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

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.