Reconstruction and stability in Gel'fand's inverse interior spectral problem

2017/02/25 by Roberta Bosi, Bosi, Roberta, Yaroslav Kurylev +3 · 5 citations
Computer Science · Mathematics · #35R30 #58J50 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1702.07937

openalex publication_date 2017/02/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Assume that M 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 Δ_g on M as well as the corresponding eigenfunctions restricted on an open set in M. We then construct a stable approximation to the manifold (M,g). Namely, we construct a metric space and a Riemannian manifold which differ, in a proper sense, just a little from M when the above data are given with a small error. We give an explicit loglog-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 Gel'fand's 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.

Cited by

• Anisotropic Calderón problem for a logarithmic Schrödinger operator of order 2+ on closed Riemannian manifolds
• A Hyperbolic Inverse Problem for lower order terms on a closed manifold with disjoint data
• Stability of Gel'fand's inverse interior spectral problem for Schrödinger operators
• Hölder stability of an inverse spectral problem for the magnetic Schrödinger operator on a simple manifold
• On Exponential Instability of an Inverse Problem for the Wave Equation

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