Equations, adjoint integral

The first one is based on a classical variation method. This approach is also known as an indirect method as it focuses on obtaining the solution of the necessary conditions rather than solving the optimization directly. Solution of these conditions often results in a two-point boundary value problem (TPBVP), which is accepted that it is difficult to solve [15], Although several numerical techniques have been developed to address the solution of TPBVP, e.g. control vector iteration (CVI) and single/multiple shooting method, these methods are generally based on an iterative integration of the state and adjoint equations and are usually inefficient [16], Another difficulty relies on the fact that it requires an analytical differentiation to derive the necessary conditions. 

The adjoint system approach requires integration of the model equations forward in time before integration of the adjoint system equations backward. 

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... 

With the results of the forward integration of the model Equation 5.4 and then by integrating backward the adjoint equations (Equation 5.18) the gradients can be determined from ... 

See Sargent and Sullivan (1979), Morison (1984) and Rosen and Luus (1991) for further details. The adjoint approach has the advantage that, in addition to the adjoint equations (Equation 5.18), only one extra equation (Equation 5.20), has to be integrated for each of the NLP optimisation variables. It is especially useful for... 

The accelerated gradient method is used because of its advantages especially when the control is constrained. The system and its adjoint equations are coupled hyperbolic partial differential equations. They can be solved numerically using the method of characteristics (Lapidus, 1962b Chang and Bankoff, 1969). This method is used with the fourth order Runge-Kutta method (with variable step size to ensure accuracy of the integration) to solve the state and adjoint equations. 

Integrate the state equations forward and the adjoint equations backward. This procedure gives the current value of the objective function, F =... 

There is a basic difficulty with this boundary condition iteration approach which often arises in practical problems the numerical integration of the equations in the same direction is very often numerically unstable. The reason for this behaviour is that the state equations are usually stable when integrated forward but are unstable in the reverse direction. Similarily, the adjoint equations are usually unstable when integrated forward, but are stable in the reverse direction. This can cause great numerical difficulties which are quite independent of the choice of the proper boundary conditions. Nevertheless, the method has been used successfully for some problems. 

Multiplying equations (D.lO), (D.ll) (D.12) and (D.13) by the adjoint variables 2, A2, A3 and //, respectively, then integrating them over z, T and combining the resulting equations with equation (D.14), gives after some rearrangement ... 

We noted that the sensitivity equations are linear. This property enables a computational approach based on adjoint operators (Koda and Seinfeld, 1982). Adjoint operators are important because they enable differential equations to be expressed as exact differentials, which can be integrated by direct quadrature. To see how adjoint operators can be used in sensitivity analysis, return to the basic model equation (4.A.3). Consider a perturbation in... 

The formulations presented in this section are for the calculation of perturbed fluxes. Analogous formulations can be derived and applied for the calculation of perturbed adjoints, kernels for the integral transport equations (see Section IV), generalized functions (see Section V), and other distribution-function perturbations. The presentation is restricted to exposition of the general formulation of the methods, without considering the technical details of the solution. 

In the collision density, the most commonly used integral transport formulation, the flux and adjoint equations can be written as follows ... 

The Usachev Gandini derivation of GPT is based on physical considerations. Their formulation is applicable to alterations that leave the reactor critical. Stacey (40) derived GPT from variational principles. His GPT formulations are also applicable to perturbations that change the static eigenvalue of the Boltzmann equation that is, that do not preserve criticality. The approach used in this work for deriving GPT expressions is neither that of the variational, nor of the physical consideration. It uses conventional perturbation techniques combined with the flux-difference and adjoint-difference methods (see Section III,B). A third version of GPT is presented in this work. Like Stacey s this new version is applicable to perturbations that do not preserve criticality. It pertains, however, to integral parameters that are related to the prompt-mode rather than to the static eigenfunctions. At the end of this section we discuss the relation between... 

The adjoint source in Eq. (259) represents some detector distribution which, when integrated over the reactor space (after weighting with the neutron distribution), yields a single measure of the reactor behavior. Physically, there is no reason why such a detector distribution should be positive definite. The existence condition [Eq. (260)] for a solution to Eq. (259) in a critical system, however, as in the neutron balance equation, requires the orthogonality of and V and since is (physically) positive definite, must be part positive, part negative. Physically, this orthogonality condition, therefore, expresses the idea that the acceptable virtual adjoint sources are detector distributions which, when reacting to the distribution of neutrons in the fundamental mode, lead to no total or net effect. 

The optimal control problem represents one of the most difficult optimization problems as it involves determination of optimal variables, which are vectors. There are three methods to solve these problems, namely, calculus of variation, which results in second-order differential equations, maximum principle, which adds adjoint variables and adjoint equations, and dynamic programming, which involves partial differential equations. For details of these methods, please refer to [23]. If we can discretize the whole system or use the model as a black box, then we can use NLP techniques. However, this results in discontinuous profiles. Since we need to manipulate the techno-socio-economic poHcy, we can consider the intermediate and integrated model for this purpose as it includes economics in the sustainabiHty models. As stated earlier, when we study the increase in per capita consumption, the system becomes unsustainable. Here we present the derivation of techno-socio-economic poHcies using optimal control appHed to the two models. 

The neutron lifetime I and effective delay fraction pi for delay group may be computed directly from Equations (40) and (41) after solving for no and mo. Alternately, they may be evaluated without resorting to the calculation of the adjoint or to the integrations involved. Thus, a perturbation comparing the standard no reactor satisfying... 

As a historical note, this distinction was not well understood until quite recently. Thus, Davison (13, p. 32) could say we shall not discuss the difficult problem of the general significance of that (adjoint) solution, since his particular solution was obtained from the integral equation introducing just such an operator. The point was clarified by Robkin and Clark (14) for the transport equation and its general effect discussed elsewhere (10). 

If we assume that F is a positive definite operator (and this is the case, on physical grounds, for production from fission) and that and N are nonnegative, then some normalization of just offsets the effect of on the lowest mode of the adjoint equation and makes Eq. (56) acceptable. In other words, we must filter out the lowest source mode the system is subcritical in higher modes and can therefore accept the higher modes of the source. The necessary normalization of this negative source term is easily expressed analytically by multiplying by N and integrating ... 

Important Note The iterative solution for this two-point boundary-value differential equation utilizing the adjoint equations is shown in Figure 4.13. Notice that this procedure may be unstable numerically and we may need to reverse the direction of the integration from the exit of the reactor toward the inlet as explained in the following subsection. 

We will integrate the adjoint equations given by equation (7.7.2) backwards in time with assumed final conditions. We perform this backward integration m times, where m is the number of states specified at the... 

This equation can either be integrated forward in time with the state equations or backward in time w ith the adjoint equations. The choice of the backward in time integration is more efficient since we do not have to store the values of p2- Instead y can be computed simultaneously while computing the adjoint variables. 

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