Functional sensitivity analysis

As such, once the scattering calculation is performed, the SC is easily constructed for a particular collision event. We see that the KVP SC places importance in regions where the scattering wavefunctions have large amplitude. 

An interesting application of functional sensitivity analysis is in determining which portion of a one dimensional barrier potential is most important for tunneling 

The functional derivative of 0(E) has three terms, because of the potential dependence of the turning points. Differentiation gives 

The first two terms in Eq. (2.14) vanish by the definition of turning points. The WKB SC thus becomes 

Some practical problems associated with the functional sensitivity analysis strategy are worth mentioning. First, it is only first order. This is a severe problem. 

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J. Chang, N. J. Brown, M. D Mello, R. E. Wyatt, and H. Rabitz, Quantum functional sensitivity analysis within the log-derivative Kohn variational method for reactive scattering, J. Chem. Phys. 97 6240 (1992). 

SPOLD format SPOLD format a lot of analytical functions (sensitivity analysis, Monte Carlo approach) A quick calculation and simple interface Cannot making LCA s process tree in the software —> import from self making excel s table... 

Kohn variational method that has been described in detail in previous publications [5, 25, 26]. We briefly review this approach for quantum reactive scattering below [27]. In addition, we demonstrate how to apply functional sensitivity analysis within the 5—matrix Kohn framework. We report the cross section calculations resulting from the two PESs for total angular momenta J = 0 and 10. We find that the theoretical cross sections do not change significantly when the LSTH PES is replaced by the DMBE [28]. This suggests that the theoretical prediction — that there is no sharp resonance in the H+H2 integraJ cross section — may be correct. 

We do not pursue functional sensitivity analysis further. Instead, we proceed with the results of the brute force comparison. 

We outlined the derivation of the 5—matrix version of the Kohn variational principle, and demonstrated how to apply functional sensitivity analysis within this computational framework. In addition, we discussed the complementarity of brute force and functional sensitivity analysis. We applied brute force sensitivity analysis to the H+H2 reaction, comparing partial cross sections resulting from the LSTH and DMBE PESs. We found that the theoretical cross sections shown in Fig. 2.1 do not change significantly when the LSTH potential is replaced by the DMBE. The computed cross sections are thus fairly insensitive to changes in the potential energy surface. We conclude that the theoretical cross sections are accurate. 

Sion to our assumptions about the initial purchase price and the cost of gasoline. Figure 1 shows the LCC of the hybrid and the conventional car over the ten-year period as a function of the cost of gasoline. When gas prices are approximately 3 per gallon, the two cars cost about the same. This value is referred to as the break-even point. If gas prices reach 3.75 per gallon, the approximate cost in Japan, the hybrid car is more economical. Sensitivity analysis can also be conducted for other input variables, such as initial purchase price, miles driven per year and actual fuel economy. 

The multimedia model present in the 2 FUN tool was developed based on an extensive comparison and evaluation of some of the previously discussed multimedia models, such as CalTOX, Simplebox, XtraFOOD, etc. The multimedia model comprises several environmental modules, i.e. air, fresh water, soil/ground water, several crops and animal (cow and milk). It is used to simulate chemical distribution in the environmental modules, taking into account the manifold links between them. The PBPK models were developed to simulate the body burden of toxic chemicals throughout the entire human lifespan, integrating the evolution of the physiology and anatomy from childhood to advanced age. That model is based on a detailed description of the body anatomy and includes a substantial number of tissue compartments to enable detailed analysis of toxicokinetics for diverse chemicals that induce multiple effects in different target tissues. The key input parameters used in both models were given in the form of probability density function (PDF) to allow for the exhaustive probabilistic analysis and sensitivity analysis in terms of simulation outcomes [71]. 

The key input parameters used in the 2 FUN model were given in the form of probability density function (PDF) to allow the exhaustive probabilistic analysis and sensitivity analysis in terms of simulation outcomes. 

