User:DVD206/Dirichlet-to-Neumann operator

The Dirichlet-to-Neumann operator is a special type of Poincaré–Steklov operator. It is the pseudo-differential operator from the Dirichlet boundary data to the Neumann boundary data of harmonic functions. It is well-defined because of uniqueness and existence of the solution of the Dirichlet problem.

Let


 * $$M=\left[\begin{matrix} A & B \\ C & D \end{matrix}\right]$$

so that M is a (p+q)&times;(p+q) matrix.

Then the Schur complement of the block D of the matrix M is the p&times;p matrix


 * $$ \Lambda(G)= A-BD^{-1}C.\,$$

The Laplace equation gives the connection between the hitting probability of the random walk started at the boundary and the value of a harmonic function at a point. The connection can be expressed using the sum of the geometric series identity applied to the blocks of the Kirchhoff matrix of the network/graph.


 * $$ \Lambda(G)= D(I-H(G)) = A-B^T\sum(I-C)^n B.$$

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