Stopping criterion for iterative regularization of large-scale ill-posed problems using the Picard parameter

Abstract

We propose a new stopping criterion for Krylov subspace iterative regularization of large-scale ill-posed inverse problems. Our stopping criterion accurately filters the data using a generalization of the Picard parameter that was originally introduced for direct regularization of small-scale problems. In the one dimension we filter the data in the discrete Fourier transform (DFT) basis using the Picard parameter, which separates noise-dominated Fourier coefficients from the signal-dominated ones. For two-dimensional problems we propose a novel vectorization scheme of the Fourier coefficients of the data based on the Kronecker product structure of the two-dimensional DFT matrix, which effectively reduces the problem to one dimension. At each iteration we compute the distance between the data reconstructed from the iterated solution and the filtered data, terminating the iterations once this distance begins to increase or to level off. The accuracy and robustness of the proposed method is demonstrated by several numerical examples and a MATLAB-based implementation is provided.