  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 $\Delta_g$
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 $\log\log$-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.
