Deterministic and Stochastic Methods

It is not uncommon for an algorithm to be able to guarantee that it will locate the optimum solution, provided that it is set to work on an appropriate problem, and this is an important benefit. However, the established methods of solution are not without their drawbacks. Even when a guarantee does exist that the optimum solution can be found, it may not be possible to find this solution within an ac- 

As we shall see, this volume concentrates on methods which are stochastic, rather than deterministic, in nature - that is, the route that they take in seeking a solution depends in some way upon chance (and, as an important consequence, the solution found may vary from one invocation of the algorithm to the next). Although stochastic methods are not as familiar to most chemists as analytical methods, examples of their use are still widespread. Monte Carlo calculations, which combine a statistical approach with the chance nature of random events, have been widely used in fields such as molecular dynamics, but Monte Carlo is only suited to a restricted range of problems. 

It is evident that no way of locating solutions is perfect, or universally applicable. Although established methods are often valuable, opportunities exist for newly developed techniques to outperform them, especially in the solution of some of the more challenging, and perhaps more interesting, problems in science. A number of approaches within the field of artificial intelligence (AI) show great promise in science, and scientific applications of AI are already numerous. Applications from within one of its principle areas, that of evolutionary algorithms (EAs), form the topic of this volume. 

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Comparison of Advantages and Disadvantages of Deterministic and Stochastic Methods... 

The problem of multivariable optimization is illustrated in Figure 3.4. Search methods used for multivariable optimization can be classified as deterministic and stochastic. 

Teegavarapu RSV, Chandramouli V (2005) Improved weighting methods, deterministic and stochastic data-driven models for estimation of missing precipitation records. J Hydrol 312 191-206... 

A number of different techniques have been developed for studying nonhomogeneous radiolysis kinetics, and they can be broken down into two groups, deterministic and stochastic. The former used conventional macroscopic treatments of concentration, diffusion, and reaction to describe the chemistry of a typical cluster or track of reactants. In contrast, the latter approach considers the chemistry of simulated tracks of realistic clusters using probabilistic methods to model the kinetics. Each treatment has advantages and limitations, and at present, both treatments have a valuable role to play in modeling radiation chemistry. 

The classic and stochastic methods used for the analysis of liquid flow inside a porous medium are strongly related. These interactions are given by the relationships between the parameters of both types of models. We show here that the analysis of the flow of a liquid through a porous medium, using a stochastic model, can describe some of the parameters used in deterministic models such as ... 

Various deterministic and stochastic sampling techniques for path ensembles have been proposed [4-6]. Here we consider only Monte Carlo methods. It is important, however, to be aware that while the path ensemble is sampled with a Monte Carlo procedure each single pathway is a fully dynamical trajectory such as one generated by molecular dynamics. 

Nonlinear dynamics of complex processes is an active research field with large numbers of publications in basic research and broad applications from diverse fields of science. Nonlinear dynamics as manifested by deterministic and stochastic evolution models of complex behaviour has entered statistical physics, physical chemistry, biophysics, geophysics, astrophysics, theoretical ecology, semiconductor physics and -optics etc. This research has induced a new terminology in science connected with new questions, problems, solutions and methods. New scenarios have emerged for spatio-temporal structures in dynamical systems far from equilibrium. Their analysis and possible control are intriguing and challenging aspects of the current research. 

We wish to introduce next a topic of increasing importance to chemical engineers, stochastic (random) simulation. In stochastic models we simulate quite directly the random nature of the molecules. We will see that the deterministic rate laws and material balances presented in the previous sections can be captured in the stochastic approach by allowing the numbers of molecules in the simulation to become large. From this viewpoint, deterministic and stochastic approaches are complementary. Deterministic models and solution methods are quite efficient when the numbers of molecules are large and the random behavior is not important. The numerical methods for solution of the nonlinear differential equations of the deterministic models are... 

Methods for solving optimization problems can be classified into two main types namely, deterministic and stochastic. Deterministic methods often require derivatives of objective... 

Henig, M.I., 1986. Extensions of the dynamic programming method in the deterministic and stochastic assembly line balancing problems. Computers and Operations Re search, 13,443 9. 

