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 studying the dynamics of the process and instantaneous rate must be used instead of 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 errors, which lead to the instability of the ordinary numerical methods of derivative 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 the regression function. Tikhonov regularization method is based on the solution of an integral equation for the derivative. It is shown that the method of Tikhonov regularization is much more flexible and powerful. The smoothness of the derivative in the Tikhonov regularization method can easily be controlled by choosing the appropriate value of the parameter δ, which represents the error of the experimental data. This method uses much more weak a priori constraints on the derivative. The method permits to reconstruct small-scale details of the derivative. Tikhonov regularization method can be successfully used even for small (N≈10) sets of data and for the reconstruction of the higher derivatives from the experimental data.