Randomized matrix approximation to enhance regularized projection schemes in inverse problems

Abstract

The authors consider a randomized solution to ill-posed operator equations in Hilbert spaces. In contrast to statistical inverse problems, where randomness appears in the noise, here randomness arises in the low-rank matrix approximation of the forward operator, which results in using a Monte Carlo method to solve the inverse problems. In particular, this approach follows the paradigm of the study N. Halko et al 2011 SIAM Rev. 53 217–288, and hence regularization is performed based on the low-rank matrix approximation. Error bounds for the mean error are obtained which take into account solution smoothness and the inherent noise level. Based on the structure of the error decomposition the authors propose a novel algorithm which guarantees (on the mean) a prescribed error tolerance. Numerical simulations confirm the theoretical findings.