Last modified on 29 October 2014, at 02:08

On 2D Inverse Problems/An infinite example

The following construction provides an example of an infinite network (featured on the cover of the book), which Dirichlet-to-Neumann operator satisfies the equation w/the operator in the title of this chapter. 
\Lambda_G = \sqrt{L}.
The matrix equation reflects the self-duality and self-symmetry of the network.

Exercise (**). Prove that the Dirichlet-to-Neumann operator of the network on the picture w/the natural boundary satisfies the equation.
The self-dual self-symmetric infinite graph w/its dual

(Hint:) Use the fact that the operator/matrix is the fixed point of the Schur complement]]:  
\Lambda_G = 
\begin{pmatrix}
2I & B \\
B^T & \Lambda + 2I
\end{pmatrix}/ (\Lambda + 2I),
where 
B =-\begin{pmatrix}
 1     &  0 & 0 & \ldots & 1 \\
 1     & 1 & 0 & \ldots & 0 \\
 0     & \vdots & \ddots & \ddots & \vdots \\
 \vdots    & \vdots & \ddots & 1 & 0 \\
 0     & 0  & \ldots & 1 & 1 \\
\end{pmatrix}
is the circulant matrix, such that L_G = 4I-BB^T.