Alexander Y. Meltzer

Eitan Levin, Alexander Y. Meltzer

Accurate determination of the regularization parameter in inverse problems still represents an analytical challenge, owing mainly to the considerable difficulty to separate the unknown noise from the signal. We present a new approach for determining the parameter for the general-form Tikhonov regularization of linear ill-posed problems. In our approach the parameter is found by approximate minimization of the distance between the unknown noiseless data and the data reconstructed from the regularized solution. We approximate this distance by employing the Picard parameter to separate the noise from the data in the coordinate system of the generalized SVD. A simple and reliable algorithm for the estimation of the Picard parameter enables accurate implementation of the above procedure. We demonstrate the effectiveness of our method on several numerical examples. A MATLAB-based implementation of the proposed algorithms can be found at https://www.weizmann.ac.il/condmat/superc/software/

Eitan Levin, Alexander Y. Meltzer

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.