Adjoint function

Define an adjoint function v,(/) satisfying the differential equation and final condition. 

This section presents perturbation theory expressions and adjoint functions that correspond to the collision probability, flux, birth-rate density, and fission density formulations [see also reference (54)]. The functional relation between different first-order approximations of perturbation theory in integral and in integrodifferential formulations is established. Specifically, the approximation of the integrodifferential formulation that is equivalent, in accuracy, to each of the first-order approximations of the integral theory formulations is identified. The physical meaning of the adjoint functions corresponding to each of the transport theory formulations and their interrelation are also discussed. 

The adjoint function (x) is the total flux of neutrons added ultimately to the critical reactor as a result of a unit flux of neutrons at x. We shall refer to as the flux importance function. ... 

Actually Eq. (50) is a homogeneous equation so that a more general definition would be that (r, E.Q) is proportional to (rather than "is") the total number of neutrons.In the present work we assume a normalization that sets the proportionality constant to unity. Moreover, all the adjoint functions considered in this section could, by a proper normalization, be referred to the same detector. 

Different first-order approximations can be derived from each integral transport theory formulation. These approximations differ in the distribution (density and adjoint) functions they use. 

The development and application of generalized perturbation theory (GPT) has made considerable progress since its introduction by Usachev (i(S). Usachev developed GPT for a ratio of linear flux functionals in critical systems. Gandini 39) extended GPT to the ratio of linear adjoint functionals and of bilinear functionals in critical systems. Recently, Stacey (40) further extended GPT to ratios of linear flux functionals, linear adjoint functionals, and bilinear functional in source-driven systems. A comprehensive review of GPT for the three types of ratios in systems described by the homogeneous and the inhomogeneous Boltzmann equations is given in the book by Stacey (41). In the present review we formulate GPT for composite functionals. These functionals include the three types of ratios mentioned above as special cases. The result is a unified GPT formulation for each type of system. 

Alternatively, the reactivity effect of the flux perturbation can be taken into account in the adjoint space. We define a generalized adjoint function 46)... 

The operator 0S [ = ( — j8) p of Eq. (22)] is the prompt part of the Hssion operator, Defining an a-mode generalized adjoint function... 

The distribution-perturbation formulation of Eq. (156) also holds for inhomogeneous systems when the flux and flux perturbation are the solutions of, respectively, Eqs. (29) and (44). The corresponding adjoint function and its perturbation are the solutions of the equations... 

The adjoint function of Eq. (205) was long ago identified as the Green s function for a detector response (35, 8L 82). Source sensitivity functions have been employed since the mid-sixties (53-55). Equation (205) is an exact expression for SR/R, as long as 5S = = 0. 

The adjoint function appearing in Eq. (233) does not represent a physical quantity. If taken, for example, as representing the group average of the importance function, provide,... 

V can be referred to as a virtual source (63). As the adjoint function of Eq. (18) (or importance-function, when referred to the critical reactor) is usually positive definite [see discussion after Eq. (264)] the virtual source requires both positive and negative components to satisfv the condition ofEq. (253). 

J. Lewins, Importance The Adjoint Function, Pergamon, Oxford, 1965. 

Also the eigenfunctions are not orthogonal to each other, but to a set of adjoint functions v>, satisfying the adjoint equation E — = 0. 

The functions and adjoint functions can then satisfy the condition /i v) = and provide the basis for bilinear expansions, so that... 

The adjoint function and neutron importance. Equation (64) leads to an interesting interpretation of the adjoint function mo which may be used to gain an intuitive understanding of the results of perturbation theory. A neutron may be regarded as having an importance to the chain reaction that is proportional to the amplitude of the persistent mode that it eventually goes into. The adjoint function mo(a ) may then be called the importance of a neutron at a . It is also known as the iterated fission probability. A natural normalization is such that the importance is unity for one neutron distributed in the persistent mode. Equivalently, the average importance is unity for neutrons in the persistent mode. In this normalization... 

