Linearized Calderón Problem: Reconstruction and Lipschitz Stability for Infinite-Dimensional Spaces of Unbounded Perturbations

SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3588-3604, June 2024.
Abstract. We investigate a linearized Calderón problem in a two-dimensional bounded simply connected [math] domain [math]. After extending the linearized problem for [math] perturbations, we orthogonally decompose [math] and prove Lipschitz stability on each of the infinite-dimensional [math] subspaces. In particular, [math] is the space of square-integrable harmonic perturbations. This appears to be the first Lipschitz stability result for infinite-dimensional spaces of perturbations in the context of the (linearized) Calderón problem. Previous optimal estimates with respect to the operator norm of the data map have been of the logarithmic type in infinite-dimensional settings. The remarkable improvement is enabled by using the Hilbert–Schmidt norm for the Neumann-to-Dirichlet boundary map and its Fréchet derivative with respect to the conductivity coefficient. We also derive a direct reconstruction method that inductively yields the orthogonal projections of a general [math] perturbation onto the [math] spaces, hence reconstructing any [math] perturbation.