On stable numerical differentiation

Abstract

ABSTRACT. Based on a regularized Volterra equation, two different approaches for numerical differentiation are considered. The first approach consists of solving a regularized Volterra equation while the second approach is based on solving a disretized version of the regularized Volterra equation. Numerical experiments show that these methods are efficient and compete favorably with the variational regularization method for stable calculating the derivatives of noisy functions.

Numerical differentiation from a viewpoint of regularization theory

“…A number of regularization parameter choice techniques have been developed for numerical differentiation. They yield satisfactory results when the smoothness of the function to be differentiated is given very precisely [1], [2], [22], [26]. However, in applications this smoothness is usually unknown, as one can see it from the following example.…”

“…Using (1.1) for reconstruction y (t) one presupposes that the value y δ (t + h), for example, is available for any sufficiently small h. Because as it has been observed in [6], [22], difference schemes may construct stable regularizing algorithms only if a stepsize h is chosen properly. An example of the method from the second category can be found in [7], where a derivative S n (t) of a natural cubic spine S n (t) solving the minimization problem…”

“…At this point we note that in the situation considered the use of finite-difference methods discussed in Section 2 may not be very satisfactory since the values y δ (t + jh) may not be available for all desired stepsizes h. Therefore, one usually takes an approach discussed, for example, in [4,19,22], and rewrites the numerical differentiation of a smooth function y as a Volterra problem…”

“…For example, it has been observed in [22] that the operator A from (3.4) acts in the Hilbert space L 2 (0, 1) as a monotone operator, i.e., ∀x ∈ L 2 (0, 1), Ax, x ≥ 0, where ·, · is the standard inner product in L 2 (0, 1).…”

Kriging-based estimation of time derivatives via FIR filtering

“…The finite-difference method is usually inapplicable, since small perturbations of the signal lead to large errors in the computed derivatives [2]. Another widespread method, Savitzky-Golay filtering [3], fits polynomials on a sliding window and uses their derivatives as estimates.…”