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

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.