Numerical sensitivity

We have chosen the steady state with Yfa = 0.872 and FCD = 1.0 giving a dense phase reactor temperature of Yrd = 1.5627 (Figure 7.14(b) and (c)) and a dense-phase gasoline yield of x-id = 0.387 (Figure 7.14(a)). This is the steady state around which we will concentrate most of our dynamic analysis for both the open-loop and closed-loop control system. We first discuss the effect of numerical sensitivity on the results. Then we address the problem of stabilizing the middle (desirable, but unstable) steady state using a switching policy, as well as a simple proportional feedback control. 

Revisit Section 4.4.6 on the numerics for the neurocycle enzyme system in light of our comments on the numerical sensitivity and chaos of I VPs on p. 462. 

Boyd DB, Smith DW, Stewart JJP, Wimmer E. Numerical sensitivity of trajectories across conformational energy hypersurfaces from geometry optimized molecular orbital calculations AMI, MNDO, and MINDO/3. J Comput Chem 1988 9 387-398. 

Let us summarize the above analysis. The initial comprehensive mechanism includes 17 elementary processes (no distinction being made between radical recombinations and disproportionations). Among these processes, 8 at least are negligible, 3 are non-determining and 3 quasi-non-determining, 3 are determining and, finally, 2 out of these three are determined. As has been noted, numerical sensitivity analyses carried out on more complex mechanisms [70, 74, 93, 94, 117] lead to the same type of conclusion. 

D. B. Boyd, Drug Inf. J., 17,121 (1983). Quantum Mechanics in Drug Design Methods and Applications. D. B. Boyd, D. W. Smith, J. J. P. Stewart, and E, Wimmer, J. Comput. Chem., 9, 387 (1988). Numerical Sensitivity of Trajectories across Conformational Energy Hypersurfaces from Geometry Optimized Molecular Orbital Calculations AMI, MNDO and MNDO/3. D. B. Boyd, in Reviews in Computational Chemistry, Vol. 1, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1990, pp. 321-354. Aspects of Molecular Modeling. 

Methods of numerical sensitivity and uncertainty analysis can be used to examine uncertainty and identify the key sources of bias and imprecision in quantitative estimates of risk. Once identified, limited resources (e.g., time, funding) can be efficiently allocated to obtain new information and data for those major sources of uncertainty and reduce it. These analyses can be repeated until uncertainties associated with the risk estimates are of an acceptable degree or until uncertainties cannot be further reduced. 

The intramolecular esterification of simple m-hydroxy-acids (for example, those containing only a saturated hydrocarbon chain) or of its co-halogeno derivative can be realized easily (Fig. 5), but the harsh reaction conditions are not compatible with the numerous sensitive functions present in the seco co-hydroxyacids of natural origin. The necessity to perform the lactonization in a smoother way has been an efficient motor for progress. The same considerations apply for the macrolactamization or the formation of macrocarbocycles of natural origins. 

Bolado, R., Castaing, W. Tarantola, S. 2009. Contribution to the sample mean plot for graphical and numerical sensitivity analysis. Reliability Engineering System Safety 94 1041-1049. 

D. B. Boyd, D. W. Smith, J. J. P. Stewart, and E. Wimmer,/. Comput. Chem., 9, 387 (1988). Numerical Sensitivity of Trajectories across Conformational Energy Hypersurfaces from Geometry Optimized Molecular Orbital Calculations AMI, MNDO, and MlNDO/3. 

The solution of the pellet-side conservation equation (Eq. 9.33) is a two-point boundary value problem but is not difficult to solve numerically so long as the intrinsic kinetics are simple and the reaction is moderately diffusion-limited. The problem becomes difficult to solve because of numerical sensitivity when the reaction is severely diffusion-limited and the accuracy of the result is not assured. However, an analytical expression can be obtained in this case for arbitrary intrinsic kinetics as shown in Chapter 4. 

The approximation of Taylor series expansion enables the avoidance of iterative response analyses such as time-history analysis for evaluating the objective function. However, the computation of full elements of the Hessian matrix requires a huge computational load when N is large, especially for numerical sensitivity analysis, i.e., the finite difference analysis using gradient vectors. A simpler approach has therefore been proposed by Chen et al. (2009) where the non-diagonal elements of the Hessian matrix are neglected. 

In the method using Taylor series expansion as explained in previous sections, the accuracy of the robustness evaluation depends directly on the reliability of the numerical sensitivity analysis. For this reason, when the evaluation of numerical sensitivities has some difficulties resulting from the elastic-plastic structural property of isolators, another URP method should be introduced where the variation of the objective function is... 

Figure 21 shows the upper bounds of the maximum drift of the base-isolation story compared for various methods (URP methods with second-order Taylor series approximation/with RSM and the Monte Carlo Simulation (MCS)). As explained before, the difference between the URP method with second-order Taylor series approximation and that with RSM is how to estimate the variation of the objective function. In the former one, the numerical sensitivities, i.e., the gradient vector and the Hessian matrix, of the objective function are needed. On the other hand, in the latter one, a kind of RSM is applied where appropriate response samplings are made and the gradient vector and the Hessian matrix are evaluated from the constmcted approximate function. 

One of the major contributions of this assessment project to the MELCOR eflFort has been the systematic search for and identification of code features which lead to time step and other numerical dependencies, as summarized in the individual task reports. Nearly all major advances in elimination of these undesirable features during the last year are the result of these systematic studies. Many of the numeric sensitivities have been traced to code problems that would not be readily detected in the single, isolated calculations that are typical of many user applications. 

The differences seen in timing of key events such as clad failure, core plate failure, lower head penetration failure, etc., in these machine-dependency and time-step studies vary by much smaller times (on the order of 10-100s) than the timestep-variation results presented by BNL to the Peer Review for their Peach Bottom station blackout analysis (which often varied by 1,000-10,000s). A large part of this reduction in numeric sensitivity probably represents the significant efforts of the code developers since the Peer Review in identifying and eliminating numeric sensitivities in MELCOR. Unfortunately, we have no comparable results of time-step studies for the more recent BNL Oconee analyses. 

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