Numerical computation of complex geometrical optics solutions to the conductivity equation

A numerical method is introduced for the evaluation of complex geometrical optics (cgo) solutions to the conductivity equation ∇⋅σ∇u(⋅,k)=0 in R2 for piecewise smooth conductivities σ. Here k is a complex parameter. The algorithm is based on the solution by Astala and Päivärinta (2006) [1] of Calderón's inverse conductivity problem and involves the solution of a Beltrami equation in the plane with an exponential asymptotic condition. The numerical strategy is to solve a related periodic problem using fft and gmres and show that the solutions agree on the unit disc. The cgo solver is applied to the problem of computing nonlinear Fourier transforms corresponding to nonsmooth conductivities. These computations give new insight into the D-bar method for the medical imaging technique of electric impedance tomography. Furthermore, the asymptotic behavior of the cgo solutions as k→∞ is studied numerically. The evidence so gained raises interesting questions about the best possible decay rates for the subexponential growth argument in the uniqueness proof for Calderón's problem with L∞ conductivities.