Linearised Calderón problem: Reconstruction and Lipschitz stability for infinite-dimensional spaces of unbounded perturbations
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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) = ⊕_k=0^∞ℋ_k and prove Lipschitz stability on each of the infinite-dimensional ℋ_k subspaces. In particular, 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 ℋ_k spaces. If the perturbation is in the subspace ⊕_k=0^K ℋ_k, then the reconstruction method only requires data corresponding to 2K+2 specific Neumann boundary values.
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