Quantitative stability of Gel'fand's inverse boundary problem

TL;DR

This paper demonstrates that the inverse boundary problem for Riemannian manifolds can be stably solved using boundary spectral data, providing explicit stability estimates and an algorithm for reconstructing the manifold.

Contribution

It establishes quantitative stability estimates for Gel'fand's inverse boundary problem and introduces an explicit reconstruction algorithm based on boundary spectral data.

Findings

Finitely many eigenvalues and boundary eigenfunctions determine a manifold close to the original in Gromov-Hausdorff distance.

Provides explicit stability estimates for the inverse problem.

Develops an algorithm for reconstructing the manifold from spectral data.

Abstract

In Gel'fand's inverse problem, one aims to determine the topology, differential structure and Riemannian metric of a compact manifold $M$ with boundary from the knowledge of the boundary $\partial M,$ the Neumann eigenvalues $\lambda_j$ and the boundary values of the eigenfunctions $\varphi_j|_{\partial M}$. We show that this problem has a stable solution with quantitative stability estimates in a class of manifolds with bounded geometry. More precisely, we show that finitely many eigenvalues and the boundary values of corresponding eigenfunctions, known up to small errors, determine a metric space that is close to the manifold in the Gromov-Hausdorff sense. We provide an algorithm to construct this metric space. This result is based on an explicit estimate on the stability of the unique continuation for the wave operator.