Quasi-random numbers

Note that Eq. (5.3) gives a deterministic error bound for integration because V(f) depends only on the nature of the function. Similarly, the discrepancy of a point set is a purely geometric property of that point set. When given a numerical quadrature problem, we must cope with whatever function we are given, it is really only the points, x(, that we control. Thus one approach to efficient integration is to seek point sets with small discrepancies. Such sets are necessarily not random but are instead referred to as quasi-random numbers (QRNs). 

This gives us a target to aim at for the construction of low-discrepancy point sets (quasi-random numbers). For comparison, the estimated error with random sequences is... 

We will now present a very brief description of quasi-random sequences. Those interested in a much more detailed review of the subject are encouraged to consult the recent work of Niederreiter [25]. An example of a onedimensional set of quasi-random numbers is the van der Corput sequence. First we choose a base, b, and write an integer n in base b as n = 2 " a, b . Then we define the van der Corput sequence as x = = " 1 1. For base b = 3, the first 12 terms of the van der Corput sequence are... 

Fig. 4. A schematic two-dimensional illustration of the idea for an information theory model of hydrophobic hydration. Direct insertion of a solute of substantial size (the larger circle) will be impractical. For smaller solutes (the smaller circles) the situation is tractable a successful insertion is found, for example, in the upper panel on the right. For either the small or the large solute, statistical information can be collected that leads to reasonable but approximate models of the hydration free energy, Eq. (7). An important issue is that the solvent configurations (here, the point sets) are supplied by simulation or X-ray or neutron scattering experiments. Therefore, solvent structural assumptions can be avoided to some degree. The point set for the upper panel is obtained by pseudo-random-number generation so the correct inference would be of a Poisson distribution of points and = kTpv where v is the van der Waals volume of the solute. Quasi-random series were used for the bottom panel so those inferences should be different. See Pratt et al. (1999).

Current experiments uniformly tend to perform a grid search on the composition and noncomposition variables. It is preferable, however, to choose the variables statistically from the allowed values. It is also possible to consider choosing the variables in a fashion that attempts to maximize the amount of information gained from the limited number of samples screened, via a quasi-random, low-discrepancy sequence (Niederreiter, 1992 Bratley et al., 1994). Such sequences attempt to eliminate the redundancy that naturally occurs when a space is searched statistically, and they have several favorable theoretical properties. An illustration of these three approaches to materials discovery library design is shown in Fig. 1. 

Niederreiter, H., Random Number Generation and Quasi-Monte Carlo Methods, Society for Industrial and Applied Mathematics, Philadelphia, 1992. 

The quantity in brackets, the variance, only depends on / Hence the standard deviation of the Monte Carlo estimate is 0(N 1/2). This is much worse than the bounds of Eq. (5.5) as a function of the number of points. It is this fact that has motivated the search for quasi-random points. 

When the number of quanta exceeds the critical value (5-18) (the so-called Chirikov stochasticity criterion Zaslavskii Chirikov, 1971), the molecular motion becomes quasi random and the modes become mixed in the vibrational quasi continuum. In the general case of polyatomic molecules, the critical excitation level decreases with number N of... 

The latter equality follows from the quasi-steady-state-assumption. Note that Pi, in a batch reactor is a function of conversion. If other transfer mechanisms are present, the denominator is extended with the corresponding contributions to the initiation process. Whether or not the pp is attached to another pp indeed follows by selecting a random number between 0 and 1 and determining whether the inequality rand(l) < Py is false or true. If connected [true) then the birth conversion of the earlier pp simply follows from the conditional distribution (given that the first sampled pp is created atx = d and grows from an earlier one) ... 

The dispersion curves theoretically calculated with these parameters are shown in Figures 2.6-2.8 by solid lines together with the experimental points. In order to estimate the error in the calculation and the stability of the method, we assmne that the relative experimental error for the energy at any quasi-momentum has a moderate value of 10% (the error are not given in [30]). Then, we scatter the experimental data (points in Figures 2.6 - 2.8) by 10% with a random number generator and reealculate the values of and 7). We have... 

In liquid metal solutions Z is normally of the order of 10, and so this equation gives values of Ks(a+B) which are close to that predicted by the random solution equation. But if it is assumed that the solute atom, for example oxygen, has a significantly lower co-ordination number of metallic atoms than is found in the bulk of die alloy, dieii Z in the ratio of the activity coefficients of die solutes in the quasi-chemical equation above must be correspondingly decreased to the appropriate value. For example, Jacobs and Alcock (1972) showed that much of the experimental data for oxygen solutions in biiiaty liquid metal alloys could be accounted for by the assumption that die oxygen atom is four co-ordinated in diese solutions. 

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