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, ||tilde{f}-f||< delta. 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 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.

Keywords

operator and integral equations of the first kind; stable differentiation; perturbation method, regularization parameter.

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