A convex lifting approach for the Calder\'on problem

Abstract

The Calder\'on problem consists in recovering an unknown coefficient of a partial differential equation from boundary measurements of its solution. These measurements give rise to a highly nonlinear forward operator. As a consequence, the development of reconstruction methods for this inverse problem is challenging, as they usually suffer from the problem of local convergence. To circumvent this issue, we propose an alternative approach based on lifting and convex relaxation techniques, that have been successfully developed for solving finite-dimensional quadratic inverse problems. This leads to a convex optimization problem whose solution coincides with the sought-after coefficient, provided that a non-degenerate source condition holds. We demonstrate the validity of our approach on a toy model where the solution of the partial differential equation is known everywhere in the domain. In this simplified setting, we verify that the non-degenerate source condition holds under certain assumptions on the unknown coefficient. We leave the investigation of its validity in the Calder\'on setting for future works.