Regularized minimal-norm solution of an overdetermined system of first kind integral equations

TL;DR

This paper introduces a regularized numerical method for solving overdetermined first kind integral equations, effectively computing minimal-norm solutions in ill-posed scenarios with boundary constraints, applicable to geophysical data interpretation.

Contribution

It develops a regularization approach based on the Riesz representation theorem and singular function expansion, with automatic parameter selection techniques for improved stability and accuracy.

Findings

Method performs well on artificial test problems.

Effective for smooth solutions, providing significant insights for non-smooth cases.

Demonstrated applicability to geophysical electromagnetic data.

Abstract

Overdetermined systems of first kind integral equations appear in many applications. When the right-hand side is discretized, the resulting finite-data problem is ill-posed and admits infinitely many solutions. We propose a numerical method to compute the minimal-norm solution in the presence of boundary constraints. The algorithm stems from the Riesz representation theorem and operates in a reproducing kernel Hilbert space. Since the resulting linear system is strongly ill-conditioned, we construct a regularization method depending on a discrete parameter. It is based on the expansion of the minimal-norm solution in terms of the singular functions of the integral operator defining the problem. Two estimation techniques are tested for the automatic determination of the regularization parameter, namely, the discrepancy principle and the L-curve method. Numerical results concerning two artificial test problems demonstrate the excellent performance of the proposed method. Finally, a particular model typical of geophysical applications, which reproduces the readings of a frequency domain electromagnetic induction device, is investigated. The results show that the new method is extremely effective when the sought solution is smooth, but gives significant information on the solution even for non-smooth solutions.