Laplacian approximation

One type of approximation is the Laplacian approximation.1 Given a complex integral, Jf(x)dx,f(x) is reexpressed as exp [Ln(f(x)] = exp [g(x)]. g(x) can then be approximated using a second-order Taylor series approximation about the point x0... 

The NLME function in S-Plus offers three different estimation algorithms a FOCE algorithm similar to NONMEM, adaptive Gaussian quadrature, and Laplacian approximation. The FOCE algorithm in S-Plus, similar to the one in NONMEM, was developed by Lindstrom and Bates (1990). The algorithm is predicated on normally distributed random effects and normally distributed random errors and makes a first-order Taylor series approximation of the nonlinear mixed effects model around both the current parameter estimates 0 and the random effects t). The adaptive Gaussian quadrature and Laplacian options are similar to the options offered by SAS. 

This was the expression for a obtained by a different route, by Fowler in 1937. By making the drastic Laplacian approximation of a sharp liquid surface he replaced the unknown function p (r,2, Zi, Z2) in the interface by the simpler function (p ) g(ri2) in the homogeneous liquid. The latter can now be determined by X-ray or neutron diftraction or by accurate and well-tested (but not exact) theoretical methods. ... 

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. 

The explicit dependence on R is not shown for equation (A.6) and will not be given in subsequent equations, it being understood that unless stated otherwise, we are working within the BO approximation. The Laplacian operator V(/)2 in Cartesian coordinates for the 7th electron is given by... 

To get an approximation for the variation of c across the tube, we argue that it must be the average (c) plus a deviation ci and, because the Laplacian of a constant is zero,... 

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. 

In order to make improvements over the LSDA, one has to assume that the density is not uniform. The approach that has been taken is to develop functionals that are dependent on not only the electron density but also derivatives of the density. This constitutes the generalized gradient approximation (GGA) and is the second rung on Jacob s Ladder. The third rung, meta-GGA functional, includes a dependence on the Laplacian of the density (V p) or on the orbital kinetic energy density (t). The fourth row, the hyper-GGA or hybrid functionals, includes a dependence on exact (HF) exchange. Finally, the fifth row incorporates the unoccupied Kohn-Sham orbitals. This is most widely accomplished within the so-called double hybrid functionals. 

Here U N) is the interaction potential energy for the complete system at a specific configuration, uniformly the same quantity that has been discussed above. U(TV) is an ejfective potential designed to be used in classical-limit partition function calculations, e.g. Eq. (3.17), p. 40, in order to include quantum mechanical effects approximately. We will call this /(TV) the quadratic Feynman-Hibbs (QFH) model. In Eq. (3.67), Mj is the mass of atom j, and V is the Laplacian of the... 

The cross Laplacian, Aj, is Important for narrow light beams, for systems with self-focusing. It is worthwhile to stress here tliat most of tlie currently cherished approaches applied for solving Eq. (10), are based on tlie power series expansion (2) of the polarization over the laser field amplitude and on an account of only the coherent contributions to the corresponding polarizabilities. However, this approximation breaks down even for fairly short laser pulses, which is the case... 

The Laplacian operator on the LHS of the pressure equation is the product of the divergence operator originating from the continuity equation and the gradient operator that comes from the momentum equations. The RHS of the pressure equation consists of a sum of derivatives of the convective terms in the three components of the momentum equation. In all these terms, the outer derivative stems from the continuity equation while the inner derivative arises from the momentum equation. In a numerical approximation, it is essential that the consistency of these operators is maintained. The approximations of the terms in the Poisson equations must be defined as the product of the divergence and gradient approximations used in the basic equations. Violation of this constraint may lead to convergence problems as the continuity equation is not appropriately satisfied. 

One may use the cutoff function technique (see Section 4.3) and change t/r fr) near the points r = R to regularize it. The modified function belongs to the domain of definition of the Laplacian operator (an alternative method is mollification, see Sect. 2.17, 2.18 in [22]) and has approximately the same mean energy. Then simple estimates (see e.g. Section V.5.3 in [25]) and relations of Equation (2.4) demonstrate that all of the values R(r), (r) are uniformly bounded, for some constant Co and large enough R, and the following simple estimates hold for any r ... 

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