One of the approaches to inverse problems based upon their relations to boundary control theory (the so-called BC method) is presented. The method gives an efficient way to reconstruct a Riemannian manifold via its response operator (dynamical Dirichlet-to-Neumann map) or spectral data (a spectrum of the Beltrami-Laplace operator and traces of normal derivatives of the eigenfunctions). The approach is applied to the problem of recovering a density, including the case of inverse data given on part of a boundary. The results of the numerical testing are demonstrated.

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The review covers the period 1997-2007 of development of the boundary control method, which is an approach to inverse problems based on their relations to control theory (Belishev 1986). The method solves the problems on unknown manifolds: given inverse data of a dynamical system associated with a manifold it recovers the manifold, the operator governing the system and the states of the system defined on the manifold. The main subject of the review is the extension of the boundary control method to the inverse problems of electrodynamics, elasticity theory, impedance tomography, problems on graphs as well as some new relations of the method to functional analysis and topology.

A planar graph consisting of strings of variable densities is considered. The spectrum of the Dirichlet problem on the graph and the values of derivatives of the (normalized) eigenfunctions at the boundary vertices form the spectral data. We show that the graph without cycles (tree) and the densities of its edges are determined by the spectral data uniquely up to a natural isometry in the plane. In the framework of our approach (boundary control method; Belishev 1986) we study the boundary controllability of the dynamical system associated with the graph and governed by the wave equation, and exploit this property for recovering the tree from its spectral data. The approach can be extended to a wide class of inverse problems on trees.

As was shown by M.Lassas and G.Uhlmann (2001), the smooth twodimensional compact orientable Riemann manifold with the boundary is uniquely determined by its Dirichlet-to-Neumann map up to conformal equivalence. We give a new proof of this fact based on relations between the Calderon problem and Function Algebras: the manifold is identi ed with the spectrum of the algebra of holomorphic functions determined by the DN-map up to isometry; as such, the manifold is recovered from the DN-map by the use of the Gelfand transform. A simple formula linking the DN-map to the Euler characteristic of the manifold is derived.