C. Maeve McCarthy, Reconstruction of an impedance in
two-dimensions from spectral data, Applicable Analysis, 81 (2002), pp. 1161-1177.
The two-dimensional spectral inverse problem
involves the reconstruction of an unknown coefficient in an elliptic partial
differential equation from spectral data, such as eigenvalues.
Projection of the boundary value problem and the unknown coefficient onto
appropriate vector spaces leads to a matrix inverse problem. Unique solutions
of this matrix inverse problem exist provided that the eigenvalue
data is close to the eigenvalues associated with the
analogous constant coefficient boundary value problem. We discuss here the
application of such a technique to the reconstruction of an impedance $p$ in
the boundary value problem
$$-\nabla .(p \nabla u) = \lambda p u \mbox{in} R, u=0 \mbox{on} \partial
R$$
where $R$ is a rectangular
domain. The matrix inverse problem, although nonstandard, is solved by a
fixed-point iterative method and an impedance function $p^*$ is constructed which
has the same $m$ lowest eigenvalues as the unknown $p$.
Numerical evidence of the success of the method will be presented.