Additional features, which are provided by existing software packages besides parameter estimation, are numerical simulation and optimization. The first option gives an opportunity to calculate, for instance, reaction rates for conditions (temperature, pressures, concentrations, etc) at which no experiments were performed. [Pg.461]

The last point deals with the numerical process of estimation of tiie parameters, which usually involves to find the minimum of a least-squares type criterion. Whatever the optimization tectoique used for this task, calling or not the Jacobian matrix of the model, repeated simulations are needed, to compute the theoretical values corresponding to each experimental point [Pg.430]

Finally, we present two examples in which parameter estimation and optimal transition between different operating conditions are solved. Finally, a comparison of numerical results, solution time, and number of variables for the resulting NLPs are provided. [Pg.569]

The flux parameters are usually estimated from the tracer studies data by minimization of the deviations between experimental and modeled labeling data corresponding to the optimized set of fluxes. In general, the isotopomer balance equations are non-linear and numerical routines are used for their so- [Pg.49]

If we decide to treat the estimation problem using the nonlinear model, the problem becomes more challenging. As we will see, the parameter estimation becomes a nonlinear optimization that must be solved numerically instead of a linear matrix inversion that can be solved analytically as in Equation 9.8. Moreover, the confidence intervals become more difficult to compute, and they lose their strict probabilistic interpretation as a-level confidence regions. As we will see, however, the approximate confidence intervals remain very useful in nonlinear problems. The numerical challenges for nonlinear models [Pg.596]

The first requirement is generally easily met as the error in the equation solution typically becomes small compared to overall modeling error for moderate values of the error control tolerances. The tradeoff between the second and third requirements is more difficult and depends on the particular numerical characteristics of the system equations and the particular values of the optimization parameters. It is desirable to have some means of estimating and adjusting the precision error to optimize this tradeoff. This requires that the precision error be estimated, its effect on the optimization assessed, and the integrator tolerances adjusted appropriately. [Pg.335]

The log-normal diffusion, log-uniform jump amplitude process has been used in many paper to solve optimal consumption and portfolio optimization and control (Hanson Westman 2002a) (Hanson Westman 2002b). Hanson and Westman (Hanson Westman 2002b) use a yearly decomposition of log-returns in order to estimate the appropriate parameters for their jump-diffusion model. The difference of model parameters between different years is noticeable. A reasonable division of the time domain is proposed according to the volatility behavior of log-return values. The estimation of the model parameters for the three predefined environment states is done using a numerical minimization method (constrained Nelder-Mead) for a least square objective function. The results show that the estimated model is suitable for [Pg.951]

To avoid the errors associated with moments analysis various more sophisticated methods such as Fourier transformation and combinations of Fourier transform and moments methods have been developed. The advantages of such methods have, however, been largely eliminated by the availability of a full analytic solution for the general model (model Ab) in the time domain and by the development of improved numerical techniques which allow the time domain solutions to be calculated rapidly, directly from the model equations. V With these developments the best approach appears to be a combination of the moments method to determine initial estimates of the parameter values coupled with a final optimization by direct matching of the response curves in the time domain. [Pg.247]

The output from the modeling tools is often not used to the extent possible. Often, the FITEQL results used are the numerical values of the optimized parameters and the overall goodness of fit. Sometimes, also the standard deviations are considered. The numerical value of the goodness-of-fit parameter and the standard deviations are dependent on the defined experimental error estimates. The values for these, which are most frequently used, are the standard values, which may not at all be reasonable for the actual equilibrium problem treated [36]. [Pg.649]

The computation of the failnre probability function FfUy ) has to be carried out repeatedly during the optimization process. From the previons formnlation it is clear that the estimator of the failure probability is completely determined by the impnlse response functions, y, i=l,. .., tir, j=l,. .., nf. At the same time, the impnlse response functions depend on the mode shapes (p]r, r=, . ..,n, and the natnral frequencies (Or, r = 1,. ..,n. These qnantities are implicit fnnctions of the vector of design variables [y] and the vector of nncertain strnctnral parameters 9, and they are available only in a numerical way. For systems of practical interest the repeated evalnation of these quantities can be very costly in terms of compntational resonrces. Hence, in order to increase the efficiency of the implementation an approximation strategy is introduced here. [Pg.569]

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