Revisiting Calderon's Problem

TL;DR

This paper explores the application of finite element methods and optimization tools to Calderon's problem, investigating the uniqueness and attainability of conductivity solutions, with findings indicating partial success in flux recovery but challenges in matching target conductivities.

Contribution

The study applies a finite element and adjoint solver framework to Calderon's problem, providing insights into the solution's uniqueness and the effectiveness of optimization in reconstructing conductivities.

Findings

Optimization reduces the cost function significantly.

Reconstructed conductivities differ from target distributions.

Normal fluxes closely match prescribed values, tangential fluxes vary widely.

Abstract

A finite element code for heat conduction, together with an adjoint solver and a suite of optimization tools was applied for the solution of Calderon's problem. One of the questions whose answer was sought was whether the solution to these problems is unique and obtainable. The results to date show that while the optimization procedure is able to obtain spatial distributions of the conductivity $k$ that reduce the cost function significantly, the resulting conductivity $k$ is still significantly different from the target distribution sought. While the normal fluxes recovered are very close to the prescribed ones, the tangential fluxes can differ considerably.