Region of convergence

A power series may be differentiated term by term and represents the function df[x)fdx within the same region of convergence... 

For models described by a set of ordinary differential equations there are a few modifications we may consider implementing that enhance the performance (robustness) of the Gauss-Newton method. The issues that one needs to address more carefully are (i) numerical instability during the integration of the state and sensitivity equations, (ii) ways to enlarge the region of convergence. 

In this section we first present an efficient step-size policy for differential equation systems and we present two approaches to increase the region of convergence of the Gauss-Newton method. One through the use of the Information Index and the other by using a two-step procedure that involves direct search optimization. 

The remedies to increase the region of convergence include the use of a pseudoinverse or Marquardt s modification that overcome the problem of ill-conditioning of matrix A. However, if the basic sensitivity information is not there, the estimated direction Ak +I) cannot be obtained reliably. 

A simple procedure to overcome the problem of the small region of convergence is to use a two-step procedure whereby direct search optimization is used to initially to bring the parameters in the vicinity of the optimum, followed by the Gauss-Newton method to obtain the best parameter values and estimates of the uncertainty in the parameters (Kalogerakis and Luus, 1982). 

For example let us consider the estimation of the two kinetic parameters in the Bodenstein-Linder model for the homogeneous gas phase reaction of NO with 02 (first presented in Section 6.5.1). In Figure 8.4 we see that the use of direct search (LJ optimization) can increase the overall size of the region of convergence by at least two orders of magnitude. 

Figure 8.4 Use of the L) optimization procedure to bring the first parameter estimates inside the region of convergence of the Gauss-Newton method (denoted by the solid line). All test points are denoted by +. Actual path of some typical runs is shown by the dotted line.

Kalogerakis, N. and R. Luus, "Increasing the Size of the Region of Convergence in Parameter Estimation", proc. American Control Conference, Arlington, Virginia, 1,358-364(1982). 

Two other important considerations are involved in the use of infinite series. Convergence may be assured only within a given range of the independent variable, or even only at a single point. Thus, the region of convergence ... 

A second question arises in practical applications, because at different points wiihin the region of convergence, the rate of convergence may be quite different. Is other words the number of terms that must be retained to yield a certain level of accuracy depends on the value of the independent variable. In this case the series is not uniformly convergent. 

A power series may be integrated term by term to represent the integral of the function within an interval of the region of convergence. Iffix) =a0 + ape + apt + , then... 

In this section, we consider briefly the accuracy and convergence aspects of the multi-mode models derived by the L-S method. We also illustrate the regularization procedure used for the local equation(s) to increase the region of convergence of the multi-mode models. 

The accuracy of low-dimensional models derived using the L S method has been tested for isothermal tubular reactors for specific kinetics by comparing the solution of the full CDR equation [Eq. (117)] with that of the averaged models (Chakraborty and Balakotaiah, 2002a). For example, for the case of a single second order reaction, the two-mode model predicts the exit conversion to three decimal accuracy when for (j>2(— pDa) 1, and the maximum error is below 6% for 4>2 20, where 2(= pDd) is the local Damkohler number of the reaction. Such accuracy tests have also been performed for competitive-consecutive reaction schemes and the truncated two-mode models have been found to be very accurate within their region of convergence (discussed below). 

When a function is defined by an infinite power series in terms of a parameter p, the traditional approach is to truncate the power series, retaining terms up to pq. However, if the power series fails to converge (i.e. outside the region of convergence of the local equation), including higher order terms does not save the truncated series from failure, and the truncated series may lead to nonphysical results in the limit of p —> oo. 

It should be noted that for non-isothermal case (and also for isothermal case with autocatalytic kinetics) the local equation may have multiple solutions. When this occurs, the averaged model obtained by the L-S method captures the complete set of solutions of the full CDR equations only within the region of convergence of the local equation. For example, for the wall-catalyzed non-isothermal reaction case, we have shown that the averaged two-mode model can capture only the three azimuthally symmetric solutions of the full CDR equation. The latter has three symmetric solutions (of which two are stable) as... 

Wang, B-C., and R. Luus, Increasing the size of region of convergence for parameter estimation through the use of shorter data-length, Int. J. Control, 31, 947-972 (1980). 

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