Numerical differentiation from a viewpoint of regularization theory

Abstract

Abstract. In this paper, we discuss the classical ill-posed problem of numerical differentiation, assuming that the smoothness of the function to be differentiated is unknown. Using recent results on adaptive regularization of general ill-posed problems, we propose new rules for the choice of the stepsize in the finite-difference methods, and for the regularization parameter choice in numerical differentiation regularized by the iterated Tikhonov method. These methods are shown to be effective for the differentiation of noisy functions, and the order-optimal convergence results for them are proved.

Regularization Inverse Method for Variable Binary Diffusivity Measurements

“…There are several common alternatives for the regularization function; each one is designed to produce certain characteristics (ex: smoothness) in the estimated data. [20][21][22][23][24][25] These are summarized below:…”

“…Several implementations have been proposed. [20,24,25] This method has been proposed to produce a reliable estimation during the numerical differentiation of noisy data. For example,…”

“…has been used for the estimation of the first [24] and second [25] spatial derivatives of the concentration by spline interpolation. The regularization formulations for smoothing and differentiation in the literature [20,24,25] can be applied to the diffusion process.…”

“…The regularization formulations for smoothing and differentiation in the literature [20,24,25] can be applied to the diffusion process. We present here an adapted formulation that specifically incorporates the physics of diffusion and what is therefore ''expected'' of the concentration profile, i.e., that it be consistent with a nonlinear diffusion process, governed by Fick's laws.…”

Extrapolation of Tikhonov Regularization Method

“…Note that in a posteriori choice of the regularization parameter in m-iterated Tikhonov method approximations are often computed for some sequence {α i } of parameters, until some condition is fulfilled, and a single approximation with maximal accuracy O(δ 2m 2m+1 ) is used. One example is the balancing principle, advocated recently in many papers [1,2,3,4,8,17,18,19,20,22,23,24,25,28]. The accuracy of the Tikhonov approximation (m = 1) is low but increasing the number m of iterations also increases the amount of computational work, since at transition from u m,αi to u m,αi+1 we have to solve m equations.…”