QUANTITATIVE STABILITY OF GEL’FAND’S INVERSE BOUNDARY PROBLEM

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 ∂M, the Neumann eigenvalues λ_j and the boundary values of the eigenfunctions ϕ_j |∂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.