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 operator à and source function f only such as ||à - A|| ≤ δ, ||f - f|| < δ. The regularizing equation equation Ãx+B(α)x = f possesses the unique solution. Here α ∈ S, S is assumed to be an open space in ℝⁿ, 0 ∈ S¯, α = α(δ). As result of proposed theory, we suggest a novel algorithm providing accurate results even in the presence of a large amount of noise.