On 2D Inverse Problems/Algorithm for solution

The Dirichlet-to-Neumann operator of a rotation invariant network is diagonal in Fourier coordinates. The direct calculation shows that, its eigenvalues are given by the values of the Stieltjes continued fraction at the eigenvalues of the discrete boundary Laplacian -L.

$\Lambda (G)={\sqrt {-L}}\beta ({\sqrt {-L}}),$

where

$\beta _{\gamma }(z)=\gamma _{n}z+{\cfrac {1}{\gamma _{n-1}z+{\cfrac {\ldots }{\ldots +{\cfrac {1}{\gamma _{1}z}}}}}}.$

Therefore, one can reduce the layered inverse problem to a Pick-Nevanlinna interpolation problem, since given the Dirichlet-to-Neumann operator of a layered network, one can find its eigenvalues and the interpolating rational function. The conductivities of the network are given then by the coefficients of continued fraction and their reciprocals and can be found by the Eucledian algorithm.

The continuous analog of the inverse problem can be transformed to the inverse problem of Krein for a string.