Estimation of the Regularization Parameter in Linear Discrete Ill-Posed Problems Using the Picard parameter

TL;DR

This paper introduces a new method for selecting the regularization parameter in linear ill-posed problems by approximating the distance between noiseless and reconstructed data using the Picard parameter, improving accuracy.

Contribution

The paper proposes a novel approach that employs the Picard parameter to estimate the regularization parameter in Tikhonov regularization, with a reliable algorithm for its computation.

Findings

Effective in numerical examples

Accurate separation of noise from data

MATLAB implementation available

Abstract

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/