The Monte Carlo Technique

Figure 1.24. Rejection of suspected outliers. A series of normally distributed values was generated by the Monte Carlo technique the mean and the standard deviation were calculated the largest normalized absolute deviate (residual) z = xi - /.i is plotted versus n (black...

In the following, an example from Chapter 4 will be used to demonstrate the concept of statistical ruggedness, by applying the chosen fitting model to data purposely corrupted by the Monte Carlo technique. The data are normalized TLC peak heights from densitometer scans. (See Section 4.2) ... 

Figure 3.11. Smoothing a noisy signal. The synthetic, noise-free signal is given at the top. After the addition of noise by means of the Monte Carlo technique, the panels in the second row are obtained (little noise, left, five times as much noise, right). A seven-point Savitzky-Golay filter of order 2 (third row) and a seven-point moving average (bottom row) filter are...

With the Monte Carlo technique, a very large number of membrane problems have been worked on. We have insufficient space to review all the data available. However, the formation of pores is of relevance for permeation. The formation of perforations in a polymeric bilayer has been studied by Muller by using Monte Carlo simulation [67] within the bond fluctuation model. In this particular MC technique, realistic moves are incorporated, such that the number of MC steps can be linked to a simulated time. 

This section has illustrated a relatively simple application of the Monte Carlo technique for simulating atmospheric diffusion. With the availability of large-scale computing capacities, Monte Carlo methods can be envi-... 

Another approach has been taken, however, by Johns, Pearson, and Brown,137 which employs either a Monte Carlo computer technique, or a random distribution method. The Monte Carlo technique is based on the formation of only a dimer and a hydrate, with cross sections for reaction according to eq. (11a)... 

The Monte Carlo technique has been used in some chemical engineer-... 

One possible method of automatic optimization was mentioned by Kahn (Kl) quite some time ago. This used the Monte Carlo technique to make a random design of cases over a broad area known to contain the optimum. After a limited number of cases had been calculated, the best case was selected and another random design of cases was made. For the second design, however, the dispersion of cases was reduced so that the investigation became more localized. After a sufficient number of repetitions of this process the optimum would be determined to a sufficient degree of accuracy. The method is perhaps the antithesis of a logical or orderly calculation procedure, but it can be programmed for automatic sequential calculation. 

The Siro model is a good tool in the development of constituent retrieval algorithms for limb scan measurements. However, the Monte Carlo technique requires a lot of computer time. Faster models need to be developed for near-real time processing of limb spectra to constituent profiles. Siro serves as a reference against which faster but more approximate methods can be validated. 

This is not a simple expression that allows us to immediately evaluate the effect of the solvent molecules on the rate constant. The integrals have to be evaluated by, for example, the Monte Carlo technique. Both p( ) and VJatra only depend on the internal 3n — r coordinates and not on the center-of-mass coordinates and rotational coordinates, as explained above. Integration over these coordinates therefore always cancels in this expression. 

The most straightforward application of the Monte Carlo technique arises where the probability of a parameter taking a particular value is constant over its entire range. For example, the initial angles between the BC inter-nuclear axis, and a line joining A to the center of mass of BC and in the plane containing A, B, and C, are distributed in this way, and a value can be selected by multiplying it by a random number between — 1 and 1. 

In this section we will discuss two applications of the Monte Carlo technique. 

The Monte Carlo technique can easily be visualized if the particles are assumed to be in a box (Fig. 6). The particles are displaced by a random amount, and the potential energy is calculated using a specified intermolecular potential, and the new configuration is either accepted or rejected, according to the following five-step criteria (24) ... 

The main advance in recent years has been the development of methods to obtain models of structures that are consistent with the total diffraction pattern. One method is the Reverse Monte Carlo (RMC) method (McGreevy and Pusztai 1988, McGreevy 1995, Keen 1997, 1998). In this method, the Monte Carlo technique is used to modify a configuration of atoms in order to give the best agreement with the data. This can be carried out using either S Q) or T(r) data, or both simultaneously. We also impose a... 

The GC results are compared in fig. 10.18 with Monte Carlo calculations of Boda et al. [32]. These were carried out assuming that the electrolyte ions are hard spheres with a diameter of 300 pm in a dielectric continuum. The estimates of < ) using the Monte Carlo technique fall below the GC estimates. They demonstrate the importance of including finite ion size in a model of the diffuse layer. 

For a more comprehensive introduction to Monte Carlo simulations we refer the interested reader to the excellent text Landau and Binder [126]. In Ref. 126 the authors discuss many applications of the Monte Carlo technique beyond the scope of the present book. 

The variance matrix of each of a collection of estimates was calculated for a variety of distributions using the Monte Carlo techniques of Andrews and co-workers1 and Relles.s The distributions were all expressed in terms of mixtures of Gaussian distributions. G(0, 1) denotes a standard Gaussian distribution with mean 0 and variance 1. The distribution formed by contaminating this with 10 percent of a Gaussian distribution with mean 0 and variance 9 is denoted by 10% G(0, 9). 

A separate random number must be chosen for each of these relations, so that they will be independent of one another. Each event will have a different CDF. Derivations of the statistical relations necessary to determine the CDF relations needed to implement the Monte Carlo technique are given in Refs. 1, 2, and 3 and are not repeated here however, the resulting relations for implementing the method are given in Table 7.5. 

Recent work on the Monte Carlo technique includes that by Farmer and Howell [98—101], who investigated the optimal strategies for implementing the Monte Carlo technique on both serial and parallel computers. For up to 32 parallel processors, the CPU time on the parallel machines was very nearly inversely proportional to the number of processors, indicating that the method is very well suited to parallel machines. 

Summary. In conclusion, some suggestions are made on how to model the problem of radiative heat transfer in porous media. First, we must choose between a direct simulation and a continuum treatment. Wherever possible, continuum treatment should be used because of the lower cost of computation. However, the volume-averaged radiative properties may not be available in which case continuum treatment cannot be used. Except for the Monte Carlo techniques for large particles, direct simulation techniques have not been developed to solve but the simplest of problems. However, direct simulation techniques should be used in case the number of particles is too small to justify the use of a continuum treatment and as a tool to verify dependent scattering models. 

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