Sensitivity decoupled direct method

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... 

In contrast to applications in structural biology where X/Y correlations are nowadays normally executed as H detected, three-dimensional experiments because of sensitivity reasons,14 many studies on inorganic or organometallic compounds are still performed as two-dimensional experiments with direct detection of one heteronucleus and under -decoupling. As compared to these two categories, one-dimensional polarisation transfer methods such as (semi) selective X/Y-INEPT or INDOR-type techniques, which had in the past been shown to be particularly useful for the characterisation of substrates with only one or two heteronuclei,11 have recently received less attention.15 NOE-based correlations, which are frequently employed for the structure elucidation of bio-molecules, remain rare, and apart from an earlier report of a 13C/6Li HOESY experiment,16 have not been further investigated. 

Similar spectra can be obtained more rapidly and with less sample if the data are acquired through the proton signals, which are much more intense. Basically, the H NMR data are acquired and the H- C coupling constant used as the delay in a pulse sequence, which enables us to obtain the carbon spectrum. This method of obtaining the data is called inverse-mode , since the carbon atoms are detected through their attached hydrogen atoms rather than by direct detection, with obvious benefits in the sensitivity and the time taken to obtain a spectrum. HMQC and HMBC are both examples of inverse-mode spectra and this method is so much quicker than CH COSY that an entire HMQC spectrum can be obtained in much less time than it takes to obtain the proton-decoupled C... 

An alternative to the simultaneous solution of (4.A.3) and (4.A.7) is to solve (4.A.3) first and store the computed values of c, (t), / = 1, 2,. .., n, for use in subsequent solution of (4.A.7). Because this method, called the direct decoupled method (DDM), solves the model and sensitivity equations separately, only n differential equations need to be solved at one time that is, the n equations (4.A.7) are solved m times for each of the values of j. Applications of the DDM are given by Dunker (1981, 1984). Milford et al. (1992) have used this method to evaluate. sensitivities of urban photochemical reaction mechanisms. The DDM is a computationally attractive alternative to the full, simultaneous. solution of (4.A.3) and (4.A.7). 

Appendix 4 Sensitivity/Uncertainty Analysis of Atmospheric Chemical Mechanisms 219 4.A. 1 Sensitivity Coefficients 222 4.A.2 The Direct Decoupled Method 223 4. A.3 Adjoint Methods 224 4.A.4 Green s Function Methods 224 References 226 Problems 230... 

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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