Infinite-Dimensional Inverse Problems with Finite Measurements

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 (EIT), inverse scattering and quantitative photoacoustic tomography (QPAT), under the assumption that the unknown belongs to a finite-dimensional subspace. In particular, we derive Lipschitz stability estimates for EIT with a matrix approximation of the Neumann-to-Dirichlet map; for the inverse scattering problem with measurements of the scattering amplitude at a finite number of directions on S^2 × S^2 ; and for QPAT with a low-pass filter of the internal energy.