Simulation iterative

The number of sampling iterations must be sufficient to give stable results for output distributions, especially for the tails. There are no simple rules, because the necessary number of runs depends on the number of variables entered as distributions, model complexity (mathematical structure), sampling technique (random or Latin hypercube), and the percentile of interest in the output distribution. There are formal methods to establish the number of iterations (Cullen and Frey 1999) however, the simulation iterations could simply be increased to a reasonable point of convergence. 

The present author prefers to use an even larger number of points, having established a function subroutine (see the collection discussed in Appendix C) GOFUNC, which implements the function Q(C, n. H) and allows up to seven points (n = 7). One can then be sure that there are no significant error terms from the approximation to G, no matter which method is used. Furthermore, although the calculation does take a little longer than the simple two-point one, this is never a lot compared with the simulation iteration itself, so it does not matter. 

Note Coverage was based on 2000 simulation iterations and represent the percent of times the Cl contained the true mean of the distribution. 

Figure 8 Box and whiskers plots of the Cl range under various distribution and sample sizes. A total of 2000 simulation iterations were used with the upper 1% of the data trimmed.

A 3x3 (NtxNr), 4x4, and 4-QAM 6x6 MIMO system is simulated. The symbols x,-and number of algorithm iterations (Nur) depends upon N, and QAM constellation size. For 3x3, 4-QAM system, x, equals 6 and it grows to 12 for 6x6,4-QAM system. Nitr is kept in the range of 10 to 20 in our simulations. Iterations are according to the system requirements. Larger Nur can result in better BER at the cost of complexity. However, the algorithm reaches saturation after a certain number of iterations and therefore Nur needs to be tuned carefully. Optimum value is taken after a number or trials to find the best BER with least complexity. 

To search for the forms of potentials we are considering here simple mechanical models. Two of them, namely cluster support algorithm (CSA) and plane support algorithm (PSA), were described in details in [6]. Providing the experiments with simulated and experimental data, it was shown that the iteration procedure yields the sweeping of the structures which are not volumetric-like or surface-like, correspondingly. While the number of required projections for the reconstruction is reduced by 10 -100 times, the quality of reconstruction estimated quantitatively remained quite comparative (sometimes even with less artefacts) with that result obtained by classic Computer Tomography (CT). 

With XRD applied to bulk materials, a detailed structural analysis of atomic positions is rather straightforward and routine for structures that can be quite complex (see chapter B 1.9) direct methods in many cases give good results in a single step, while the resulting atomic positions may be refined by iterative fitting procedures based on simulation of the diffraction process. 

Molecular dynamics simulations ([McCammon and Harvey 1987]) propagate an atomistic system by iteratively solving Newton s equation of motion for each atomic particle. Due to computational constraints, simulations can only be extended to a typical time scale of 1 ns currently, and conformational transitions such as protein domains movements are unlikely to be observed. 

Fig. 3. Stepsize r used in the simulation of the collinear photo dissociation of ArHCl the adaptive Verlet-baaed exponential integrator using the Lanczos iteration (dash-dotted line) for the quantum propagation, and a stepsize controlling scheme based on PICKABACK (solid line). For a better understanding we have added horizontal lines marking the collisions (same tolerance TOL). We observe that the quantal H-Cl collision does not lead to any significant stepsize restrictions.

Using different types of time-stepping techniques Zienkiewicz and Wu (1991) showed that equation set (3.5) generates naturally stable schemes for incompressible flows. This resolves the problem of mixed interpolation in the U-V-P formulations and schemes that utilise equal order shape functions for pressure and velocity components can be developed. Steady-state solutions are also obtainable from this scheme using iteration cycles. This may, however, increase computational cost of the solutions in comparison to direct simulation of steady-state problems. 

Adding time-dependent terms to the equations the simulation is treated as an initial value problem in which at a given reference time all stresses are zero the steady-state solution can be found iteratively. 

Iterative solution methods are more effective for problems arising in solid mechanics and are not a common feature of the finite element modelling of polymer processes. However, under certain conditions they may provide better computer economy than direct methods. In particular, these methods have an inherent compatibility with algorithms used for parallel processing and hence are potentially more suitable for three-dimensional flow modelling. In this chapter we focus on the direct methods commonly used in flow simulation models. 

The size of the move in step 3 of the above procedure will affect the elhciency of the simulation. In this case, an inefficient calculation is one that requires more iterations to obtain a given accuracy result. If the size is too small, it will take many iterations for the atom locations to change. If the move size is too large, few moves will be accepted. The efficiency is related to the acceptance ratio. This is the number of times the move was accepted (step 5 above) divided by the total number of iterations. The most efficient calculation is generally obtained with an acceptance ratio between 0.5 and 0.7. 

Monte Carlo simulations require less computer time to execute each iteration than a molecular dynamics simulation on the same system. However, Monte Carlo simulations are more limited in that they cannot yield time-dependent information, such as diffusion coefficients or viscosity. As with molecular dynamics, constant NVT simulations are most common, but constant NPT simulations are possible using a coordinate scaling step. Calculations that are not constant N can be constructed by including probabilities for particle creation and annihilation. These calculations present technical difficulties due to having very low probabilities for creation and annihilation, thus requiring very large collections of molecules and long simulation times. 

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