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.
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 .
Exercise (**). Let G be a network, with the Kirchhoff matrix
For a positive vector x, let Dx denote the diagonal matrix with the vector x on its diagonal. That is Dx1=x. Find a new conductivity on the network 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(!)