Using regularizing algorithms for the reconstruction of growth rate from the experimental data

It is shown that the average rate is not a convenient parameter for studyin g the dynamics of the process and instantaneous rate must be used instead o f the average rate. The problem of reconstruction of the instantaneous rate of the process from the experimental data is an ill-posed inverse problem. The main problem is the following. The experimental data always have error s, which lead to the instability of the ordinary numerical methods of deriv ative reconstruction. Special regularizing algorithms, which stabilize the process of solution, must be used to solve this problem. Two methods based on approximating spline regression and Tikhonov regularization are compared . The idea of the spline regression method consists in the approximation of data by means of cubic splines and further analytical differentiation of t he regression function. Tikhonov regularization method is based on the solu tion of an integral equation for the derivative. It is shown that the metho d of Tikhonov regularization is much more flexible and powerful. The smooth ness of the derivative in the Tikhonov regularization method can easily be controlled by choosing the appropriate value of the parameter delta, which represents the error of the experimental data. This method uses much more w eak a priori constraints on the derivative. The method permits to reconstru ct small-scale details of the derivative. Tikhonov regularization method ca n be successfully used even for small (N approximate to 10) sets of data an d for the reconstruction of the higher derivatives from the experimental da ta. (C) 2000 Elsevier Science R.V. All rights reserved.