Asymptotic algorithms

Under these conditions, we can develop an asymptotic algorithm generalizing that in Section IV.A. We will not do it here as it is done in detail in [8]. 

Let us now return to the system (3.1). Suppose that equation F(z, y, t) = 0 has several solutions (roots) z = (Pi y, t). Previously, we considered the case when only one such root was used in the construction of the asymptotic expansion. But some other types of asymptotic behavior, when several such solutions are used in the asymptotic algorithm, are also possible. In these cases, we can obtain a solution with transitions from one root to another or solutions with, so-called, interior layers (internal layers). 

For the first order, we have , = 0, and is an arbitrary function. At the next step of the asymptotic algorithm, we obtain (a typical situation for problems in the critical case) the equation for the as yet unknown function... 

A more detailed description of the low-frequency behavior of the algorithm is obtained by an asymptotic analysis. In spite of the explicit expression (32) it is probably simpler to obtain a power series representation of z directly from the characteristic equation, see Krenk, (2008). This leads to the asymptotic algorithmic damping ratio... 

Because the pseudo-inverse filter is chosen from the class of additive filters, the regularization can be done without taking into account the noise, (n). At the end of this procedure the noise is transformed to the output of the pseudo-inverse filter (long dashed lines on Fig. 1). The regularization criteria F(a,a) has to fulfill the next conditions (i) leading to an additive filter algorithm, (ii) having the asymptotic property a, —> a, for K,M... 

The comparison of curve 1 and 2 in Fig. 3 yields, that the convergence with respect to the number of projections k is not strongly influenced by noise because of the properties of the reconstruction algorithm. Nevertheless, the noise increases the asymptotic value of o(n)/0 ... 

HarF75b Harary, F. Twenty step algorithm for determining the asymptotic number of trees of various species. J. Austral. Math. Soc. 20 (1975) 483-503. 

FIGURE 11.9 Outliers, (a) Dose-response curve fit to all of the data points. The potential outlier value raises the fit maximal asymptote, (b) Iterative least squares algorithm weighting of the data points (Equation 11.25) rejects the outlier and a refit without this point shows a lower-fit maximal asymptote. 

However, since and -5 asymptote to the same function, one might approximate (U) = S dJ) in (3.57) so that the acceptance probability is a constant.3 The procedure allows trial swaps to be accepted with 100% probability. This general parallel processing scheme, in which the macrostate range is divided into windows and configuration swaps are permitted, is not limited to density-of-states simulations or the WL algorithm in particular. Alternate partition functions can be calculated in this way, such as from previous discussions, and the parallel implementation is also feasible for the multicanonical approach [34] and transition-matrix calculations [35],... 

According to the results, it is determined that the asphericities can be described in terms of polynomials in Forni et al. [140] also used an off-lattice model and an MC Pivot algorithm to determine the star asphericity for ideal, theta, and EV 12-arm star chains. They also found that the EV stars chains are more spherical than the ideal and theta star chains. In these simulations the theta chains exhibit a remarkable variation of shape with arm length, so that short chains (where core effects are dominant for all chains with intramolecular interactions) have asphericities closer to those to those found with EV, while longer chains asymptotically approach the ideal chain value(see Fig. 10). 

Real space algorithms (section 4) allow for mappings between present day computer programs and strict molecular quantum mechanics [10,11]. It is the separability of base molecular states that permits characterizing molecular states in electronic Hilbert space and molecular species in real space. This feature eliminates one of the shortcomings of the standard BO scheme [6,7,12]. Confining and asymptotic GED states are introduced. In section 5 the concept of conformation states in electronic Hilbert space is qualitatively presented. 

Diversification of catalysts proposed by the algorithm tends to decrease more quickly with increasing asymptotic probability ratio of quantitative mutation to any modification, and with increasing coefficient of quantitative mutation. Again, these are only general tendencies, really apparent only for particular combinations of the remaining parameters. 

An implementation of the above algorithm in this recursive fashion would have an exponential asymptotic runtime behavior. A simple observation shows that such high computing demand is unnecessary. If two trees A and B with nA and nB nodes, respectively, are compared, only 4(nA-l)(nB-l) different calls of... 

In principle, Equation (3.5) represents an infinite set of coupled equations. In practice, however, we must truncate the expansion (3.4) at a maximal channel n which turns (3.5) into a finite set that can be numerically solved by several, specially developed algorithms (Thomas et al. 1981). The required basis size depends solely on the particular system. The convergence of the close-coupling approach must be tested for each system and for each total energy by variation of n until the desired cross sections do not change when additional channels are included. Expansion (3.4) should, in principle, include all open channels (k > 0) as well as some of the closed vibrational channels (k% < 0). Note, however, that because of energy conservation the latter cannot be populated asymptotically. 

The second approach is a fractional-step method we call asymptotic timestep-splitting. It is developed by consideration of the specific physics of the problem being solved. Stiffness in the governing equations can be handled "asymptotically" as well as implicitly. The individual terms, including those which lead to the stiff behavior, are solved as independently and accurately as possible. Examples of such methods include the Selected Asymptotic Integration Method (4,5) for kinetics problems and the asymptotic slow flow algorithm for hydrodynamic problems where the sound speed is so fast that the pressure is essentially constant (6, 2). ... 

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