CGO solutions for coupled conductivity equations

Code repository for the CGO solutions for coupled conductivity equations paper.

Lecaros R., Montecinos G., Ortega J.H. and Ramírez-Ganga J. (2022) CGO solutions for coupled conductivity equations. Math. Rep. (Bucur.)

Code

The code implemented in Matlab computes the CGO solutions for coupled conductivity equations by solving a Beltrami system equivalent using Fast Fourier Transform (FFT) and GMRES.

Manuscript

The final manuscript is available from Link

Abstract

This paper is devoted to study of complex geometrical optics (CGO) solutions to the coupled conductivity equations written in a matrix form $\mathrm{div}\left(Q \cdot \nabla U\right)=0$ in $\mathbb{R}^2$ for symmetric, positive definite matrix functions $Q$. The CGO solutions were introduced by Faddeev in 1966 to prove the uniqueness in the inverse potential scattering problem for Schödinger equation, later Sylvester and Uhlmann in 1987 use the CGO functions to study the uniqueness of the Calder'on's inverse problem.

Following the ideas of Astala and Päivärinta (2006), we compute CGO solutions considering the vectorial solutions of an associated Beltrami system. In this work, we first prove the existence of CGO solution and then use a numerical strategy based on the method introduced by Huhtanem and Perämäki (2012) for the Beltrami equation. Numerical experiments are considered to show the influence of coupled equations.