Adaptive optimal regularization of the linear ill posed integral equations

TL;DR

This paper develops an adaptive method for optimally estimating solutions to linear ill-posed integral equations of the first kind, providing asymptotic optimality and confidence regions in the L2 sense.

Contribution

It introduces an adaptive estimation procedure that achieves asymptotic optimality for solving linear ill-posed integral equations, including confidence region construction.

Findings

Establishes an asymptotically optimal estimator in L2 sense.

Provides a method for adaptive regularization with confidence regions.

Demonstrates theoretical optimality of the proposed approach.

Abstract

We construct an adaptive asymptotically optimal in order in the $ L(2) $ sense a solution (estimation) of an integral linear equation of a first kind and energy of this solution with the confidence region building, also adaptive.