Analytical models can be classified into deterministic and stochastic. The former formulates the relationship between the known and unknown factors in the form of equations, the solution of which often requires application of numerical methods. By following prescribed rules the same result can always be obtained from the same starting conditions and initial values of known factors. In the latter, the model contains a degree of uncertainty caused by random events or variations in the values of factors, thus leading to potentially different results even when starting from the same initial conditions. 

This chapter provides an overview of the most frequently applied numerical methods for the simulation of polymerization processes, that is, die calculation of the polymer microstructure as a function of monomer conversion and process conditions such as the temperature and initial concentrations. It is important to note that such simulations allow one to optimize the macroscopic polymer properties and to influence the polymer processability and final polymer product application range. Both deterministic and stochastic modeling techniques are discussed. In deterministic modeling techniques, time variation is seen as a continuous and predictable process, whereas in stochastic modeling techniques, a random-walk process is assumed instead. 

Sensitivity analysis The main task of sensitivity analysis is to find the impact of changes in input parameters on the result. This is done by making variations in the input parameters over a range to see if the impact on cost can help highlight the major factors affecting costs. There are several methods available for sensitivity analysis. Mainly deterministic and stochastic approach are used. 

In general, deterministic and probabilistic methods should be seen not as competitive but rather as complementary. For example, if a surge is generated by a tropical cyclone, the evaluation is usually carried out by deterministic methods with account taken of the more symmetrical characteristics of the generating storm. Surges generated by extra-tropical storms have been evaluated mainly by stochastic methods since such storms are usually very... 

Both deterministic and stochastic simulations can be used for response-history dynamic analysis, but only stochastic simulations can be utilized for stochastic dynamic (i.e., random vibration) analysis, because the latter analysis method requires a random process model of the earthquake ground motion. Synthetic ground motions are particularly useful for nonlinear dynamic analysis due to the scarcity of recorded motions for large-magnitude earthquakes that are capable of causing nonlinear responses. Two approaches are available for nonlinear dynamic analysis of structures subjected to earthquakes (1) nonlinear response-history analysis by the use of a selected set of ground motion time series and (2) nonlinear stochastic dynamic analysis by the use of a stochastic representation of the ground motion. 

During the past three decades, a number of theoretical studies have been carried out to describe field-scale dispersive mixing as a function of soil heterogeneity and develop upscaling methods for the estimation of macrodispersivities. These upscaling methods can usually be categorized into deterministic or stochastic methods. 

There are several approaches to this interpolation problem approaches that are not equivalent. It is, however, generally agreed that some probabilistic element is required. Having said that, it is also the case that deterministic interpolation procedures are in widespread use. Thus, before reviewing statistical and stochastic methods, a survey of deterministic methods is given. Later sections show that these methods are closely related to kriging. This is not a new result [89] but does not seem to be widely known. 

Stochastic methods do not need auxiliary information, such as derivatives, in order to progress. They only require an objective function for the search. This means that stochastic methods can handle problems in which the calculation of the derivatives would be complex and cause deterministic methods to fail. 

Also, it is possible to combine stochastic and deterministic methods as hybrid methods. For example, a stochastic method can be used to control the structural changes and a deterministic method to control the changes in the continuous variables. This can be useful if the problem involves a large number of integer variables, as for such problems, the tree required for branch and bound methods explodes in size. 

The procedures discussed so far take as fundamental variables the species concentration and specific rates, the latter obtained from homogeneous experiments. Such procedures are called deterministic—that is, admitting no fluctuation in the number of reactant species—as opposed to stochastic methods where statistical variation is built in. 

The different theoretical models for analyzing particle deposition kinetics from suspensions can be classified as either deterministic or stochastic. The deterministic methods are based on the formulation and solution of the equations arising from the application of Newton s second law to a particle whose trajectory is followed in time, until it makes contact with the collector or leaves the system. In the stochastic methods, forces are freed of their classic duty of determining directly the motion of particles and instead the probability of finding a particle in a certain place at a certain time is determined. A more detailed classification scheme can be found in an overview article [72]. 

The present investigation applies deterministic methods of continuous mechanics of multiphase flows to determine the mean values of parameters of the gaseous phase. It also applies stochastic methods to describe the evolution of polydispersed particles and fluctuations of parameters [4]. Thus the influence of chaotic pulsations on the rate of energy release and mean values of flow parameters can be estimated. The transport of kinetic energy of turbulent pulsations obeys the deterministic laws. 

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