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

TL;DR

This paper introduces a new stopping criterion for iterative regularization of large-scale ill-posed inverse problems, based on a generalized Picard parameter, improving the filtering of noise and signal separation during the iterative process.

Contribution

It extends the Picard parameter concept to large-scale problems using a novel Fourier coefficient vectorization, enabling effective stopping criteria for Krylov subspace methods.

Findings

Accurately filters noise using the generalized Picard parameter.

Effectively reduces large-scale 2D problems to 1D filtering.

Demonstrates robustness and accuracy through numerical examples.

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.