Stochastic variation coefficient

In order to solve the multi-objectives optimization problem proposed, several numerical applications have been carried out for specific levels of the main system and filter characteristics. These parameters, stochastically expressed by the mean and the variation coefficient, are considered deterministically known. The principle aim is to incorporate uncertainties in both the load and the structural model parameters. All data with certain and uncertain parameters are listed in Table 1, as above ... 

To interpret the solutions obtained from the stochastic models, we propose to investigate their corresponding coefficient of variation Cv. Cv for a set of values is defined as the ratio of the standard deviation to the expected value or mean and is usually expressed as a percentage. It is calculated as ... 

Table 6.1 Determination of the coefficient of variation Q, for the deterministic and stochastic models.

The performed calculations demonstrate that a type of the asymptotic solution of a complete set of the kinetic equations is independent of the initial particle concentrations, iVa(0) and 7Vb(0). Variation of parameters a and (3 does not also result in new asymptotical regimes but just modifies there boundaries (in t and k). In the calculations presented below the parameters 7Va(0) = 7Vb(0) =0.1 and a = ft = 0.1 were chosen. The basic parameters of the diffusion-controlled Lotka-Volterra model are space dimension d and the ratio of diffusion coefficients k. The basic results of the developed stochastic model were presented in [21, 25-27],... 

In all previous dissolution models described in Sections 5.1 and 5.2, the variability of the particles (or media) is not directly taken into account. In all cases, a unique constant (cf. Sections 5.1, 5.1.1, and 5.1.2) or a certain type of time dependency in the dissolution rate constant (cf. Sections 5.1.3, 5.2.1, and 5.2.2) is determined at the commencement of the process and fixed throughout the entire course of dissolution. Thus, in essence, all these models are deterministic. However, one can also assume that the above variation in time of the rate or the rate coefficient can take place randomly due to unspecified fluctuations in the heterogeneous properties of drug particles or the structure/function of the dissolution medium. Lansky and Weiss have proposed [130] such a model assuming that the rate of dissolution k (t) is stochastic and is described by the following equation ... 

The stochastic error is expressed in (9.23) by the variance Var [Aj (t)] and co-variance Cov [Nj (t) Nk (t)] that did not exist in the deterministic model. This error could also be named spatial stochastic error, since it describes the process uncertainty among compartments for the same t and it depends on the number of drug particles initially administered in the system. For the sake of simplicity, assume riQi = uq for each compartment i. From the previous relations, the coefficient of variation CVj (t) associated with a time curve Nj (t) in compartment 3 is... 

As previously, initial conditions for the compartmental model and the enzymatic reaction were set to tiq = [100 50], and so = 100, eo = 50, and cq = 0, respectively. Figures 9.31 and 9.32 show the deterministic prediction, a typical run, and the average and confidence corridor for 100 runs from the stochastic simulation algorithm for the compartmental system and the enzyme reaction, respectively. Figures 9.33 and 9.34 show the coefficient of variation for the number of particles in compartment 1 and for the substrate particles, respectively. 

The example is a 32-feet long, fixed ended beam with stochastic flexural rigidity, EI(x which is subject to a distributed load with stochastic intensity W(jc), as shown in Fig. 1. Both processes are assumed to be homogeneous and Gaussian with the means and coefficients of variation listed in Table 1. The autocorrelation coefficient functions are assumed to be of the form... 

The perturbation method starts with a Taylor series expansion of the solution, the external loading, and the stochastic stiffness matrix in terms of the random variables introduced by the discretization of the random parameter field. The unknown coefficients in the expansion of the solution are obtained by equating terms of equal order in the expansion. From this, approximations of the first two statistical moments can be obtained. The perturbation method is computationally more efficient than direct Monte Carlo simulation. However, higher-order approximations will increase the computational effort dramatically, and therefore accurate results are obtained for small coefficients of variation only. 

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