Abstract. We prove a global logarithmic stability estimate for the Gel'fand-Calderón inverse problem on a two-dimensional domain.

We prove that an L ∞ potential in the Schrödinger equation in three and higher dimensions can be uniquely determined from a finite number of boundary measurements, provided it belongs to a known finite dimensional subspace W. As a corollary, we obtain a similar result for Calderón's inverse conductivity problem. Lipschitz stability estimates and a globally convergent nonlinear reconstruction algorithm for both inverse problems are also presented. These are the first results on global uniqueness, stability and reconstruction for nonlinear inverse boundary value problems with finitely many measurements. We also discuss a few relevant examples of finite dimensional subspaces W, including bandlimited and piecewise constant potentials, and explicitly compute the number of required measurements as a function of dim W.

Abstract. We consider two inverse problems for the multi-channel two-dimensional Schrödinger equation at fixed positive energy, i.e. the equation −∆ψ + V (x)ψ = Eψ at fixed positive E, where V is a matrixvalued potential. The first is the Gel'fand inverse problem on a bounded domain D at fixed energy and the second is the inverse fixed-energy scattering problem on the whole plane R 2 . We present in this paper two algorithms which give efficient approximate solutions to these problems: in particular, in both cases we show that the potential V is reconstructed with Lipschitz stability by these algorithms up to O(E −(m−2)/2 ) in the uniform norm as E → +∞, under the assumptions that V is m-times differentiable in L 1 , for m ≥ 3, and has sufficient boundary decay.