Computational performance

Several years ago Baer proposed the use of a mahix A, that hansforms the adiabatic electronic set to a diabatic one [72], (For a special twofold set this was discussed in [286,287].) Computations performed with the diabatic set are much simpler than those with the adiabatic set. Subject to certain conditions, A is the solution of a set of first order partial diffei ential equations. A is unitary and has the form of a path-ordered phase factor, in which the phase can be formally written as... 

In order to compare the efficiency of the SISM with the standard LFV method, we compared computational performance for the same level of accuracy. To study the error accumulation and numerical stability we monitored the error in total energy, AE, defined as... 

Path integral Monte Carlo simulations were performed [175] for the system with Hamiltonian (Eq. (25)) for uj = ujq/J = A (where / = 1) with N = 256 particles and a Trotter dimension P = 64 chosen to achieve good computer performance. It turned out that only data with noise of less than 0.1% led to statistically reliable results, which were only possible to obtain with about 10 MC steps. The whole study took approximately 5000 CPU hours on a CRAY YMP. 

A Compound Distribution.—We now give an illustration of computations performed on a stochastic process. 

In recent years, density-functional theory has emerged as the computational quantum chemistry method of choice for biological problems of medium size range (up to a few hundreds of atoms) in applications that do not require extensive conformational sampling. The field continues to advance in the accuracy of new functionals, the improvement of algorithms and the functionality and computational performance of software [81]. 

A special tearing procedure known as diakoptics (K10) was investigated by Brameller et al. (BIO) and by Gay and Middleton (Gl). According to Brameller et al. (BIO), this procedure lends itself very readily to the exercise of engineering judgment. However, computational performance of this method reported by Gay and co-workers (Gl, G2) has not been too impressive. The more recent work of Gay and Preece (G2) indicates that the nodal method of diakoptics is outperformed by an alternative method based on formulation D for problems up to 100 edges and 50 cycles. 

A detailed description of the discrete merge algorithm and an analysis of its performance characteristics may be found elsewhere (C8). It suffices to point out that experience with this algorithm to date has been most encouraging. Cheng and Mah (C8) reported that the discrete merge method is about 2-4 times faster than the best previously known procedure (M9) and requires only about as much storage. Table VII summarizes the computation performance data based on the CDC 6400 computer. 

Computational performance data on the Galerkin method were reported by Rachford and Dupont (Rl). Time increments of the order of 5 minutes are typical. According to these authors an average of 120 pipe steps per CPU second on a CDC 6600 computer was achieved. Approximately 100 words of storage per pipe were required for array storage and 12,000 words of storage for the program and subroutines. On that basis the authors estimated that a 1000-pipe network can be simulated on a full-size CDC 6600 computer at about 100 CPU seconds per simulated hour. 

Computations performed on an N-methylated imidazole showed that the presence of the methyl group increases the proton affinity of the other nitrogen. In contrast, a peptide substituent at a carbon adjacent to the nitrogen decreases its proton affinity. This can be explained by the conjugation stabilization of the peptide-substituted neutral form, which is absent in the protonated form. This result seems to indicate that heteroatomic rings linked by... 

Thus, the user must make a trade-off between the amount of computation performed and the exactness of the calculated nonlinearity measure, taking into account the actual amount of nonlinearity in the data. However, if sufficient points are used, the results are stable and depend only on the amount of nonlinearity in the original data set. 

The computation performed in this study is based on the model equations developed in this study as presented in Sections II.A, III.A, III.B, and III.C These equations are incorporated into a 3-D hydrodynamic solver, CFDLIB, developed by the Los Alamos National Laboratory (Kashiwa et al., 1994). In what follows, simple cases including a single air bubble rising in water, and bubble formation from a single nozzle in bubble columns are first simulated. To verify the accuracy of the model, experiments are also conducted for these cases and the experimental results are compared with the simulation results. Simulations are performed to account for the bubble-rise phenomena in liquid solid suspensions with single nozzles. Finally, the interactive behavior between bubbles and solid particles is examined. The bubble formation and rise from multiple nozzles is simulated, and the limitation of the applicability of the models is discussed. 

The previous general continuous-time formulations are mostly oriented towards arbitrary network processes. On the other hand, different continuous-time formulations focused their attention on particular features of a wide variety of sequential processes. One of the first contributions following this direction is based on the concept of time slots, which stand for a set of predefined time intervals with unknown durations. The main idea is to postulate an appropriate number of time slots for each processing unit in order to allocate them to the batches to be processed. The definition of the number of time slots required is not a trivial decision and represents an important trade-offbetween optimality and computational performance. Other alternative approaches for sequential processes were developed based on the concept of batch precedence. Model variables defining the processing sequence of batch tasks are explicitly embedded into these formulations and, consequently,... 

Different measures of the quality of the solution can be used for scheduling problems. However, the criterion selected to be optimized usually has a direct effect on the model computational performance. In addition, some objective functions can be very hard to implement for some event representations, requiring additional variables and complex constraints. 

For the solubility of TPA in prepolymer, no data are available and the polymer-solvent interaction parameter X of the Flory-Huggins relationship is not accurately known. No experimental data are available for the vapour pressures of dimer or trimer. The published values for the diffusion coefficient of EG in solid and molten PET vary by orders of magnitude. For the diffusion of water, acetaldehyde and DEG in polymer, no reliable data are available. It is not even agreed upon if the mutual diffusion coefficients depend on the polymer molecular weight or on the melt viscosity, and if they are linear or exponential functions of temperature. Molecular modelling, accompanied by the rapid growth of computer performance, will hopefully help to solve this problem in the near future. The mass-transfer mechanisms for by-products in solid PET are not established, and the dependency of the solid-state polycondensation rate on crystallinity is still a matter of assumptions. 

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