Objective function integration

Secondly, the linearized inverse problem is, as well as known, ill-posed because it involves the solution of a Fredholm integral equation of the first kind. The solution must be regularized to yield a stable and physically plausible solution. In this apphcation, the classical smoothness constraint on the solution [8], does not allow to recover the discontinuities of the original object function. In our case, we have considered notches at the smface of the half-space conductive media. So, notche shapes involve abrupt contours. This strong local correlation between pixels in each layer of the half conductive media suggests to represent the contrast function (the object function) by a piecewise continuous function. According to previous works that we have aheady presented [14], we 2584... 

Minimization of S(k) can be accomplished by using almost any technique available from optimization theory, however since each objective function evaluation requires the integration of the state equations, the use of quadratically convergent algorithms is highly recommended. The Gauss-Newton method is the most appropriate one for ODE models (Bard, 1970) and it presented in detail below. 

Having the smoothed values of the state variables at each sampling point, Xj and the integrals, P(t,), we have essentially transformed the problem to a "usual" linear regression problem. The parameter vector is obtained by minimizing the following LS objective function... 

If we wish to avoid the additional objective function evaluation at p=qa/2, we can use the extra information that is available at p=0. This approach is preferable for differential equation models where evaluation of the objective function requires the integration of the state equations. It is presented later in Section 8.7 where we discuss the implementation of Gauss-Newton method for ODE models. 

It should be emphasized here that it is unnecessary to integrate the state equations for the entire data length for each value of u. Once the objective function becomes greater than S(k0)). a smaller value for p can be chosen. By this procedure, besides the savings in computation time, numerical instability is also avoided since the objective function often becomes large very quickly and integration is stopped well before computer overflow is threatened. 

The main difference with differential equation systems is that every evaluation of the objective function requires the integration of the state equations, In this section we present an optimal step size policy proposed by Kalogerakis and Luus (1983b) which uses information only at g=0 (i.e., at k ) and at p=pa (i.e., at... 

Note in Table 5.10 that many of the integrals are common to different kinetic models. This is specific to this reaction where all the stoichiometric coefficients are unity and the initial reaction mixture was equimolar. In other words, the change in the number of moles is the same for all components. Rather than determine the integrals analytically, they could have been determined numerically. Analytical integrals are simply more convenient if they can be obtained, especially if the model is to be fitted in a spreadsheet, rather than purpose-written software. The least squares fit varies the reaction rate constants to minimize the objective function ... 

A computer can do only three things add, subtract, and decide whether some value is positive, negative, or zero. The last capacity allows the computer to decide which of two alternatives is best when some quantitative objective function has been selected. The ability to add and subtract permits multiplication and division, plus the approximation of integration and differentiation. 

A mathematical formulation based on uneven discretization of the time horizon for the reduction of freshwater utilization and wastewater production in batch processes has been developed. The formulation, which is founded on the exploitation of water reuse and recycle opportunities within one or more processes with a common single contaminant, is applicable to both multipurpose and multiproduct batch facilities. The main advantages of the formulation are its ability to capture the essence of time with relative exactness, adaptability to various performance indices (objective functions) and its structure that renders it solvable within a reasonable CPU time. Capturing the essence of time sets this formulation apart from most published methods in the field of batch process integration. The latter are based on the assumption that scheduling of the entire process is known a priori, thereby specifying the start and/or end times for the operations of interest. This assumption is not necessary in the model presented in this chapter, since water reuse/recycle opportunities can be explored within a broader scheduling framework. In this instance, only duration rather start/end time is necessary. Moreover, the removal of this assumption allows problem analysis to be performed over an unlimited time horizon. The specification of start and end times invariably sets limitations on the time horizon over which water reuse/recycle opportunities can be explored. In the four scenarios explored in... 

A uneven discretization of time mathematical formulation for direct heat integration of multipurpose batch plants has been presented. The formulation results in smaller problems compared to the discrete-time formulation, which renders it applicable to large-scale problems. Application of the formulation to an industrial case study showed an 18.5% improvement in objective function for the heat-integrated scenario relative to the standalone scenario. 

The economic values in the objective function are treated and structured from an operations research perspective as variables calculated bottom up considering underlying volume decision variables. These result variables are integrated in the model to make the objective function more readable and easier to communicate to stakeholders such as planners, top-management, marketing and/or controlling. 

Heinzle, E., Weirich, D., Brogli, F., Hoffmann, V., Roller, G., Verduyn, M.A. and Hungerbuhler, K. (1998) Ecological and Economic Objective Functions for Screening in Integrated Development of Fine Chemical Processes. 1. Flexible and Expandable Framework Using... 

Khogeer (2005) developed an LP model for multiple refinery coordination. He developed different scenarios to experiment with the effect of catastrophic failure and different environmental regulation changes on the refineries performance. This work was developed using commercial planning software (Aspen PIMS). In his study, there was no model representation of the refineries systems or clear simultaneous representation of optimization objective functions. Such an approach deprives the study of its generalities and limits the scope to a narrow application. Furthermore, no process integration or capacity expansions were considered. 

In this final chapter, we study the multisite refinery and petrochemical integration problem, explained in Chapter 5, under uncertainty. The randomness considered includes both the objective function and right-hand side parameters of inequality constraints. As pointed out in the previous chapters, considering such strategic planning decisions requires proper handling of uncertainties as they play a maj or role in the final decision making process. 

You are certainly aware that the compartmental model is a simplified representation of the real physicological process. Therefore, it is completely adequate to use a simplified objective function by approximating the integrals in (2.17). Me divide the interval [ , t] of integration into NM equal subintervals of length At = /NM, and approximate c(t) by its midpoint value c = c[(i - 1/2)At]. The objective function is approximated by... 

Before listing the output, recall that the objective function to be minimized is based on the approximate response (5.60). The minimum of this function is 1.01B602, whereas solving the differential equation (5.63) at the final estimate of the parameters gives the value 1.060729. The direct integral estimares are acceptable only if these two values do not significantly differ, see (ref. IB). 

The rate expressions Rj — Rj(T,ck,6m x) typically contain functional dependencies on reaction conditions (temperature, gas-phase and surface concentrations of reactants and products) as well as on adaptive parameters x (i.e., selected pre-exponential factors k0j, activation energies Ej, inhibition constants K, effective storage capacities i//ec and adsorption capacities T03 1 and Q). Such rate parameters are estimated by multiresponse non-linear regression according to the integral method of kinetic analysis based on classical least-squares principles (Froment and Bischoff, 1979). The objective function to be minimized in the weighted least squares method is... 

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