On 2D Inverse Problems/Riemann mapping theorem

In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex plane C which is not all of the complex plane, then there exists an analytic mapping f from U onto the open unit disk D. This mapping is known as a Riemann mapping.

Since the composition of a harmonic and analytic function is harmonic, the Riemann mapping provides a bijection between harmonic functions defined on the set U and on the disc D. Therefore, one can transfer a solution of a Dirichlet problem on the set D to the set U.

Let

$f: U \rightarrow D$

be a Riemann mapping 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,

$K_U(f(\phi),f(\theta)) = |f'(\phi)|K_D(f(\phi),f(\theta))|f'(\theta)|$

off the diagonal.

It is a remarkable fact that a discrete/network version of the statement of the exercise above is true, see also [5].

Exercise (**). Let G be a network, with the Kirchhoff matrix

$K(G) = \begin{pmatrix} A & B \\ B^{T} & C \end{pmatrix} .$

For a positive vector x, let D_x denote the diagonal matrix with the vector x on its diagonal. That is D_x1=x. 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 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 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 matrix by uniform sampling of the kernel off the diagonal, and defining the diagonal entries by the property that the constant vector is in the kernel of Dirichlet-to-Neumann operator. 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(!)

$2n, 4n-2, 6n-6, \ldots, n(n+1)$

with multiplicity 2 and 0 w/multiplicity 1. (Hint) The matrix is Toeplitz, circulant and symmetric.