Algorithms simultaneous estimation

Thus, we take advantage of the accuracy, robustness and efficiency of the direct problem solution, to tackle the associated inverse heat transfer problem analysis [26, 27] towards the simultaneous estimation of momentum and thermal accommodation coefficients in micro-channel flows with velocity slip and temperature jump. A Bayesian inference approach is adopted in the solution of the identification problem, based on the Monte Carlo Markov Chain method (MCMC) and the Metropolis-Hastings algorithm [28-30]. Only simulated temperature measurements at the external faces of the channel walls, obtained for instance via infrared thermography [30], are used in the inverse analysis in order to demonstrate the capabilities of the proposed approach. A sensitivity analysis allows for the inspection of the identification problem behavior when the external wall Biot number is also included among the parameters to be estimated. 

Simultaneous estimation of object space coordinates with measures for algorithmic quality control. 

When a Monte Carlo algorithm is used for estimation of any physical quantity B), a simple and fast additional procedure can be implemented that simultaneously estimates sensitivity of B to any parameter (Delatorre et al., 2014). This practically means that when Monte Carlo code is available that computes B, only a few additional lines of code are needed so that partial derivatives of B are also computed with respect to all the parameters ofinter-est. We are interested either in physical analysis (how does B evolve when a parameter is modified ) or in optimal design (what is the optimal value of the parameter for a target value of E ). A general overview of sensitivity estimation is available in Delatorre et al. (2014). This methodology was implemented in Dauchet et al. (2013) and Delatorre et al. (2014) to evaluate sensitivity of the radiation field within a DiCoFluV photobioreactor (see... 

In this section, we present an iterative algorithm in the spirit of the generalized least squares approach (Goodwin and Payne, 1977), for simultaneous estimation of an FSF process model and an autoregressive (AR) noise model. The unique features of our algorithm are the application of the PRESS statistic introduced in Chapter 3 for both process and noise model structure selection to ensure whiteness of the residuals, and the use of covariance matrix information to derive statistical confidence bounds for the final process step response estimates. An important assumption in this algorithm is that the noise term k) can be described by an AR time series model given by... 

This approach is useful when dealing with relatively simple partial differential equation models. Seinfeld and Lapidus (1974) have provided a couple of numerical examples for the estimation of a single parameter by the steepest descent algorithm for systems described by one or two simultaneous PDEs with simple boundary conditions. 

FIGURE 6 Convergence characteristics of the algorithm selecting the channel A, both parameters estimated simultaneously A, channel for parameter h opened first O, Channel for parameter X opened first (from Bortolotto et al., 1985). 

Historically, factorial designs were introduced by Sir R. A. Fisher to counter the then prevalent idea that if one were to discover the effect of a factor, all other factors must be held constant and only the factor of interest could be varied. Fisher showed that all factors of interest could be varied simultaneously, and the individual factor effects and their interactions could be estimated by proper mathematical treatment. The Yates algorithm and its variations are often used to obtain these estimates, but the use of least squares fitting of linear models gives essentially identical results. 

We choose three test molecules formaldehyde, proline and 2-phenylphenoxide. The structure of these systems is shown in Figure 1.8. The calculations were performed in vacuo and in water solution, with the C and the D versions of PCM with the standard and the simultaneous approaches. Here we note that we used the same solute-shaped cavity for all the optimizations of each system. The force field we used for all the calculations, both in vacuo and in solution, is the UFF [35] and the nuclear charges at the initial point were estimated with the QEq [36] algorithm. As we are not interested in obtaining results comparable with experimental data or with other calculations, but only in the PCM results with the different optimization schemes, the choice of the force field is not a critical point. The only requirement is that we performed all the calculations with the same force field. 

Fitting a two-component PARAFAC model using a least squares simultaneous algorithm provides (unique) estimates of A, B, and C that give a perfect model of X... 

Clearly, these residuals are far from perfect, and thus, even though the data are known to be perfectly trilinear with two components, the sequential PARAFAC algorithm fails to find a reasonable estimate of the parameters. However, this difference between sequential and simultaneous fitting is not related to the three-way nature of the PARAFAC model. Rather it is the orthogonality of the components in principal component analysis that enables the components to be calculated sequentially. A simple two-way example will help in illustrating this. 

Another common fitting algorithm found in the pharmacokinetic literature is extended least-squares (ELS) wherein 0, the structural model parameters, and 4>, the residual variance model parameters, are estimated simultaneously (Sheiner and Beal, 1985). The objective function in ELS is the same as the objective function in PL... 

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