Assume that is a compact Riemannian manifold of bounded geometry given by restrictions on its diameter, Ricci curvature and injectivity radius. Assume we are given, with some error, the first eigenvalues of the Laplacian on as well as the corresponding eigenfunctions restricted on an open set in . We then construct a stable approximation to the manifold . Namely, we construct a metric space and a Riemannian manifold which differ, in a proper sense, just a little from when the above data are given with a small error. We give an explicit --type stability estimate on how the constructed manifold and the metric on it depend on the errors in the given data. Moreover a similar stability estimate is derived for the Gelfand inverse problem. The proof is based on methods from geometric convergence, a quantitative stability estimate for the unique continuation and a new version of the geometric boundary control method.
"Reconstruction and stability in Gelfand’s inverse interior spectral problem." Anal. PDE 15 (2) 273 - 326, 2022. https://doi.org/10.2140/apde.2022.15.273