Infinite-dimensional Lipschitz stability in the Calder\'on problem and general Zernike bases

TL;DR

This paper demonstrates that Lipschitz stability in Calderón's inverse conductivity problem can be achieved for infinite-dimensional classes of conductivities using Zernike bases, expanding the understanding of stability in inverse problems.

Contribution

The paper introduces a new elementary proof showing Lipschitz stability for infinite-dimensional conductivity classes and constructs general Zernike bases to exemplify this stability in any dimension.

Findings

Lipschitz stability holds for certain infinite-dimensional classes of conductivities.

Zernike bases provide explicit examples of infinite-dimensional stability.

Stability depends on the decay rate of basis coefficients.

Abstract

Calder\'on'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.