In this example with only three components, the optimum could have been determined by simply overlaying the individual response contour plots. This approach would be difficult, if not impossible, if the formulation would have many responses or contain four or more components. By contrast, the combination of the desirability function and the Complex algorithm permits an optimization of a multiresponse formulation having many constrained components in addition to providing the basis for sensitivity analysis. 

The last entry in Table 1.1 involves checking the candidate solution to determine that it is indeed optimal. In some problems you can check that the sufficient conditions for an optimum are satisfied. More often, an optimal solution may exist, yet you cannot demonstrate that the sufficient conditions are satisfied. All you can do is show by repetitive numerical calculations that the value of the objective function is superior to all known alternatives. A second consideration is the sensitivity of the optimum to changes in parameters in the problem statement. A sensitivity analysis for the objective function value is important and is illustrated as part of the next example. 

Step 6. You should always be aware of the sensitivity of the optimal answer, that is, how much the optimal value of C changes when a variable such as D changes or a coefficient in the objective function changes. Parameter values usually contain errors or uncertainties. Information concerning the sensitivity of the optimum to changes or variations in a parameter is therefore very important in optimal process design. For some problems, a sensitivity analysis can be carried out analytically, but in others the sensitivity coefficients must be determined numerically. 

Finally, we should mention that in addition to solving an optimization problem with the aid of a process simulator, you frequently need to find the sensitivity of the variables and functions at the optimal solution to changes in fixed parameters, such as thermodynamic, transport and kinetic coefficients, and changes in variables such as feed rates, and in costs and prices used in the objective function. Fiacco in 1976 showed how to develop the sensitivity relations based on the Kuhn-Tucker conditions (refer to Chapter 8). For optimization using equation-based simulators, the sensitivity coefficients such as (dhi/dxi) and (dxi/dxj) can be obtained directly from the equations in the process model. For optimization based on modular process simulators, refer to Section 15.3. In general, sensitivity analysis relies on linearization of functions, and the sensitivity coefficients may not be valid for large changes in parameters or variables from the optimal solution. 

Fiacco, A. V. Sensitivity Analysis of Nonlinear Programming Using Penalty Function Methods. Math Program 10 287-311 (1976). 

Part I comprises three chapters that motivate the study of optimization by giving examples of different types of problems that may be encountered in chemical engineering. After discussing the three components in the previous list, we describe six steps that must be used in solving an optimization problem. A potential user of optimization must be able to translate a verbal description of the problem into the appropriate mathematical description. He or she should also understand how the problem formulation influences its solvability. We show how problem simplification, sensitivity analysis, and estimating the unknown parameters in models are important steps in model building. Chapter 3 discusses how the objective function should be developed. We focus on economic factors in this chapter and present several alternative methods of evaluating profitability. 

Nalewajski, R. F. 1995. Charge sensitivity analysis as diagnostic tool for predicting trends in chemical reactivity. In Proceedings of the NATO ASI on Density Functional Theory, (Eds.) E. K. U. Gross, and R. M. Dreizler, pp. 339-389. New York Plenum Press. 

Nalewajski, R. F. 2000. Coupling relations between molecular electronic and geometrical degrees of freedom in density functional theory and charge sensitivity analysis. Computers Chem. 24 243-257. 

Nalewajski, R. F. and A. Michalak. 1995. Use of charge sensitivity analysis in diagnosing chemisorption clusters Minimum-energy coordinate and Fukui function study of model toluene-[V205] Systems. Int. J. Quantum Chem. 56 603-613. 

The relationship between temperature sensitivity and burning rate is shown in Fig. 7.21 as a function of AP particle size and burning rate catalyst (BEFP).li31 The temperature sensitivity decreases when the burning rate is increased, either by the addition of fine AP particles or by the addition of BEFP. The results of the temperature sensitivity analysis shown in Fig. 7.22 indicate that the temperature sensitivity of the condensed phase, W, defined in Eq. (3.80), is higher than that of the gas phase, 5), defined in Eq. (3.79). In addition, 4> becomes very small when the propel-... 

Table 2.16 Sensitivity analysis for the objective function coefficients.

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