Laplacian method

The FO method was the first algorithm available in NONMEM and has been evaluated by simulation and used for PK and PD analysis [9]. Overall, the FO method showed a good performance in sparse data situations. However, there are situations where the FO method does not yield adequate results, especially in data rich situations. For these situations improved approximation methods such as the first-order conditional estimation (FOCE) and the Laplacian method became available in NONMEM. The difference between both methods and the FO method lies in the way the linearization is done. 

The NONMEM program implements two alternative estimation methods, the first-order conditional estimation and the Laplacian methods. The first-order conditional estimation (FOCE) method uses a first-order expansion about conditional estimates (empirical Bayes estimates) of interindividual random effects, rather than about zero. In this respect, it is like the conditional first-order method of Lindstrom and Bates.f Unlike the latter, which is iterative, a single objective function is minimized, achieving a similar effect as with iteration. The Laplacian method uses second-order expansions about the conditional estimates of the random effects. ... 

Jonsson (25) showed from a simulation study that the use of the standard mixed effects modeling approach may produce biased parameter estimates when ordered categorical data with a skewed distribution are analyzed using the Laplacian method. Increasing interindividual variability and skewness in the distribution of the data increase the bias associated with the estimation of those parameters. The conse-... 

A much less basis set dependent method is to analyze the total electron density. This is called the atoms in molecules (AIM) method. It is designed to examine the small effects due to bonding in the primarily featureless electron density. This is done by examining the gradient and Laplacian of electron density. AIM analysis incorporates a number of graphic analysis techniques as well as population analysis. The population analysis will be discussed here and the graphic techniques in the next chapter. 

Several methods have been used for analyzing the electron density in more detail than we have done in this paper. These methods are based on different functions of the electron density and also the kinetic energy of the electrons but they are beyond the scope of this article. They include the Laplacian of the electron density ( L = - V2p) (Bader, 1990 Popelier, 2000), the electron localization function ELF (Becke Edgecombe, 1990), and the localized orbital locator LOL (Schinder Becke, 2000). These methods could usefully be presented in advanced undergraduate quantum chemistry courses and at the graduate level. They provide further understanding of the physical basis of the VSEPR model, and give a more quantitative picture of electron pair domains. 

Because the conditional scalar Laplacian is approximated in the FP model by a non-linear diffusion process (6.91), (6.145) will not agree exactly with CMC. Nevertheless, since transported PDF methods can be easily extended to inhomogeneous flows,113 which are problematic for the CMC, the FP model offers distinct advantages. 

Using Jacobi s method to compute the inverse of the Laplacian is rather slow. Faster convergence may be achieved using successive over-relaxation (SOR) (Bronstein et al. 2001 Demmel 1996). The iterative solver can also be written in the Gauss-Seidel formulation where already computed results are reused. 

The electron density and its Laplacian have been correlated exponentially with the HB distance. These results are in agreement with other reports that show the generality of this rule [120-122]. The dimers of a series of chiral and nonchiral a-aminoalcohols were studied by means of DFT methods (B3LYP/6-311++G ) (Scheme 3.14). Due to the flexibility of the molecules, twelve possible complexes can be proposed in the case of the chiral molecules. Half of them are mirror image of the other half, and thus, only six where considered. The six unique configurations studied for each dimer (Scheme 3.15) correspond to three homochiral dimers (RR or SS) and three heterochiral ones (RS or SR), and can be grouped in three chairs, Cl-3, and three boats, Bl-3, respectively. 

The FOCE method uses a first-order Taylor series expansion around the conditional estimates of the t] values. This means that for each iteration step where population estimates are obtained the respective individual parameter estimates are obtained by the FOCE estimation method. Thus, this method involves minimizations within each minimization step. The interaction option available in FOCE considers the dependency of the residual variability on the interindividual variability. The Laplacian estimation method is similar to the FOCE estimation method but uses a second-order Taylor series expansion around the conditional estimates of the 77 values. This method is especially useful when a high degree of nonlinearity occurs in the model [10]. 

The following are the steps used in the method of separation of variables in solving the Laplacian equation. 

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