Equations, adjoint

It was reported that the convergence of the Krotov iteration method [81, 93] was four or five times faster than that of the gradient-type methods. The formulation of Rabitz and others, [44, 45, 92], designed to improve the convergence of the above algorithm, introduces a further nonlinear propagation step into the adjoint equation (i.e., the equation for the undetermined Lagrange multiplier % t)) and is expressed as... 

Since the wave function may be interpreted as a column vector with four components, may be defined to be a row matrix with components and which satisfies the adjoint equation... 

Multiplying the equation on the left by and the adjoint equation on the right by and taking the difference of the two results, it is found that... 

These methods are efficient for problems with initial-value ODE models without state variable and final time constraints. Here solutions have been reported that require from several dozen to several hundred model (and adjoint equation) evaluations (Jones and Finch, 1984). Moreover, any additional constraints in this problem require a search for their appropriate multiplier values (Bryson and Ho, 1975). Usually, this imposes an additional outer loop in the solution algorithm, which can easily require a prohibitive number of model evaluations, even for small systems. Consequently, control vector iteration methods are effective only when limited to the simplest optimal control problems. 

The density rt M is closely related to the Green s function, GN lll, which satisfies the self-adjoint equation... 

The most frequent use, however, of the adjoint equation is in connection with absorbing boundaries and first-passage problems, see chapter XII. 

Hint To avoid excessive calculations use the adjoint equation.] Compare (IX.4.14). 

The approach through the adjoint equation - Discrete case... 

The less ambitious approach in 2 through the adjoint equation aims only at the splitting probabilities and the mean first-passage times. We construct an equation analogous to (2.2) and (3.1). If at time t the value of Y equals x, it will have at t + At another value x with probability At JT(x x), or it has the same value x. The probability 7rK(x) that Y, starting at x, will exit through R obeys therefore the identity... 

This leads to the adjoint equation satisfied by G in the source coordinates... 

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. 

In standard quantum field theory, particles are identified as (positive frequency) solutions ijj of the Dirac equation (p — m) fj = 0, with p = y p, m is the rest mass and p the four-momentum operator, and antiparticles (the CP conjugates, where P is parity or spatial inversion) as positive energy (and frequency) solutions of the adjoint equation (p + m) fi = 0. This requires Cq to be linear e u must be transformed into itself. Indeed, the Dirac equation and its adjoint are unitarily equivalent, being linked by a unitary transformation (a sign reversal) of the y matrices. Hence Cq is unitary. 

The Hamiltonian, the adjoint equations and the optimal reflux ratio correlation will be same as those in Equations P.10-P.13 (Diwekar, 1992). However, note that the final conditions (stopping criteria) for the minimum time and the maximum distillate problems are different. The stopping criterion for the minimum time problem is when (D, xq) is achieved, while the stopping criterion for the maximum distillate problem is when t, xo) is achieved. See Coward (1967) for an example problem. 

Similar to those presented in section 5.5 the adjoint equations for the model represented by Equation 5.4 can be written as ... 

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... 

While Huang and Ozisik solved the spacewise variation of wall heat flux for laminar forced convection problem, Silva Neto and Ozisik [57] used the conjugate gradient method and the adjoint equation simultaneously to solve for the timewise-varying strength of a two plane heat source. 

Results from this model were verified by Neupauer and Wilson [51] using the adjoint method. In this method, the forward governing equation, with concentration as the dependent variable, is replaced by the adjoint equation, with the adjoint state as the dependent variable. They showed that backward-in-time location and travel time probabilities are adjoint states of the forward-in-time resident concentration. In this and the follow-up paper, Neupauer and Wilson [51,52] presented the adjoint method as a formal framework for obtaining the backward-in-time probabilities for multidimensional problems and more complex domain geometries. 

A combination of the forward and inverse modelling approaches allows to solve some environmental and nuclear risk problems more effectively compared with the traditional ways based on the forward modelling. For the inverse modelling problem, most of the western scientists (Persson et al., 1987 [491] Prahm et al., 1980 [509] Seibert, 2001 [569]) use the common back- trajectory techniques, suitable only for the Lagrangian models. The Novosibirsk scientific school established by G.I. Marchuk in Russia has suggested a fruitful theoretical method for inverse modelling, based on adjoint equations (Marchuk, 1982 [391], 1995 [392] Penenko, 1981 [486]) and suitable for the Eulerian models. This approach has further been used and improved by several authors (Baklanov, 1986 [20], 2000 [25] Pudykiewicz, 1998 [512] Robertson and Lange, 1998 [538]) for estimation of source-term parameters in the atmospheric pollution problems. 

Adjoint equations with an original algorithm of illumination / smoothing of measurement functions are considered in (Issartel, 2003 [309]). 

Marchuk, G.I. (1995) Adjoint equations and analysis of complex systems, Kluber Academic Publication. 

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