Linearised Calderón problem: Reconstruction and Lipschitz stability for infinite-dimensional spaces of unbounded perturbations
Henrik Garde, Nuutti Hyvönen
TL;DR
The paper addresses the linearised Calderón problem in a two-dimensional setting, showing that perturbations in $L^2(\Omega)$ can be stably reconstructed by exploiting an orthogonal decomposition into infinite-dimensional harmonic subspaces $\mathcal{H}_k$ and the Hilbert–Schmidt structure of the Neumann-to-Dirichlet derivative $F=D\Lambda(1)$. It provides an exact reconstruction procedure on the unit disk using Zernike polynomials to recover the coefficients of any perturbation from $F\eta$, with a data-efficient scheme when restricting to $\mathcal{W}_K$. The authors prove Lipschitz stability on the infinite-dimensional subspaces $\mathcal{H}_k$ and on finite sums $\mathcal{W}_K$ through explicit bounds involving combinatorial constants (e.g., binomial coefficients), and they demonstrate how these results extend to general simply connected domains via conformal mapping. The results hinge on the two-dimensional Hilbert–Schmidt setting, enabling stability results far stronger than the classical logarithmic-type bounds in infinite dimensions, and they include an explicit reconstruction algorithm and an injectivity proof. Overall, the work suggests that reframing inverse problems in terms of ND maps and HS topologies can yield robust, scalable stability and practical reconstruction methods in 2D.
Abstract
We investigate a linearised Calderón problem in a two-dimensional bounded simply connected $C^{1,α}$ domain $Ω$. After extending the linearised problem for $L^2(Ω)$ perturbations, we orthogonally decompose $L^2(Ω) = \oplus_{k=0}^\infty \mathcal{H}_k$ and prove Lipschitz stability on each of the infinite-dimensional $\mathcal{H}_k$ subspaces. In particular, $\mathcal{H}_0$ 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 (linearised) 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 $L^2(Ω)$ perturbation onto the $\mathcal{H}_k$ spaces, hence reconstructing any $L^2(Ω)$ perturbation.