Iterative Problem Solving Strategy

Depending on the space discretisation techniques used, the set of equations to be solved may be different, but for FD- and FV- based methods, the discretisation results in a set of linear or non-linear algebraic equations. These depend on the nature of these partial differential equations and how they are derived. For linear equations, it is well known that a Gauss elimination method can be used as a basic method to solve them. Further details of the Gauss method can be found in [60], 

Assume that a solution after the nth iteration is represented as M which contains a set of values of interest (e.g. temperature, velocity, etc.) for all elements 

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The classic methods use an ODE solver in combination with an optimization algorithm and solve the problem sequentially. This solution strategy is referred to as a sequential solution and optimization approach, since for each iteration the optimization variables are set and then the differential equation constraints are integrated. Though straightforward, this approach is generally inefficient because it requires the accurate solution of the model equations at each iteration within the optimization, even when iterates are far from the final optimal solution. 

The indeterminacy of the 2-CSE had caused this equation to be overlooked for many years. In 1992 Valdemoro proposed a method to approximate the 2-RDM in terms of the 1-RDM [108], which was extended in order to approximate the 3- and 4-RDMs in terms of the lower-order matrices [46, 47] and in 1994 Colmenero and Valdemoro [48] applied these approximate constructing algorithms to avoid the indeterminacy problem and solve iteratively the 2-CSE. Since then, the smdy of improved constructing algorithms as well as alternative strategies in the iterative procedure have been proposed [1,6, 15, 18, 36, 49-57, 59, 60, 62-65, 68, 70, 71, 79-85, 109-111]. Also, good results of several calculations have been reported [6, 49-52, 55, 56, 68, 70, 88, 109, 111]. 

Several attempts have been made to devise simpler optimization methods than the lull second order Newton-Raphson approach. Some are approximations of the full method, like the unfolded two-step procedure, mentioned in the preceding section. Others avoid the construction of the Hessian in every iteration by means of update procedures. An entirely different strategy is used in the so called Super - Cl method. Here the approach is to reach the optimal MCSCF wave function by annihilating the singly excited configurations (the Brillouin states) in an iterative procedure. This method will be described below and its relation to the Newton-Raphson method will be illuminated. The method will first be described in the unfolded two-step form. The extension to a folded one-step procedure will be indicated, but not carried out in detail. We therefore assume that every MCSCF iteration starts by solving the secular problem (4 39) with the consequence that the MC reference state does not... 

In contrast to the sequential solution method, the simultaneous strategy solves the dynamic process model and the optimization problem at one step. This avoids solving the model equations at each iteration in the optimization algorithm as in the sequential approach. In this approach, the dynamic process model constraints in the optimal control problem are transformed to a set of algebraic equations which is treated as equality constraints in NLP problem [20], To apply the simultaneous strategy, both state and control variable profiles are discretized by approximating functions and treated as the decision variables in optimization algorithms. 

The aim of the optimisation is to determine the spatial pattern of retrofit flue gas desulphurisation FGD which, for a given total installed capacity of abatement, minimises the magnitude of the difference between the deposition loads and the critical loads for total sulphur deposition at the receptor sites. Such differences between deposition loads and ciritical loads are termed critical load exceedences. For a near continuous distribution of emission controls at about 50 power stations and 11 receptor sites, there are a large number of possible strategies to work through in an exhaustive analysis. The problem was solved using optimisation by simulated annealing [8-10], a specialised iterative improvement technique. 

Equations 17.18 through 17.28 may be solved using different strategies, including simultaneous or iterative methods. Distefano (1968) describes a computational procedure that solves the equations one at a time. Boston et al. (1981) describe a more flexible and efficient method that can handle a variety of specifications and column configurations, including multiple feeds, side draws, and side heaters. The method uses advanced numerical procedures to handle nonlinearity and stability problems. The object here is not to describe the details of the methods, which may be found in the references. The Distefano method is outlined here to illustrate briefly some of the potential numerical computational problems. 

As foreshadowed in Section 9.7.1, the iterative strategy is based on an outer iteration on discrete local current densities. For given current densities, the coupled initial value problem for the channel fluxes and temperatures are solved by marching, with some terms handled implicitly. Details of the iterative strategy are given below. 

An important aspect of the solution strategy is that channel saturation states are not changed during the quasi-Newton channel iterations. The iterations proceed to convergence, and only then are the saturation states changed if needed. In this way, only smooth problems are solved, and faster and more robust convergence using quasi-Newton methods can be obtained. 

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