Iterative method, description

Appendix E. Detailed Description of Krotov Iterative Method for Solving the Optimal Control Equations... 

APPENDIX E DETAILED DESCRIPTION OF KROTOV ITERATIVE METHOD FOR SOLVING THE OPTIMAL CONTROL EQUATIONS... 

At this point, it may be of interest to briefly review the status of various types of iterative methods, in the Boolean as well as in the continuous description. 

The situation is rather similar if one applies iterative methods to the Boolean description. As noticed by, for example, Robert39 and by Goles,40 Boolean iterations in parallel and in series correspond, respectively, to the Jacobi and Gauss-Seidel iterations used in the quantitative description. In the first case (Jacobi), from an initial Boolean state (ot0 Jo yo.. . . ) one computest the values of the functions a, b, c,. . . which are reintroduced, respectively, as ot,Pi-y,.. . , and so on in the second case (Gauss-Seidel), the new value of each variable is reintroduced in a defined (but arbitrary) order. 

Having filled in all the elements of the F matr ix, we use an iterative diagonaliza-tion procedure to obtain the eigenvalues by the Jacobi method (Chapter 6) or its equivalent. Initially, the requisite electron densities are not known. They must be given arbitrary values at the start, usually taken from a Huckel calculation. Electron densities are improved as the iterations proceed. Note that the entire diagonalization is carried out many times in a typical problem, and that many iterative matrix multiplications are carried out in each diagonalization. Jensen (1999) refers to an iterative procedure that contains an iterative procedure within it as a macroiteration. The term is descriptive and we shall use it from time to time. 

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

Since the composition of the unknown appears in each of the correction factors, it is necessary to make an initial estimate of the composition (taken as the measured lvalue normalized by the sum of all lvalues), predict new lvalues from the composition and the ZAF correction factors, and iterate, testing the measured lvalues and the calculated lvalues for convergence. A closely related procedure to the ZAF method is the so-called ())(pz) method, which uses an analytic description of the X-ray depth distribution function determined from experimental measurements to provide a basis for calculating matrix correction factors. 

Section II discusses the real wave packet propagation method we have found useful for the description of several three- and four-atom problems. As with many other wave packet or time-dependent quantum mechanical methods, as well as iterative diagonalization procedures for time-independent problems, repeated actions of a Hamiltonian matrix on a vector represent the major computational bottleneck of the method. Section III discusses relevant issues concerning the efficient numerical representation of the wave packet and the action of the Hamiltonian matrix on a vector in four-atom dynamics problems. Similar considerations apply to problems with fewer or more atoms. Problems involving four or more atoms can be computationally very taxing. Modern (parallel) computer architectures can be exploited to reduce the physical time to solution and Section IV discusses some parallel algorithms we have developed. Section V presents our concluding remarks. 

Shown in Figure 8 is essentially the same plot as Figure 7, but one of the expansions is made using the first order theory. This figure shows that although the first order theory is not adequate for an accurate description of the x2 surface even for a 5% variation around the true solution, the first order theory nevertheless results in a solution which is closer to the true solution compared with its starting point and that the method of iteration may be applied to improve the accuracy of the solution. 

LPSVD is a non-iterative linear fitting procedure where all model parameters are estimated in a single step without need of any start values. This makes this method very fast and operator-independent. Disadvantages of this method are the very limited incorporation of prior knowledge and the limitation to exponentially decaying sinusoids. For this reason, LPSVD is not always appropriate for in vivo spectra. A more detailed description of LPSVD can be found in Ref 33. 

SC (simultaneous correction) method. The MESH equations are reduced to a set of N(2C +1) nonlinear equations in the mass flow rates of liquid components ltJ and vapor components and the temperatures 2J. The enthalpies and equilibrium constants Kg are determined by the primary variables lijt vtj, and Tf. The nonlinear equations are solved by the Newton-Raphson method. A convergence criterion is made up of deviations from material, equilibrium, and enthalpy balances simultaneously, and corrections for the next iterations are made automatically. The method is applicable to distillation, absorption and stripping in single and multiple columns. The calculation flowsketch is in Figure 13.19. A brief description of the method also will be given. The availability of computer programs in the open literature was cited earlier in this section. 

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