be used to linearize and solve the inverse problem for potential reconstruction.

Findings

Neumann to Dirichlet map converges to that of the homogenized medium.

Equivalent coefficient becomes negative and controllable.

Potential can be reconstructed explicitly using CGOs.

Abstract

The original Calder\'on problem consists in recovering the potential (or the conductivity) from the knowledge of the related Neumann to Dirichlet map (or Dirichlet to Neumann map). Here, we first perturb the medium by injecting small-scaled and highly heterogeneous particles. Such particles can be bubbles or droplets in acoustics or nanoparticles in electromagnetism. They are distributed, periodically for instance, in the whole domain where we want to do reconstruction. Under critical scales between the size and contrast, these particles resonate at specific frequencies that can be well computed. Using incident frequencies that are close to such resonances, we show that 1) the corresponding Neumann to Dirichlet map of the composite converges to the one of the homogenized medium. In addition, the equivalent coefficient, which consist in the sum of the original potential and the effective coefficient, is negative valued with a controlable amplitude. 2) as the equivalent coefficient is negative valued, then we can linearize the corresponding Neumann to Dirichlet map using the effective coefficient's amplitude. 3) from the linearized Neumann to Dirichlet map, we reconstruct the original potential using explicit complex geometrical optics solutions (CGOs).