Nonlinear Monte Carlo method

Dondi et al. developed a stochastic approach to nonlinear chromatography based on the Monte Carlo method [69]. The Monte Carlo method consists in simulating the migration of an ensemble of molecules through the chromatographic column that contains a finite number of adsorption sites. The random sequence of adsorption-desorption events is modeled at the molecular level with the stochastic terms and concepts discussed in Chapter 6, Section 6.5. 

Monte-Carlo methods are able to simulate rather complicated nonlinear phenomena, like periodic oscillations and formation of waves on the catalyst surfaces, mirntcking experimental observations (Figuresw 3.21-3-23). 

Prediction of molecular weight distributions for reversible, linear, alternating polycondensation is discussed in Section 3.4.5. Mathematical difficulties grow considerably in the presence of SSSEs. There seems to be no alternative to Monte Carlo methods for dealing with reversible nonlinear polycondensations or even linear polycondensations where more than two kinds of bonds are present. 

Myers et al. (2007, 2009) introduced a Bayesian nonlinear inversion framework to multiple-event analysis (Bayesloc). The nonlinear Markov chain Monte Carlo method enables Bayesloc to simultaneously assess the joint posterior distribution spanning event locations, travel-time corrections, phase names, and arrival-time measurement precision. Myers et al. (2011) demonstrated that Bayesloc can be applied to data sets of tens of thousands of events and millions of arrivals because the solution does not involve direct inversion of a matrix and computation demands grow linearly with the number of arrivals. 

The probabilistic stracture and the statistical moments of the response of any type of nonlinear mechanical systems can be evaluated using simulation techniques, tike the Monte Carlo method. This method operates in the time domain by repeating a great number of times deterministic... 

The original proposal of the approach, supported by a Monte Carlo simulation study [36], has been further validated with both pre-clinical [38, 39] and clinical studies [40]. It has been shown to be robust and accurate, and is not highly dependent on the models used to fit the data. The method can give poor estimates of absorption or bioavailability in two sets of circumstances (i) when the compound shows nonlinear pharmacokinetics, which may happen when the plasma protein binding is nonlinear, or when the compound has cardiovascular activity that changes blood flow in a concentration-dependent manner or (ii) when the rate of absorption is slow, and hence flip-flop kinetics are observed, i.e., when the apparent terminal half-life is governed by the rate of drug input. 

First-order error analysis is a method for propagating uncertainty in the random parameters of a model into the model predictions using a fixed-form equation. This method is not a simulation like Monte Carlo but uses statistical theory to develop an equation that can easily be solved on a calculator. The method works well for linear models, but the accuracy of the method decreases as the model becomes more nonlinear. As a general rule, linear models that can be written down on a piece of paper work well with Ist-order error analysis. Complicated models that consist of a large number of pieced equations (like large exposure models) cannot be evaluated using Ist-order analysis. To use the technique, each partial differential equation of each random parameter with respect to the model must be solvable. 

On- and off-lattice Monte Carlo simulations are among the best available methods to analyze complex nonlinear polymerizations, particularly those presenting a high extent of intramolecular cyclization (Somvarsky and Dusek, 1994 Anseth and Bowman, 1994). 

The TPE-HNC/MS theory reduces to an integral form of the nonlinear Poisson-Boltzmann equation in the limit of point ions [8,44]. Hence, in that limit agreement between the two methods is exact. For a 0.1 M, 1 1 electrolyte separating plates with surface potentials of 70 mV, Lozada-Cassou and Diaz-Herrera [8] show excellent agreement between the TPE-HNC/MS theory and the Poisson-Boltzmann equation. The agreement becomes very poor, however, at a higher concentration of 1 M. In addition, like the Monte Carlo and AHNC results, the TPE-HNC/MS theory predicts attractive interactions at sufficiently high potentials and/or salt concentrations, and such effects are missed entirely by the Poisson-Boltzmann equation. 

Polymerization rate represents the instantaneous status of reaction locus, but the whole history of polymerization is engraved within the molecular weight distribution (MWD). Recently, a new simulation tool that uses the Monte Carlo (MC) method to estimate the whole reaction history, for both hnear [263-265] and nonlinear polymerization [266-273], has been proposed. So far, this technique has been applied to investigate the kinetic behavior after the nucleation period, where the overall picture of the kinetics is well imderstood. However, the versatility of the MC method could be used to solve the complex problems of nucleation kinetics. 

Donaldson and Schnabel (1987) used Monte Carlo simulation to determine which of the variance estimators was best in constructing approximate confidence intervals. They conclude that Eq. (3.47) is best because it is easy to compute, and it gives results that are never worse and sometimes better than the other two, and is more stable numerically than the other methods. However, their simulations also show that confidence intervals obtained using even the best methods have poor coverage probabilities, as low as 75% for a 95% confidence interval. They go so far as to state confidence intervals constructed using the linearization method can be essentially meaningless (Donaldson and Schnabel, 1987). Based on their results, it is wise not to put much emphasis on confidence intervals constructed from nonlinear models. 

More difficult is the estimation of errors for the nonlinear parameters, since no variance-covariance matrix exists. Frequently, the error estimations are restricted to a locally linear range. In the linearization range, the confidence bands for the parameters are then calculated as in the linear case (Eqs. (6.25)-(6.27)). An alternative consists in error estimations on the basis of Monte Carlo simulations or bootstrapping methods (cf. Section 8.2). 

If used for modeling, it should be noted that estimation of confidence intervals for the weights (parameters) or for predictions cannot be performed analytically because of the nonlinearity of the networks. As with other nonhnear methods, the confidence intervals have to be estimated by means of Monte Carlo simulations or bootstrapping methods. 

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