Outer iteration convergence

Modification of Describing Equations. In the simple K-value and enthalpy models just described, there are Ng+6 parameters, a, A, B, C, D, E and F, which have characteristics that make them excellent choices for the iteration variables of an outer iteration loop. This is in fact the essence of the inside-out concept. Its success, however, rests on the ability to transform and rearrange the describing equations to properly accommodate these variables. The result should be an efficient and well-behaved inner iteration loop in which values of the primitive variables - now regarded as dependent variables - are calculated. When the inside loop is converged, the actual K-value and enthalpy models can be employed to calculate new values of the simple model parameters. 

There are two "do loops" in each of these equilibrium calculations. When the unknown is either temperature or pressure, the inner "do loop" of 30 iterations converges on the unknown phase compositions and the outer "do loop" of 50 iterations... 

DEW P. The calculational scheme here is shown in Fig. 12.13. We read and store T and yk, along with appropriate constants. Since we can calculate neither the d> nor the yk, all values of each are set equal to unity. Values of P are found from the Antoine equation, and Eqs. (12.29) and (12.27) are then solved for P and xk. Evaluation of y now allows recalculation of P by Eq. (12.29). With this rather good estimate of P, we evaluate and enter an inner iteration loop that converges on values for xk and (yj. Subsequent recalculation of P by Eq. (12.29) leads to the outer iteration loop that establishes the final value of P. Since the xk calculated within the inner loop are not constrained to sum to... 

For unsteady flows the system of non-linear equations are linearized in the iteration process within each time step, since all the solvers are limited to linear systems. The iterative process is thus performed on two different levels. The solver iterations are performed on provisional linear systems with fixed coefficients and source terms until convergence. Then, the system coefficients and sources are updated based on the last provisional solution and a new linearized system is solved. This process is continued until the non-linear system is converged, meaning that two subsequent linear systems give the same solution within the accuracy of a prescribed criterion. A standard notation used for the different iterations within one time step is that the coefficient and source matrices are updated in the outer iterations, whereas the inner iterations are performed on provisionally linear systems with fixed coefficients. On each outer iteration, the equations solved are on the form ... 

Figure 7.7 shows the convergence of the penalty function method. At each outer iteration r, the plotted objective functional value is the optimal with corresponding constraint violation quantified as q. It was 17.2 at the very beginning (corresponding to the initial controls) and dropped finally to 5.9 x 10 at the convergence, which was attained in 11 outer iterations. 

The constraint violation in terms of q was 1182 at the very beginning with initial controls and converged finally to 1.1 x 10 in five outer iterations. Upon convergence, the final values of yi, y, and 2/3 were... 

With initial controls, the constraint violation in terms of q was 55.7, which reduced and converged to 3.3x10 in 10 outer iterations. At convergence, the optimal objective functional was —4.45, which corresponds to the final product concentration of 4.45 g/cm. The optimal final time reduced from 60 to 33.3 min. 

Interesting mathematical problems arise in the convergence of the outer iteration where it is desirable to use some procedure to accelerate the convergence. We can write the pth iterative solution of the multigroup equation in the form... 

Other methods for accelerating the convergence of the outer iterations exist, and are interesting in their own right. With the definition of the matrices in 3, we now write our discrete time independent eigenvalue problem in the matrix form... 

As a third method for accelerating the convergence of the outer iterations, we consider an iterative method which stems from the work of Sheldon [32]. From (5.12), we have... 

In the previous subsections, we developed Newton s method for the optimization of Cl wave functions and energies. Newton s method requites only a few macro (outer) iterations, but, in each macro iteration (i.e. in each Newton step), a relatively large number of micro (inner) iterations are needed for solving the linear equations (11.5.3). Each micro iteration requires the multiplication of the Hamiltonian matrix by a trial vector (11.5.1). The total number of micro iterations needed for convergence may therefore become quite large with Newton s method. 

The optimization an MCSCF wave function is carried out as a sequence of inner and outer iterations. The outer iterations are those discussed in Sections 12.3 and 12.4 - the individual iterations of the second-order methods carried out to converge the wave function. The inner iterations are those carried out to solve the linear equations (12.5.1) or the eigenvalue equations (12.5.2). The inner and outer iterations are also referred to as micro and macro iterations. 

To estimate the maximum error R that will not impede the convergence of the outer iterations, we recall that the second-order convergence of the exact solution X is characterized by the relationship ... 