As a final comment, the importance interpretation of the adjoint function may be used to write the adjoint equation by similar physical reasoning as is used to write the equation for the neutron density n. Thus the importance of a particular neutron is the same as the total importance of the neutron distribution that results from the original neutron at any later stage in the neutron cycle. Stating this relation in mathematical form results in a correct equation for the adjoint function m. 

If the adjoint function satisfies these equations and boundary conditions, Lis a stationary expression, insensitive to small errors in the density, whose numerical value will yield C. Inspection shows that the Lagrangian has a certain symmetry such that, if N satisfies its equation and boundary conditions, then the Lagrangian is stationary to errors in iV (stationary, in fact, to large errors, since /o and M are not functions of the costate variable). In practice, both equations are perturbed by a change in the control variable and simultaneous errors are made in both functions. For small control perturbations, we anticipate small perturbations in the state and costate variables and that the resulting expression is in error in the cost function only through terms involving the product of small errors. We write... 

The derivation in terms of the Lagrangian and its density adds considerable insight to the nature of the optimum theorem. For example, it is evident from the Lagrangian that the adjoint function N is the weighting given to the source in evaluating the cost function. We shall take up this idea later in a derivation of the adjoint equation from a physical concept. 

The term in Nf dNf vanishes, for it is just the transversality condition imposed as a final boundary condition on the adjoint function such that the Lagrangian should be stationary. For an optimum system, SL is nonnegative. If, as is true with a free end time, St is arbitrary in sign, then Hf must be zero. Being zero at the final time and constant during the control period, the Hamilton density of an autonomous optimum free end time system is identically zero. 

For free end time problems, H (and H ) will be zero on the boundary of restraint, along which the rate of change of the xenon concentration vanishes. We therefore have the solution for the iodine adjoint function by considering... 

In the previous discussion, it will perhaps have become apparent that the generalized Lagrange multiplier or adjoint function plays a significant role in the theory of optimal processes. Furthermore, it becomes as necessary to solve the adjoint or costate equations as the state equations if we are to analyze or synthesize optimal systems. We have also noted that the adjoint functions appear in the Lagrangian as a weighting given to the source density 5. In this section, we shall take up this idea to develop a physical interpretation of the adjoint function which should help us understand its role and perhaps find the adjoint equations, boundary conditions, and even solutions more easily. This physical interpretation as an importance function follows closely the interpretation given to the adjoint function in reactor theory 54). 

Since the adjoint function is to be obtained in physical terms, we shall confine ourselves immediately to an example system, selecting again the xenon shutdown problem. It is clear that the density functions, described... 

As a further example of the way in which the importance concept can help to establish the adjoint function, we consider the problem with state... 

A number of similar results can be established for the energy optimal problem. We may say there is some practical value in this interpretation of the adjoint function, since it quite often happens that the adjoint boundary conditions, etc., arising from the stationary condition are not a complete specification of the adjoint function nevertheless, such a specification may be necessary. 

To demonstrate a connection between dynamic programming and optimum control theory, we suppose that — dCjdN is the adjoint function N. Then dynamic programming takes the form... 

In the b form, limiting ourselves to a time optimal problem for simplicity, the control period is fixed, and the Hamilton density does not now vanish. Since the end of the trajectory is not bound to any particular target curve, we must take both adjoint functions to vanish at the end time if the Lagrangian is to be stationary for arbitrary errors in the density. On the other hand, the cost functional is now the post-shutdown xenon peak, which is determined only by the end state Nf(tf). Thus, the integrand of the cost function has a delta function form ... 

Finite systems, described by differential equations, require some sort of boundary conditions at the edges of the system for full specification. From our point of view, it is sufficient to recognize that they exist and to say that nonzero boundary conditions can be included in the form of Eq. (87) by means of a suitable delta function. We must also introduce a distributed (generalized) Lagrange multiplier or adjoint function, N (x, t). 

For this reason, it is necessary to impose adjoint boundary conditions on the adjoint function 54). 

J. Lewins, Importance the Adjoint Function. PergamonPress,Oxford(1965). 

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