  We consider abstract inverse problems between infinite-dimensional Banach
spaces. These inverse problems are typically nonlinear and ill-posed, making
the inversion with limited and noisy measurements a delicate process. In this
work, we assume that the unknown belongs to a finite-dimensional manifold: this
assumption arises in many real-world scenarios where natural objects have a low
intrinsic dimension and belong to a certain submanifold of a much larger
ambient space. We prove uniqueness and H\"older and Lipschitz stability results
in this general setting, also in the case when only a finite discretization of
the measurements is available. Then, a Landweber-type reconstruction algorithm
from a finite number of measurements is proposed, for which we prove global
convergence, thanks to a new criterion for finding a suitable initial guess.
  These general results are then applied to several examples, including two
classical nonlinear ill-posed inverse boundary value problems. The first is
Calder\'on's inverse conductivity problem, for which we prove a Lipschitz
stability estimate from a finite number of measurements for piece-wise constant
conductivities with discontinuities on an unknown triangle. A similar stability
result is then obtained for Gel'fand-Calder\'on's problem for the Schr\"odinger
equation, in the case of piece-wise constant potentials with discontinuities on
a finite number of non-intersecting balls.