Let us first consider the calculations on the H2 system. In Table 12.2, the MP2 natural orbitals are used in the first iteration in Table 12.3, we use the canonical Hartree-Fock orbitals. Because of the different choices of orbitals, the optimizations proceed rather differently. For the optimization based on the MP2 natural orbitals, the optimization begins in the local region each step corresporxls to a Newton step with no step-length restrictions. Quadratic convergence is therefore observed in all outer iterations - see the reduction in the gradient and step norms in Table 12.2. The ratio parameter r (12.3.21), which probes the quadratic dominance of the energy function, is close to 1 in all iterations. 

At this point in the inside-out method, the revised column profiles of temperature and phase compositions are used in the outer loop with the complex SRK thermodynamic models to compute updates of the approximate K and H constants. Then only one inner-loop iteration is required to obtain satisfactory convergence of the energy equations. The K and H constants are again updated in the outer loop. After one inner-loop iteration, the approximate K and H constants are found to be sufficiently close to the SRK values that overall convergence is achieved. Thus, a total of only 3 outer-loop iterations and 4 inner-loop iterations are required. 

To illustrate the efficiency of the inside-out method to converge this example, the results from each of the three outer-loop iterations are summarized in the following tables ... 

From these tables, it is seen that the stage temperatures and total liquid flows are already close to the converged solution after only one outer-loop iteration. However, the composition of the bottoms product, specifically with respect to the lightest component, C, is not close to the converged solution until after two iterations. The inside-out method does not always converge so dramatically, but is usually quite efficient,... 

However, this is in general a quite impractical method, as it requires the iterative series of dress-then-diagonalize steps to get convergence for each individual state. Moreover, even if no particular numerical problems arise in the case of states that are not dominated by one particular excitation, the method does not seem to be well-adapted from a formal point of view to such cases because they do not satisty the condition stated in eq. (2). Notwithstanding, this procedure can be practical for the calculation of outer-valence ionization potentials of closed-shell molecules. In such cases, one must to deal with the doublet states of the cation that are well dominated by a unique Koopmans determinant. 

The basic idea in OA is similar to the one in GBD that is, at each iteration we generate an upper bound and a lower bound on the MINLP solution. The upper bound results from the solution of the problem which is problem (6.13) with fixed y variables (e.g., y = yk). The lower bound results from the solution of the master problem. The master problem is derived using primal information which consists of the solution point xk of the primal and is based upon an outer approximation (linearization) of the nonlinear objective and constraints around the primal solution xk. The solution of the master problem, in addition to the lower bound, provides information on the next set of fixed y variables (i.e., y = yt+ ) to be used in the next primal problem. As the iterations proceed, two sequences of updated upper bounds and lower bounds are generated which are shown to be nonincreasing and nondecreasing respectively. Then, it is shown that these two sequences converge within e in a finite number of iterations. 

For the reformer we assume that the outer wall temperature profile of the reformer tubes decouples the heat-transfer equations of the furnace from those for the reformer tubes themselves. The profile is correct when the heat flux from the furnace to the reformer tube walls equals the heat flux from the tube walls to the reacting mixture. We must carry out sequential approximating iterations to find the outer wall temperature profile Tt,o that converges to the specific conditions by using the difference of fluxes to obtain a new temperature profile T) o for the outer wall and the sequence of calculations is then repeated. In other words, a T) o profile is assumed to be known and the flux Q from the furnace is computed from the equations (7.136) and (7.137), giving rise to a new Tt o-This profile is compared with the old temperature profile. We iterate until the temperature profiles become stationary, i.e., until convergence. 

The one level optimal control formulation proposed by Mujtaba (1989) is found to be much faster than the classical two-level formulation to obtain optimal recycle policies in binary batch distillation. In addition, the one level formulation is also much more robust. The reason for the robustness is that for every function evaluation of the outer loop problem, the two-level method requires to reinitialise the reflux ratio profile for each new value of (Rl, xRI). This was done automatically in Mujtaba (1989) using the reflux ratio profile calculated at the previous function evaluation in the outer loop so that the inner loop problems (specially problem P2) could be solved in a small number of iterations. However, experience has shown that even after this re-initialisation of the reflux profile sometimes no solutions (even sub-optimal) were obtained. This is due to failure to converge within a maximum limit of function evaluations for the inner loop problems. On the other hand the one level formulation does not require such re-initialisation. The reflux profile was set only at the beginning and a solution was always found within the prescribed number of function evaluations. 

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