Last modified on 25 February 2013, at 01:18

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 operator equation 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, w/

L_G = 4I-BB^T.