Sensitivity parameter

The value of the reaction rate constants is usually uncertain. The object of sensitivity analysis is to systematically determine the effect of uncertain parameters on system solutions. A recent review paper has been written by Rabitz et al. (1983). The methods of sensitivity analysis can be divided into global and local methods. 

Global methods are aimed at providing information on the change of solutions in a large domain of parameters, i.e. rate constants. As in this case the induced kinetic differential equation should be solved tens of thousands of times in the case of larger systems these methods are only of rather limited use. 

Let us consider the solution of the induced kinetic differential equation at a fixed time point as the function of the rate constants and let us consider the Taylor series of the solution with respect to the parameters around the nominal value of the parameters. The coefficients of this Taylor series are the local sensitivity coefficients. In general only the first order coefficients are considered. 

Several methods are used to calculate these coefficients, such as the finite difference approximation, the direct method and its modifications, the Green-function method, and the method of polynomial approximations. 

The (local) sensitivity analysis can be applied in several areas. 

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Minimum exposure times must be observed in order to reach the requisite S/N ratio. As per EN 1435 and EN 584-1, for the different ranges of utilization (energy, wall thickness), definite film elasses are prescribed. They are characterized by the minimum gradient-to-noise ratios. Based on this, one can calculate the minimum values for the S/N ratio based on the IP systems. The exposure time and the device parameter sensitivity and dynamics (latitude) must be adjusted accordingly, with an availability of an at least 12 bit system for the digitalization. 

Numerical simulations are designed to solve, for the material body in question, the system of equations expressing the fundamental laws of physics to which the dynamic response of the body must conform. The detail provided by such first-principles solutions can often be used to develop simplified methods for predicting the outcome of physical processes. These simplified analytic techniques have the virtue of calculational efficiency and are, therefore, preferable to numerical simulations for parameter sensitivity studies. Typically, rather restrictive assumptions are made on the bounds of material response in order to simplify the problem and make it tractable to analytic methods of solution. Thus, analytic methods lack the generality of numerical simulations and care must be taken to apply them only to problems where the assumptions on which they are based will be valid. 

We have been discussing a class of penetration problems that are accurately modeled by two-dimensional calculations. There are many three-dimensional problems, however, that can be well approximated by two-dimensional analyses, and the greatly reduced computer memory and time requirements for such calculations make them attractive alternatives for scoping studies, or for parameter sensitivity studies. Although good quantitative predictions may not be obtained with such approximations, the calculations can be expected to reveal trends and qualitative results that will carry over to the full three-dimensional problem. 

Measures surface crystal structure parameters, sensitive to structural defects... 

G. H. Bu-Abbud, N. M. Bashara, and J. A. Woollam. Thin Solid Films. 138,27, 1986. Description of Marquardt algorithm and parameter sensitivity correlation in ellipsometry. 

Parameter variation and optimization If parameter sensitivity is investigated, simulation is cheaper and quicker than experiments. Trends are usually well predicted by computer models. 

The SimuSolv optimize function can be used for minimization of a user defined cost expression. However, at this early stage of process development, no attempt was made to determine parameters for an economic optimum. Instead, many simulations were run to determine parameter sensitivities and malce comparisons of various possibilities which satisfied a necessary objective of reaching the desired low level of residual monomer. Attention was given to minimizing total finishing time, the amount of initiator required and residual initiator level at the end of the finishing step. Based on simulation results for... 

The parameter sensitivity matrix G(t) can be obtained as shown in the previous section by solving the matrix differential equation,... 

Similar to the parameter sensitivity matrix, the initial state sensitivity matrix, P(t), cannot be obtained by a simple differentiation. P(t) is determined by solving a matrix differential equation that is obtained by differentiating both sides of Equation 6.1 (state equation) with respect to p( ... 

Thus, the error in the solution vector is expected to be large for an ill-conditioned problem and small for a well-conditioned one. In parameter estimation, vector b is comprised of a linear combination of the response variables (measurements) which contain the error terms. Matrix A does not depend explicitly on the response variables, it depends only on the parameter sensitivity coefficients which depend only on the independent variables (assumed to be known precisely) and on the estimated parameter vector k which incorporates the uncertainty in the data. As a result, we expect most of the uncertainty in Equation 8.29 to be present in Ab. 

Step 2. Given the current estimate of the parameters, k , compute the parameter sensitivity matrix, G the response variables f(x k[Pg.161]

The main consideration for the choice of the most appropriate time interval is the availability of parameter sensitivity information. Kalogerakis and Luus (1984) proposed to choose the time interval over which the output vector (measured variables) is most sensitive to the parameters. In order to obtain a measure of the available sensitivity information with respect to time, they proposed the use of... 

Jury, W. A., Farmer, W. J., Spencer, W. F. (1984a) Behavior assessment model for trace organics in soil II. Chemical classification and parameter sensitivity. J. Environ. Qual. 13, 567-572. 

With regard to sensor sensitivity, Fig. 3.26 shows the wavelength shift and amplitude changes vs. chloroform concentration with quadratic interpolation for the 160 and 260 nm coated LPG compared with the uncoated one. As expected, without the sensing overlay negligible variations have been observed in both the measured parameters. Sensitivities of —0.130 nm/ppm and 0.163 dB/ppm were observed for the thinner in the range of 0 10 ppm. Sensitivities of —0.85 nm/ppm and 0.26 dB/ppm were measured for the thicker overlay, in the same range. 

The discussions in Sections 3.1 and 3.2 show that the interaction among enthalpies of reaction, reaction kinetics, and surrounding conditions is of paramount importance relative to the existence of potential thermal hazards such as runaways. Whereas valuable information on parameter sensitivity can be estimated by a theoretical approach, it remains of vital importance to evaluate hazards by appropriate and adequate laboratory tests to obtain information on the rates of heat and gas generation, and the maximum quantities of heat and gas involved. Materials which are real to the process should be used in tests to assure that the effects of any contaminants are recognized. 

Model parameters are usually determined from experimental data. In doing this, sensitivity analysis is valuable in identifying the best experimental conditions for the estimation of a particular model parameter. Sensitivity analysis is easy effected with MADONNA, and sensitivity analysis is also provided in other more advanced software packages, such as ACSL-OPTIMIZE. 

Rose KA, Smith EP, Gardner RH, Brenkert AL, Bartell SM. 1991. Parameter sensitivities, Monte Carlo filtering, and model forecasting uncertainty. J Forecasting 10 117-133. 

Figure 10.4 Parameter sensitivity analysis performed around the 700mg dose. RadPart particle radius RefSol reference solubility.

B. KanyAr, Parameter sensitivity analysis for designing of experiments in kinetics, Acta Biochim. Biophys. Acad. Sci. Hung. 12 (1978) 24-28. 

Example 5.3 Parameter sensitivities in the microbial growth model... 

In these spectra, a parameter sensitive to chromium and lead substitution into Fe304 is the area ratio S, defined as the area of the octahedral iron component divided by the corresponding peak for the tetrahedral component. For Fe304 ((5 = 0), S should equal approximately 2, as experimentally observed however, for the PbCr-Fe304 catalyst the value of S is about... 

As a result of a few parameter sensitivity analyses carried out in the case of the ED recovery of some sodium salts of hydrogen chloride (Fidaleo and Moresi, 2005a), lactic (Fidaleo and Moresi, 2004), acetic (Fidaleo and... 

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