Iterative approach

Joo T and Albrecht A C 1993 Inverse transform in resonance Raman scattering an iterative approach J. Phys. Chem. 97 1262-4... 

An alternative to split operator methods is to use iterative approaches. In these metiiods, one notes that the wavefiinction is fomially "tt(0) = exp(-i/7oi " ), and the action of the exponential operator is obtained by repetitive application of //on a function (i.e. on the computer, by repetitive applications of the sparse matrix... 

Iterative approaches, including time-dependent methods, are especially successfiil for very large-scale calculations because they generally involve the action of a very localized operator (the Hamiltonian) on a fiinction defined on a grid. The effort increases relatively mildly with the problem size, since it is proportional to the number of points used to describe the wavefiinction (and not to the cube of the number of basis sets, as is the case for methods involving matrix diagonalization). Present computational power allows calculations... 

Presented in Table 1 is a list of the parameters in Eqs. (2) and (3) and the type of target data used for their optimization. The infonnation in Table 1 is separated into categories associated with those parameters. It should be noted that separation into the different categories represents a simplification in practice there is extensive correlation between the different parameters, as discussed above for example, changes in bond parameters that affect the geometry may also have an influence on AGsoivation for a given model compound. These correlations require that parameter optimization protocols include iterative approaches, as will be discussed below. 

Eor many problems, the ideal umbrella potential would be one that completely flattens the free energy profile along q, i.e., UXq) = W(q). Such a potential cannot be determined in advance. However, iterative approaches exist that are known as adaptive... 

The maximum stress or strain is not specified so an iterative approach is needed. From the 1 year isochronous for PP the initial modulus is 370 MN/m2... 

The use of dendritic cores in star polymer synthesis by NMP, ATRP and RAFT polymerization was mentioned in Section 9.9.1, In this section wc describe the synthesis of multi-generation dendritic polymers by an iterative approach. 

The RWP method also has features in common with several other accurate, iterative approaches to quantum dynamics, most notably Mandelshtam and Taylor s damped Chebyshev expansion of the time-independent Green s operator [4], Kouri and co-workers time-independent wave packet method [5], and Chen and Guo s Chebyshev propagator [6]. Kroes and Neuhauser also implemented damped Chebyshev iterations in the time-independent wave packet context for a challenging surface scattering calculation [7]. The main strength of the RWP method is that it is derived explicitly within the framework of time-dependent quantum mechanics and allows one to make connections or interpretations that might not be as evident with the other approaches. For example, as will be shown in Section IIB, it is possible to relate the basic iteration step to an actual physical time step. 

It is beyond the scope of this book to go though all the specifics of catalyst testing and to discuss all pitfalls that may arise. Instead we list the Ten Commandments for the Testing of Catalysts. This is a set of guidelines that have been provided by experts of a company called Catalytica [F.M. Dautzenberg in Characterization of Catalyst Development An Iterative Approach (Eds. S.A. Bradley, M.J. Gattuso, R.J. Ber-tolacini), ACS Symposium Series, Vol. 411 (1989)]. 

One of the drawbacks of the first iteration, however, is that computation of energy quantities, e.g. orbital and total energies, requires to evaluate the integrals occurring in Eq. 3 on the basis of the ( )il )(p)- Unfortunately, the transcendental functions in terms of which the (]>il Hp) are expressed at the end of the first iteration do not lead to closed form expressions for these integrals and a numerical procedure is therefore needed. This constitutes a barrier to carry out further iterations to improve the orbitals by approaching the HE limit. A compromise has been proposed between a fully numerical scheme and the simple first iteration approach based on the fact that at the end of each iteration the < )j(k)(p) s entail the main qualitative characteristics of the exact solution and most... 

If we consider the limiting case where p=0 and q O, i.e., the case where there are no unknown parameters and only some of the initial states are to be estimated, the previously outlined procedure represents a quadratically convergent method for the solution of two-point boundary value problems. Obviously in this case, we need to compute only the sensitivity matrix P(t). It can be shown that under these conditions the Gauss-Newton method is a typical quadratically convergent "shooting method." As such it can be used to solve optimal control problems using the Boundary Condition Iteration approach (Kalogerakis, 1983). 

Furthermore, iterative approaches are useful methods to construct polyhydroxy chains with 1,2- or 1,3-diol units of any length as chiral precursors for the synthesis of complex natural products [57] because automated synthesis becomes feasible. A preparation of trans-fused polytetrahydropyranes as structural unit for polycyclic ether biotoxines by repeated reaction sequences was recently named reiterative synthesis [58]. 

The measured value of k Sg is 0.716 cm3/(sec-g catalyst) and the ratio of this value to k ltTueSg should be equal to our assumed value for the effectiveness factor, if our assumption was correct. The actual ratio is 0.175, which is at variance with the assumed value. Hence we pick a new value of rj and repeat the procedure until agreement is obtained. This iterative approach produces an effectiveness factor of 0.238, which corresponds to a differs from the experimental value (0.17) and that calculated by the cylindrical pore model (0.61). In the above calculations, an experimental value of eff was not available and this circumstance is largely responsible for the discrepancy. If the combined diffusivity determined in Illustration 12.1 is converted to an effective diffusivity using equation 12.2.9, the value used above corresponds to a tortuosity factor of 2.6. If we had employed Q)c from Illustration 12.1 and a tortuosity factor of unity to calculate eff, we would have determined that rj = 0.65, which is consistent with the value obtained from the straight cylindrical pore model in Illustration 12.2. 

An iterative approach involving coupling reactions of living anionic polymers followed by functionalization, leads to three generation homo- and block copolymers. [208]. The reactions used are shown in Scheme 115. 

In the iterative approach using a constant bias F, the free energy barrier is reduced each successive iteration and therefore produces complete sampling of important configurations along the coupling parameter (i.e., reaction coordinates). Furthermore, more complicated umbrella potentials (e.g. see Equation 21) may also be applied with the iterative procedure. 

Fig. 4.1. Newton s method for solving a nonlinear equation with one unknown variable. The solution, or root, is the value of x at which the residual function R(x) crosses zero. In (a), given an initial guess. vl0,), projecting the tangent to the residual curve to zero gives an improved guess v( l ). By repeating this operation (b), the iteration approaches the root.

In the absence of detailed structural information about GPCRs, much of the efforts to interpret experimental results in a structural context has focused on creating molecular representations of these proteins that can incorporate directly and consistently the many types of function-related information (for a recent review, see ref. [5]). In turn, such molecular models serve as hypotheses-generators for experimental probing of functional inferences, and are continuously refined by the data obtained from such experiments. Listed below are some of the main advantages of such an iterative approach, as illustrated in this chapter ... 

Data on water leaching fluxes have been calculated using iteration approaches (Priputina, 2004). Water percolation parameters have been accounted (Manual, 2004). Annual mean air temperature and precipitation data have been obtained from IWMI World Water and Climate Atlas (2002). Two iteration versions of the map of water leaching parameters are shown in Figure 7. 

Scheme 4.117 Retrosynthetic analysis of heptaglucan phytoalexin synthesis achieved using an iterative approach.

The time frames available to solve our problems are limited by practical and economic factors. This frequently means that there is no time to repeat a critical study if the first attempt fails. So a true iterative approach is not possible. 

In this chapter, we concentrate on the fundamental physical-chemical law of mass action. It forms the basis of many chemical investigations including kinetics and equilibrium studies. Importantly, the solutions are usually not explicit and thus require iterative approaches. 

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