Sensitivity equation

The sensitivity equation can be developed by differentiating equation 4 with respect to J. Since the signal is proportional to J and detection is defined as to when the measured signal to noise ratio equals 3, the gas detection sensitivity (ppm) in terms of the NEP for gas detection becomes ... 

The mercury film electrode has a higher surface-to-volume ratio than the hanging mercury drop electrode and consequently offers a more efficient preconcentration and higher sensitivity (equations 3-22 through 3-25). hi addition, the total exhaustion of thin mercury films results in sharper peaks and hence unproved peak resolution in multicomponent analysis (Figure 3-14). 

If the dimensionality of the problem is not excessively high, simultaneous integration of the state and sensitivity equations is the easiest approach to implement the Gauss-Newton method without the need to store x(t) as a function of time. The latter is required in the evaluation of the Jacobeans in Equation 6.9 during the solution of this differential equation to obtain G(t). 

The four sensitivity equations (Equations 6.56a-d) should be solved simultaneously with the two state equations (Equation 6.52). Integration of these six [=nx(p+1 )=2x(2+1)] equations yields x(t) and G(t) which are used in setting up matrix A and vector b at each iteration of the Gauss-Newton method. 

For models described by a set of ordinary differential equations there are a few modifications we may consider implementing that enhance the performance (robustness) of the Gauss-Newton method. The issues that one needs to address more carefully are (i) numerical instability during the integration of the state and sensitivity equations, (ii) ways to enlarge the region of convergence. 

Solution of Equation 8.65 yields the optimum value for the step-size. The solution can be readily obtained by Newton s method within 3 or 4 iterations using jLta as a starting value. This optimal step-size policy was found to yield very good results. The only problem that it has is that one needs to store the values of state and sensitivity equations at each iteration. For high dimensional systems this is not advisable. 

In our opinion the above formulation does not provide any computational advantage over the approach described next since the PDEs (state and sensitivity equations) need to be solved numerically. 

Therefore, efficient computation schemes of the state and sensitivity equations are of paramount importance. One such scheme can be developed based on the sequential integration of the sensitivity coefficients. The idea of decoupling the direct calculation of the sensitivity coefficients from the solution of the model equations was first introduced by Dunker (1984) for stiff chemical mechanisms... 

Figure 10.1 Schematic diagram of the sequential solution of model and sensitivity equations. The order is shown for a three parameter problem. Steps l, 5 and 9 involve iterative solution that requires a matrix inversion at each iteration of the fully implicit Euler s method. All other steps (i.e., the integration of the sensitivity equations) involve only one matrix multiplication each.

The procedure for the selection of the most appropriate time interval requires the integration of the state and sensitivity equations and the computation of lj(t), j=l,...,p at each grid point of the operability region. Next by plotting I/t), j=l,...,p versus time (preferably on a log scale) the time interval [t, tNp] where the information indices are excited and become large in magnitude is determined. This is the time period over which measurements of the output vector should be obtained. 

Step 4. Integrate the state and sensitivity equations and compute matrix A"ew. 

Since the quantities df/d(Mj) are generally required during the solution of Eq. (2.69), the sensitivity equations are conveniently solved simultaneously with the species concentration equations. The initial conditions for Eq. (2.70) result from mathematical consideration versus physical consideration as with Eq. (2.69). Here, the initial condition [c M /da, ]f=0 is the zero vector, unless a, is the initial concentration of the /th species, in which case the initial condition is a vector whose components are all zero except the y th component, which has a value of unity. Various techniques have been developed to solve Eq. (2.70) [22, 23],... 

The solution procedure to this equation is the same as described for the temporal isothermal species equations described above. In addition, the associated temperature sensitivity equation can be simply obtained by taking the derivative of Eq. (2.87) with respect to each of the input parameters to the model. The governing equations for similar types of homogeneous reaction systems can be developed for constant volume systems, and stirred and plug flow reactors as described in Chapters 3 and 4 and elsewhere [31-37], The solution to homogeneous systems described by Eq. (2.81) and Eq. (2.87) are often used to study reaction mechanisms in the absence of mass diffusion. These equations (or very similar ones) can approximate the chemical kinetics in flow reactor and shock tube experiments, which are frequently used for developing hydrocarbon combustion reaction mechanisms. 

These partial derivatives provide a lot of information (ref. 10). They show how parameter perturbations (e.g., uncertainties in parameter values) affect the solution. Identifying the unimportant parameters the analysis may help to simplify the model. Sensitivities are also needed by efficient parameter estimation procedures of the Gauss - Newton type. Since the solution y(t,p) is rarely available in analytic form, calculation of the coefficients Sj(t,p) is not easy. The simplest method is to perturb the parameter pj, solve the differential equation with the modified parameter set and estimate the partial derivatives by divided differences. This "brute force" approach is not only time consuming (i.e., one has to solve np+1 sets of ny differential equations), but may be rather unreliable due to the roundoff errors. A much better approach is solving the sensitivity equations... 

The sensitivity equations (5.39) are derived by differentiating (5.37) with respect to Pj, and changing the order of differentiation on the left hand side. The initial values to (5.39) are given by... 

In this section we deal with estimating the parameters p in the dynamical model of the form (5.37). As we noticed, methods of Chapter 3 directly apply to this problem only if the solution of the differential equation is available in analytical form. Otherwise one can follow the same algorithms, but solving differential equations numerically whenever the computed responses are needed. The partial derivations required by the Gauss - Newton type algorithms can be obtained by solving the sensitivity equations. While this indirect method is... 

Since spline interpolation and integration is mucht faster than solving the sensitivity equations and the original differential equations, the direct method is superior to the indirect one in terms of numerical efficiency, whenever it is feasible. 

The first term on the right-hand side is the product of the physical problem s current Jacobian matrix and the sensitivity-coefficient matrix (i.e., the dependent variable). Assuming that the underlying physical problem (i.e., Eq. 15.58) is solved by implicit methods, the Jacobian evaluation is already part of the solution algorithm. The second term, which is the matrix that describes the explicit dependence of f on the parameters, must be evaluated to form the sensitivity equation. Note that all terms on the right-hand side are time dependent, as are the sensitivity coefficients S(t). 

In solving the underlying model problem, the Jacobian matrix is an iteration matrix used in a modified Newton iteration. Thus it usually doesn t need to be computed too accurately or updated frequently. The Jacobian s role in sensitivity analysis is quite different. Here it is a coefficient in the definition of the sensitivity equations, as is 3f/9a matrix. Thus accurate computation of the sensitivity coefficients depends on accurate evaluation of these coefficient matrices. In general, for chemically reacting flow problems, it is usually difficult and often impractical to derive and program analytic expressions for the derivative matrices. However, advances in automatic-differentiation software are proving valuable for this task [36]. 

The overlay to the upper of two curves labeled blue adapted (short dashes) is also based on the overall spectral sensitivity equation of this work. The function as drawn includes contributions from all three spectral channels... 

On the other hand, the trajectory sensitivity equations method requires simultaneous integration of a greater number of equations than the adjoint system approach. However, it is more stable than the adjoint system approach due to the requirement of forward integration only. It is usually preferred in the area of parameter estimation and sensitivity (Kalogerakis and Luus, 1983 Caracotsios and... 

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