Infinite-dimensional Lipschitz stability in the Calderón problem and general Zernike bases
Henrik Garde, Markus Hirvensalo, Nuutti Hyvönen
TL;DR
The paper addresses the Calderón inverse conductivity problem on the unit ball and demonstrates Lipschitz stability on infinite-dimensional conductivity classes. It introduces a d-dimensional Zernike basis built from radial Zernike polynomials and spherical harmonics, and proves that, under a decay condition on basis coefficients, boundary data determines interior conductivities with a Lipschitz bound. The authors verify the necessary Assumption for the Zernike basis, provide explicit 2D and 3D basis formulas, and derive sharp bounds for the radial polynomials. This work extends Lipschitz stability results to nonlinear Calderón problems in higher dimensions and offers a framework for stable reconstruction in infinite-dimensional settings.
Abstract
Calderón's inverse conductivity problem has, so far, only been subject to conditional logarithmic stability for infinite-dimensional classes of conductivities and to Lipschitz stability when restricted to finite-dimensional classes. Focusing our attention on the unit ball domain in any spatial dimension $d\geq 2$, we give an elementary proof that there are (infinitely many) infinite-dimensional classes of conductivities for which there is Lipschitz stability. In particular, Lipschitz stability holds for general expansions of conductivities, allowing all angular frequencies but with limited freedom in the radial direction, if the basis coefficients decay fast enough to overcome the growth of the basis functions near the domain boundary. We construct general $d$-dimensional Zernike bases and prove that they provide examples of infinite-dimensional Lipschitz stability.