Stochastic variations

Juxtaposition seal can be complicated by a number of factors. These include unsystematic lateral lithologic variations, for instance those of channel belts. While it is common to model such lithologic variations stochastically for input to reservoir simulations, this is not to our knowledge used in fault seal analysis. For small fault throws any continuous (i.e., not wedged out) sand horizons may provide across-fault connectivity which could be misinterpreted as a seal if interpreted by juxtaposition analysis alone. Other complications relate to so-called damage and relay zones. 

In stochastical methods the random kick is typically somewhat larger, and a standard minimization is carried out starting at the perturbed geometry. This may or may not produce a new minimum. A new perturbed geometry is then generated and minimized etc. There are several variations on how this is done. 

The problems of operations research have stimulated new developments in several mathematical fields various aspects of game theory, stochastic processes, the calculus of variations, graph theory, and numerical analysis, to name a few. 

The polymeric material produced in a single stirred-tank reactor will, except for stochastic variations, be of uniform composition. This polymer composition can be significantly different from the composition in the monomer feed mixture unless the conversion is high. If several tanks are connected in series the composition of the polymer produced in each reactor can be quite different. Since most particles are formed in the first reactor this change in composition in the following reactors can yield polymer particles in which composition varies with radius within the particles. 

Variational electrostatic projection method. In some instances, the calculation of PMF profiles in multiple dimensions for complex chemical reactions might not be feasible using full periodic simulation with explicit waters and ions even with the linear-scaling QM/MM-Ewald method [67], To remedy this, we have developed a variational electrostatic projection (VEP) method [75] to use as a generalized solvent boundary potential in QM/MM simulations with stochastic boundaries. The method is similar in spirit to that of Roux and co-workers [76-78], which has been recently... 

Another important property of PMTs is the pulse height distribution. The amplification of individual photoelectrons by the PMT is a stochastic process that causes variations in the gain of individual photoelectrons. As a result significant jitter in the amplitude of the output pulses is observed, see Fig. 3.6. These pulse height variations can be more than a factor of 10. The lowest pulse heights mainly consist of (thermal) noise, indicated by the dashed line in Fig. 3.6. The pulse height distribution exhibits a peak corresponding to detected photons. The threshold level of the... 

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. 

In random functions, each realization of the function can be conceived as the sum of a structured component and an erratic or stochastic component. The structured component is the one assuring that the observed values have a systematic variation or, in other words, that if we are, for instance, in an area where different measurements above a normal value have been obtained, there exist a high probability for additional measurements to be high as well. On the other hand, the random component is the one making difficult the exact prediction of these hypothetical measurements, since unpredictable fluctuations exist. 

Of course, the role of the artificially introduced stochastics for mimicking the effect of all eddies in a RANS-based particle tracking is much more pronounced than that for mimicking the effect of just the SGS eddies in a LES-based tracking procedure. In addition, the random variations may suffer from lacking the spatial or temporal correlations the turbulent fluctuations exhibit in real life. In RANS-based simulations, these correlations are not contained in the steady spatial distributions of k and e and (if applicable) the Reynolds stresses from which a typical turbulent time scale such as k/s may be derived. One may try and cure the problem of missing the temporal coherence in the velocity fluctuations by picking a new random value for the fluid s velocity only after a certain period of time has lapsed. 

Due to stochastic demand in China, stochastic production yields in Europe and some stochastic variations in transport times between the two it was decided to support the decision between these alternatives by means of simulation. The structure of the simulation model is shown in Figure 2.2. 

In particular results on tank capacity are a typical output of simulations on the plant engineering level. It could be argued that these results may be obtained without simulation as well and this is true as long as the stochastic impact on supply and demand is within certain boundaries. As soon as the facility needs to be able to handle stochastic supply and demand with significant variations static calculations reach their limits. These limitations become even more critical if a multi-product process is analyzed as quite often is the case. 

The performance ofthe deterministic and of the stochastic scheduler is compared in Figure 9.8. The figure shows the objective for all scenarios and the average objective. The stochastic scheduler improves the average objective by approximately 10% and for five out of eight scenarios. On the other hand, the stochastic scheduler produces a larger variation in the objective of the scenarios. 

In die literature on stochastic processes, the above Fokker-Planck equation describes a multi-variate Omstein— Uhlenbeck process. For a discussion on the existence of Gaussian solutions to this process, see Gardiner (1990). 

Between-subject variations are excluded from the experimental and stochastic errors. 

More sophisticated calculations (14,20), using either stochastic Monte Carlo or deterministic methods, are able to consider not only different Irradiating particles but also reactant diffusion and variations In the concentration of dissolved solutes, giving the evolution of both transient and stable products as a function of time. The distribution of species within the tracks necessitates the use of nonhomogeneous kinetics (21,22) or of time dependent kinetics (23). The results agree quite well with experimental data. 

Y. Suzuki and K. Varga, Stochastic Variational Approach to Quantum Mechanical Few-Body Problems, Springer-Verlag, Berlin, 1998. 

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