On Perturbation Method for the First Kind Equations: Regularization and Application

TL;DR

This paper presents a perturbation-based regularization method for solving first kind linear equations, effectively computing derivatives from noisy data with improved accuracy and stability.

Contribution

It introduces a novel regularization algorithm for ill-posed first kind equations, handling noisy data and providing stable solutions with theoretical guarantees.

Findings

The method achieves accurate derivative computation despite noise.

The regularization algorithm ensures unique solutions under specified conditions.

The approach is applicable to scientific problems involving noisy experimental data.

Abstract

One of the most common problems of scientific applications is computation of the derivative of a function specified by possibly noisy or imprecise experimental data. Application of conventional techniques for numerically calculating derivatives will amplify the noise making the result useless. We address this typical ill-posed problem by application of perturbation method to linear first kind equations $Ax=f$ with bounded operator $A.$ We assume that we know the operator $\tilde{A}$ and source function $\tilde{f}$ only such as $||\tilde{A} - A||\leq \delta_1,$ $||\tilde{f}-f||< \delta_2.$ The regularizing equation $\tilde{A}x + B(\alpha)x = \tilde{f}$ possesses the unique solution. Here $\alpha \in S,$ $S$ is assumed to be an open space in $\mathbb{R}^n,$ $0 \in \overline{S},$ $\alpha= \alpha(\delta).$ As result of proposed theory, we suggest a novel algorithm providing accurate results even in the presence of a large amount of noise.