Lagrangian method

Penalty functions with augmented Lagrangian method (an enhancement of the classical Lagrange multiplier method)... 

Another popular approach to the isothennal (canonical) MD method was shown by Nose [25]. This method for treating the dynamics of a system in contact with a thennal reservoir is to include a degree of freedom that represents that reservoir, so that one can perform deterministic MD at constant temperature by refonnulating the Lagrangian equations of motion for this extended system. We can describe the Nose approach as an illustration of an extended Lagrangian method. Energy is allowed to flow dynamically from the reservoir to the system and back the reservoir has a certain thermal inertia associated with it. However, it is now more common to use the Nose scheme in the implementation of Hoover [26]. 

To apply the Lagrangian method, this problem must be expressed mathematically as follows ... 

Fig. 6 Contour plots for the Lagrangian method (a) tablet hardness (b) dissolution (t50%) (c) feasible solution space indicated by crosshatched area. (From Ref. 15.)... 

Although the Lagrangian method was able to handle several responses or dependent variables, it was generally limited to two independent variables. A search method of optimization was also applied to a pharmaceutical system and was reported by Schwartz et al. [17], It takes five independent variables into... 

An important advance in making explicit polarizable force fields computationally feasible for MD simulation was the development of the extended Lagrangian methods. This extended dynamics approach was first proposed by Sprik and Klein [91], in the sipirit of the work of Car and Parrinello for ab initio MD dynamics [168], A similar extended system was proposed by van Belle et al. for inducible point dipoles [90, 169], In this approach each dipole is treated as a dynamical variable in the MD simulation and given a mass, Mm, and velocity, p.. The dipoles thus have a kinetic energy, JT (A)2/2, and are propagated using the equations of motion just like the atomic coordinates [90, 91, 170, 171]. The equation of motion for the dipoles is... 

Because this method avoids iterative calculations to attain the SCF condition, the extended Lagrangian method is a more efficient way of calculating the dipoles at every time step. However, polarizable point dipole methods are still more computationally intensive than nonpolarizable simulations. Evaluating the dipole-dipole interactions in Eqs. (9-7) and (9-20) is several times more expensive than evaluating the Coulombic interactions between point charges in Eq. (9-1). In addition, the requirement for a shorter integration timestep as compared to an additive model increases the computational cost. 

Although a direct comparison between the iterative and the extended Lagrangian methods has not been published, the two methods are inferred to have comparable computational speeds based on indirect evidence. The extended Lagrangian method was found to be approximately 20 times faster than the standard matrix inversion procedure [117] and according to the calculation of Bernardo et al. [208] using different polarizable water potentials, the iterative method is roughly 17 times faster than direct matrix inversion to achieve a convergence of 1.0 x 10-8 D in the induced dipole. 

Binder, J. L., and T. J. Hanratty, 1992, Use of Lagrangian Methods to Describe Drop Deposition and Distribution in Horizontal Gas-Liquid Annular Flows, Int. J. Multiphase Flow 7S(6) 803 821. (3)... 

The penalty term of an augmented Lagrangian method is designed to add positive curvature so that the Hessian of the augmented function is positive-definite. 

Murray, W., and Wright, M., Projected Lagrangian methods based on trajectories of barrier and penalty functions, SOL Report 78-23, Stanford University, Stanford, California (1978). 

Considering such recent relevance of SDP in quantum chemistry, this chapter discusses some practical aspects of this variational calculation of the 2-RDM formulated as an SDP problem. We first present the definition of an SDP problem, and then the primal and dual SDP formulations of the variational calculation of the 2-RDM as SDP problems (Section II), an efficient algorithm to solve the SDP problems the primal-dual interior-point method (Section III), a brief section about alternative and also efficient augmented Lagrangian methods (Section IV), and some computational aspects when solving the SDP problems (Section V). 

Software based on the augmented Lagrangian method (Section IV) is also available PENSDP by Kocvara and Stingl [24] (unique commercial code) and SDPLR by Burer, Monteiro, and Choi [26-28]. 

On the basis of the Lagrangian method (Lagrangian multipliers) and the conservation of total number of particles and total energy of the system, show that the Maxwell-Boltzmann distribution can take the form... 

In actual applications, the gas flow in a gravity settler is often nonuniform and turbulent the particles are polydispersed and the flow is beyond the Stokes regime. In this case, the particle settling behavior and hence the collection efficiency can be described by using the basic equations introduced in Chapter 5, which need to be solved numerically. One common approach is to use the Eulerian method to represent the gas flow and the Lagrangian method to characterize the particle trajectories. The random variations in the gas velocity due to turbulent fluctuations and the initial entering locations and sizes of the particles can be accounted for by using the Monte Carlo simulation. Examples of this approach were provided by Theodore and Buonicore (1976). 

Farhat et al. (1990) used augmented Lagrangian method to solve the optimisation problem presented above. See the original reference for further details. 

A different predictive procedure is to use the extended Lagrangian method, in which each dipole is treated as a dynamical variable and given a mass M and velocity (i. The dipoles thus have a kinetic energy, and are propagated using the equations of motion just like the atomic coordi-nates. The equation of motion for the dipoles is... 

In most electronegativity equalization models, if the energy is quadratic in the charges (as in Eq. [36]), the minimization condition (Eq. [41]) leads to a coupled set of linear equations for the charges. As with the polarizable point dipole and shell models, solving for the charges can be done by matrix inversion, iteration, or extended Lagrangian methods. 

In the extended Lagrangian method, as applied to a fluctuating charge system,the charges are given a fictitious mass, M, and evolved in time according to Newton s equation of motion, analogous to Eq. [23],... 

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