  Several applications in medical imaging and non-destructive material testing
lead to inverse elliptic coefficient problems, where an unknown coefficient
function in an elliptic PDE is to be determined from partial knowledge of its
solutions. This is usually a highly non-linear ill-posed inverse problem, for
which unique reconstructability results, stability estimates and global
convergence of numerical methods are very hard to achieve.
  The aim of this note is to point out a new connection between inverse
coefficient problems and semidefinite programming that may help addressing
these challenges. We show that an inverse elliptic Robin transmission problem
with finitely many measurements can be equivalently rewritten as a uniquely
solvable convex non-linear semidefinite optimization problem. This allows to
explicitly estimate the number of measurements that is required to achieve a
desired resolution, to derive an error estimate for noisy data, and to overcome
the problem of local minima that usually appears in optimization-based
approaches for inverse coefficient problems.
