A Discrete Regularization Method for Ill-Posed Operaror Equations

TL;DR

This paper introduces a novel discrete regularization technique for ill-posed operator equations that extends existing methods by using finite rank projection-like operators on subspaces of the codomain, including quadrature-based collocation.

Contribution

It proposes a new regularization method employing finite rank projection-like operators on subspaces of the codomain, broadening the scope of existing projection-based approaches.

Findings

Includes existing projection methods as special cases

Incorporates a quadrature-based collocation method

Provides stable approximate solutions for ill-posed equations

Abstract

Discrete regularization methods are often applied for obtaining stable approximate solutions for ill-posed operator equations $Tx=y$, where $T: X\to Y$ is a bounded operator between Hilbert spaces with non-closed range $R(T)$ and $y\in R(T)$. Most of the existing such methods involve finite rank bounded projection operators on either the domain space $X$ or on codomain space $Y$ or on both. In this paper, we propose a discrete regularization based on finite rank projection-like operators on some subspace of the codomain space such that their ranges need not be subspaces of the codomain space. This method not only incudes some of the exisiting projection based methods but also a quadrature based collocation method considered by the author in \cite{mtn-acm} for integral equations of the firt kind.