# On 2D Inverse Problems/Riemann mapping theorem

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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

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,

off the diagonal.

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

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

For a positive vector *x*, let *D _{x}* denote the diagonal matrix with the vector

*x*on its diagonal. That is

*D*. Find a new conductivity on the network

_{x}1=x*G*, such that

(Hint). , where

is the solution of the Dirichlet problem and

Compare to the continuous case.

**Exercise (*)** Prove that the Cayley transform

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*.

Note, that the formula can also be derived by taking the radial derivative of the Poisson kernel for solving Dirichlet problem on the disc.

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:

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(!)

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