The Decoupled Direct Method

If the local sensitivity coefficients are calculated using Eq. (5.6) (this is called the direct method), then the error of calculation can be estimated and limited. The 

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Dunker, A.M., "The Decoupled Direct Method for Calculating Sensitivity Coefficients in Chemical Kinetics", J. Chem. Phys.. 81,2385-2393 (1984). 

A.M Dunker, The decoupled direct method for calculating sensitivity coefficients in chemical kinetics, J. Chem. Phys. 81 (1984) 2385-2393. 

The most efficient algorithm for the solution of the sensitivity differential equations is called the decoupled direct method (ddm), which was first applied in chemical kinetics by Dunker [67, 68]. He drew attention to the fact that equations (4.1) and (4.6) have the same Jacobian, so that a stiff ode solver will use the same step size and order of approximation in the solution of both odes. The ddm method first takes a step for the solution of equation (4.1) and then performs steps for the solution of equation (4.6) for / = 1,. . . , m. The procedure is repeated in the subsequent steps. Since the Jacobian of the equations is the same, it has to be triangularized only once for each time interval. This method is applied in the program SENKIN [69]. 

For a long time the main topic of research in the area of sensitivity analysis was to find an accurate and effective method for the calculation of local concentration sensitivities. This question now seems to be settled, and the decoupled direct method (ddm) is generally considered the best numerical method. All the main combustion simulation packages such as CHEMKIN, LSENS, RUNIDL and FACSIMILE calculate sensitivities as well as the simulation results and, therefore, many publications contain sensitivity calculations. However, usually very little information is actually deduced from the sensitivity results. It is surprising that the application of principal component analysis is not widespread, since it is a simple postprocessing method which can be used to extract a lot of information from the sensitivities about the structure of the kinetic mechanism. Also, methods for parameter estimation should always be preceded by the principal component analysis of the concentration sensitivity matrix. 

The EDA system corresponding to the model is solved by a modified version of the LSODI routine, which is based on Gear s method. The version implemented performs the solution of the EDA system concomitantly to the evaluation of the parameters sensitivities based on the decoupled direct method (4). As a matter of fact, the simulation of the system is... 

Another example of the use of AD is in the calculation of the Jacobian used by the decoupled direct method. AD can provide a more automatic approach compared to using an analytic or symbolic expression for the definition of the Jacobian based on the RHS of the kinetic system of differential equations, but one which is more accurate than defining the Jacobian numerically using finite-difference methods. This approach is implemented in the freely available KPP package for atmospheric chemical simulations (Damian et al. 2002 Sandu et al. 2003 Daescu et al. 2003 KPP). 

The first step in the assignment process after considering the chemical shifts themselves is to obtain one or more off-resonance decoupled spectra to determine the number.of protons attached to the carbons and to correlate the carbon and proton resonances. (191, 192) [It remains to be seen whether or not the two-dimensional FT technique of producing chemical shift correlation maps directly (159, 501) will supersede the more conventional methods. ] Only then should more sophisticated experiments be contemplated. Many of the papers concerned with correlation have concentrated on deriving... 

In 1968, J. S. Waugh et al reported a new method for removal of the homonuclear dipolar interaction using multiple pulses (WAHUHA-4 Waugh, Huber and Haeberlen 4 pulses/cycle), which is shown in Fig. 2. Homonuclear dipolar decoupling by multiple-pulse irradiation is termed MAS in spin space . In general, we consider that the magnetization of the z direction is... 

In fact, this approach was used to identify the resonant frequencies of heteronuclei prior to the advent of the pulse-FT methods [7] which made their direct observation possible. Thus, the proton spectrum was recorded whilst applying a second if field to the heteronucleus. Successively stepping the decoupler frequency and repeating the measurement would ultimately indicate the X resonance position when the X— H coupling was seen to disappear from the proton spectrum. 

The pulse calibration on the decoupler channel cannot be performed using the direct method since the pulse which is to be calibrated is transmitted on a different rf channel and resonance frequency to the observed nucleus. To calibrate the rf pulse on the decoupler channel it is necessary to determine the pulse length from the effect the pulse has on the nucleus observed on the transmitter/receiver channel. This indirect calibration is achieved by using a coupled IS spin system and transferring the detectable antiphase... 

An example of such spectra is shown in Fig. 16 (n-hexane). The traces plotted for various values of co how that parallel to the coj-axis, the full multiplet structure is retained whereas in the coj direction the completely decoupled spectrum results. The undecoupled spectrum is to be considered as a projection of the spectra for various values of onto the coj-axis. This rather involved technique is less sensitive than ordinary FTNMR. It has been used mainly in C-NMR spectroscopy. A number of applications as well as the solution to problems connected with this method have been reported -307>... 

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