On 2D Inverse Problems/Riemann mapping theorem

In complex analysis, the Riemann mapping theorem states that for every non-empty simply connected open subset U of the complex plane C, which is not all of the complex plane, there exists a 1-to-1 conformal Riemann map from U onto the open unit disk D. Since the composition of a harmonic and analytic function is harmonic, the Riemann map provides a 1-to-1 correspondence b/w harmonic functions defined on the set U and on the disc D. Therefore, one can transfer a solution of a Dirichlet boundary problem on the set D to the set U.

Let $f: U \rightarrow D$ be a Riemann map for the region U, then the kernel of the Dirichlet-to-Neumann map for the region U can be expressed in terms of the Dirichlet-to-Neumann map for the disc.

Exercise (*). Proof that,

$K_U(\phi,\theta) = |f'(\phi)|K_D(f(\phi),f(\theta))|f'(\theta)|$ off the diagonal.

It is a remarkable fact that a discrete/network version of the statement above is true, see also [Ca].

Exercise (**). Let G be a network w/the Kirchhoff matrix

$K(G) = \begin{pmatrix} A & B \\ B^{T} & C \end{pmatrix} .$ Find a new conductivity $\tilde{\gamma}$ on the network G, such that $\Lambda(G, \tilde{\gamma}) = D_x \Lambda(G) D_x + D_z.$

(Hint). $\tilde{\gamma}_{kl} = y(v_k)\gamma_{kl}y(v_l)$, where $y = -C^{-1}B^T x > 0$

is the solution of the Dirichlet boundary problem and

$z = -D_x\Lambda x.$

Compare to the continuous case.

Exercise (*)

Prove that the Cayley transform $\tau(z) = \frac{1-z}{1+z}$

is a Riemann mapping of the complex right half-plane C+ onto the unit disc D

Exercise (**) Use statements above to derive the formula for the kernel of the Dirichlet-to-Neumann map for the unit disc D.

$K_D(\phi,\theta) = \frac{-1}{\pi(1-cos(\phi-\theta))}.$

Note, that the formula can also be derived by taking the radial derivative of the Poisson kernel for solving a Dirichlet problem on the disc. $K_P(z,\theta)=\frac{1-|z|^2}{2\pi|1-ze^{-i\theta}|}.$

In order to solve a continuous inverse problem by data discretization, one can define a Dirichlet-to-Neumann (DN) matrix by uniform sampling of the kernel off the diagonal, and defining the diagonal entries by the fact that rows and columns of a DN matrix sum up to zero. This leads to the following definition of the matrix in the case of the unit disc: $\Lambda_{kl} = \begin{cases}\frac{2n(n+1)}{3}, \mbox{ }k = l, \\ \frac{-1}{1-\cos\frac{2\pi(k-l)}{2n+1}}, \mbox{ }k \ne l,\end{cases}$

where n is a natural number and k,l = 1,2, ... 2n+1.

Exercise (***). Prove that the eigenvalues of the matrix above are natural numbers(!)

$2n, 4n-2, 6n-6, \ldots, n(n+1)$ w/multiplicity 2 and 0 w/multiplicity 1.