  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.
