A general method of computing the derivative of experimental data

Abstract

Many scientific and engineering investigations require the extraction of the first derivative from experimental data. Direct numerical differentiation is usually impractical because this amplifies the noise in the data, leading to unreliable results. This investigation shows that the problem of differentiating experimental data can be converted into one of solving an integral equation of the first kind. Tikhonov regularization is used to solve this integral equation, leading to a smooth first derivative. By using the built‐in regularization parameter in the method noise amplification is kept under control. The performance of this method is demonstrated by applying it to data taken from the literature. © 2005 American Institute of Chemical Engineers AIChE J, 2006

Investigating the Properties of Aqueous Monoisopropanolamine Using Density Data from 283.15K to 353.15K

“…The Tikhonov regularization procedure used to compute the partial derivative ∂v/∂x W in this research has a built-in regularization parameter λ that is able to keep the noise amplification under control [7]. The mathematical principles of the Tikhonov regularization procedure can be found in [7,8]. Some of its advantages over existing curve fitting methods, when applied to evaluate isothermal partial molar volumes of a number of different binary solutions, are discussed in [8,11,12].…”

“…The aim of this research is to apply the recently reported Tikhonov regularization technique [7,8] in order to convert the density data of aqueous MIPA reported by Mokraoui et al [4] into partial molar volumes V MIPA and V W of MIPA and water, respectively. This new technique allows the partial molar volumes to be obtained more easily and more reliably.…”

Alkali-activation potential of biomass-coal co-fired fly ash

“…To enable these comparisons, the first and second derivatives of each dilatometric curve were plotted to easily detect the onset temperature of this reaction for each gel (i.e., the second derivative has a maximum at the location of the onset of this expansion peak). The derivative curves were determined using numerical differentiation based on Tikhonov regularization [45], which incorporates smoothing techniques to mitigate the inherent scatter in the numerical data upon differentiation. An example of the derivative curves for the A-FA-0 Na-1-7 mix is shown in Figure 8, with the expansion peak of interest marked.…”

A method for computing the partial derivatives of experimental data

“…In Tikhonov regularization, instead of minimizing (1), (2), and (3) separately, their linear combination…”

“…1 Tikhonov regularization is a specialized technique for handling ill-posed inverse problems. Lubansky et al 2 applied this technique to evaluate the ordinary derivatives of experimental data that depend on a single independent variable. The built-in regularization parameter in Tikhonov regularization is able to keep noise amplification under control.…”