On 2D Inverse Problems/Electrical networks

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An electrical network can be defined as a weighted graph with boundary

G(V, \partial G, E, \gamma),
\partial G\subset V.

The weight function defined on the edges of the graph is called conductivity. The Laplacian matrix of an electrical network is symmetric. It is often called Kirchhoff matrix. The boundary measurements for inverse problems can be expressed conveniently in terms of the matrix blocks.

The Dirichlet-to-Neumann operator is a special type of Poincaré–Steklov operator. On a surface w/boundary it is the pseudo-differential operator from the Dirichlet boundary values (potential) to the Neumann boundary values (current) of the harmonic functions. It is well-defined because of uniqueness and existence of the solution of the Dirichlet problem.

Definition of the Dirichlet-to-Neumann operator for a domain
Exercise (*). Prove that the Dirichlet-to-Neumann operator of an electrical network is self-adjoint and equal to the Schur complement of its Kirchhoff matrix.

\Lambda_G = 
 A & B \\
 B^T  & C 
/ (C) =  A-BC^{-1}B^T.

The Laplace equation gives the direct connection between the hitting probability of the random walk started at the boundary and the value of a harmonic function at a vertex/point, see [10]. The connection follows from the sum of the geometric series identity applied to the blocks of the Kirchhoff matrix of the network/graph.

 \Lambda_G= D_A(I-W_G) = A-B^T D_C \sum_k(I-D_C^{-1}C)^k B.

This is a special case of the convergent geometric Neumann series applied to the diagonally dominated matrix.

The effective conductivity between two distinct nodes of an electrical network G without the boundary is equal to the ratio between the total current flowing between the nodes and the difference in potentials between the two nodes. The effective conductivities can be calculated in terms of the Schur complement of the Kirchhoff matrix of the network.

Exercise (*). Prove that the effective conductivities between all pairs of nodes of a subset ∂G of a network G. determine the Dirichlet-to-Neumann operator of {G,∂G}, and vice versa.
Exercise (***). Prove that the effective conductivities between all pairs of nodes of a network G determine the individual conductivities of the edges of the network G, and vice versa.

The following problem is classical:

Exercise (**). Find the effective conductivity between two neighboring nodes of the electrical network that is an infinite rectangular grid Zn w/all conductivities equal to 1.