  We present a general framework to study uniqueness, stability and
reconstruction for infinite-dimensional inverse problems when only a
finite-dimensional approximation of the measurements is available. For a large
class of inverse problems satisfying Lipschitz stability we show that the same
estimate holds even with a finite number of measurements. We also derive a
globally convergent reconstruction algorithm based on the Landweber iteration.
This theory applies to nonlinear ill-posed problems such as electrical
impedance tomography, inverse scattering and quantitative photoacoustic
tomography, under the assumption that the unknown belongs to a
finite-dimensional subspace.
