The Calderón problem revisited: Reconstruction with resonant perturbations

Abstract

The original Calderón 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 homogenised medium. In addition, the equivalent coefficient, which consists in the sum of the original potential and the effective coefficient, is negative valued with a controllable amplitude; (2) as the equivalent coefficient is negative valued, then we can linearise the corresponding Neumann to Dirichlet map using the effective coefficient’s amplitude; (3) from the linearised Neumann to Dirichlet map, we reconstruct the original potential using explicit complex geometrical optics solutions (CGOs).