Stochastic techniques

Other methods which are applied to conformational analysis and to generating multiple conformations and which can be regarded as random or stochastic techniques, since they explore the conformational space in a non-deterministic fashion, arc genetic algorithms (GA) [137, 1381 simulation methods, such as molecular dynamics (MD) and Monte Carlo (MC) simulations 1139], as well as simulated annealing [140], All of those approaches and their application to generate ensembles of conformations arc discussed in Chapter II, Section 7.2 in the Handbook. 

Monte Carlo search methods are stochastic techniques based on the use of random numbers and probability statistics to sample conformational space. The name Monte Carlo was originally coined by Metropolis and Ulam [4] during the Manhattan Project of World War II because of the similarity of this simulation technique to games of chance. Today a variety of Monte Carlo (MC) simulation methods are routinely used in diverse fields such as atmospheric studies, nuclear physics, traffic flow, and, of course, biochemistry and biophysics. In this section we focus on the application of the Monte Carlo method for... 

This book aims at providing the reader with a detailed understanding of the planning, integration and coordination of multisite refinery and petrochemical networks using proper deterministic and stochastic techniques. The book consists of three parts ... 

The correction for non-uniform instrumental broadening in SEC is solved through a non-recursive matrix stochastic technique. To this effect, Tung s equation ( ) must be reformulated in matrix form, and the measurements assumed contaminated with zero-mean noise. 

M.E. Kainourgiakis, E.S. Kikkinides and A.K. Stubos, Diffusion and flow in porous domains constructed using process-based and stochastic techniques , J. Porous Mat., in press... 

Several attempts to describe replication-mutation networks by stochastic techniques were made in the past. We cannot discuss them in detail here, but we shall brieffy review some general ideas that are relevant for the quasispecies model. The approach that is related closest to our model has been mentioned already [51] the evolutionary process is viewed as a sequence of stepwise increases in the populations mean fitness. Fairly long, quasi-stationary phases are interrupted by short periods of active selection during which the mean fitness increases. The approach towards optimal adaptation to the environment is resolved in a manner that is hierarchical in time. Evolution taking place on the slow time scale represents optimization in the whole of the sequence space. It is broken up into short periods of time within which the quasi-species model applies only locally. During a single evolutionary step only a small part of sequence space is explored by the population. There, the actual distributions of sequences resemble local quasispecies confined to well-defined regions. Error thresholds can be defined locally as well. 

Gantt, J. A. Gatzke, E. P. 2006 A stochastic technique for multidimensional granulation modeling. AIChE Journal 52, 3067-3077. 

It is possible to eliminate all mass effects and all dynamical information in determining the ensemble averages by the use of Monte Carlo simulation procedures. The direct application of such fully stochastic techniques is not common in the field of macromolecular simulations because the presence of... 

For a more extensive overview of descriptive mnltivariate stochastic techniques it is referred to Hair et al. (1995), Berthouex and Brown (2002), and Schabenberger and Pierce (2002). For scaling methods in aquatic ecology it is referred to Seurant and Strntton (2004). 

An alternative proposed in previous works (for example, Bdrard et al 2(X)0) consists in coupling a Discrete Event Simulator (DES) (in order to evaluate the feasibility of the production at medium term scheduling) with a master optimization procedure generally based on stochastic techniques such as GA (to take into account the combinatorial feature due to the large number of discrete variables in the optimization problem). These ideas... 

PSO is a swarm-intelligence-based, approximate, nondeterministic optimization technique. It is a robust stochastic technique based on the movement and intelligence of swarms. It applies the concept of social interaction to problem-solving (people.scs.carleton.ca). 

Figure 1 shows the canonical GA as developed by Holland. The canonical GA encodes the problem within binary string individuals. Evolutionary pressure is applied in step 3, where the stochastic technique of roulette wheel parent selection is used to pick parents for the new population. The concept is illustrated in Figure 2, using a trivial example with a population of four individuals. Each individual is assigned a sector of a roulette wheel that is proportional to its fitness and the wheel is spun to select a parent. While selection is random and any individual has the capacity to become a parent, selection is clearly biassed towards fitter individuals. Parents are not required to be unique and, in each iteration, fit individuals may produce many offspring. 

When modeling the droplet size, evolution, and other dynamical phenomena in a hquid spray, one quickly encounters the problem that the number of droplets in a reafistic spray is very large, typically 10 or an order of magnitude more. Deterministic modeling of each and every droplet is therefore out of the question. Fortunately, liquid sprays are usually also very dilute, making them perfect for modeling through stochastic techniques... 

Many problems in chemical engineering are expressed mathematically as optimization problems, and involve finding the particular x that minimizes some cost function F x). Each component ofx may vary either continuously or discretely. In this chapter, we assume that eachvy varies continuously. In Chapter 7, we consider stochastic techniques that can be used with discretely-varying parameters. 

In this section, we will present results of microldnetics simulations based on elementary reaction energy schemes deduced from quantum chemical studies. We use an adapted scheme to enable analysis of the results in terms of the values of elementary rate constants selected. For the same reason, we ignore surface concentration dependence of adsorption energies, whereas this can be readily implemented in the simulations. We are interested in general trends and especially the temperature dependence of overall reaction rates. The simulations will also provide us with information on surface concentrations. In the simulations to be presented here, we exclude product readsorption effects. Microldnetics simulations are attractive since they do not require an assumption of rate-controlling steps or equilibration. Solutions for overall rates are found by solving the complete set of PDFs with proper initial conditions. While in kinetic Monte Carlo simulations these expressions are solved using stochastic techniques, which enable formation... 

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