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.
Библиографическая ссылка
Muftahov I. R. , Sidorov D.N., Sidorov N.A. On perturbation method for the first kind equations: Regularization and application // Bulletin of the South Ural State University, Series: Mathematical Modelling, Programming and Computer Software. Vol.8. No.2. 2015. P.69-80. DOI: 10.14529/mmp150206
