Lagrange multiplicator

Now, by using the Lagrange multiplicators method, the equation governing the maximalization of the entropy required in order to obtain the statistical equilibrium, in view of the four constraints ... 

Function L(pj,p2,. ..pL, i, supports the same minimization as (Pi, P2, Pl) but here, the number of parameters is increased with, > 2 > > which are called the Lagrange multiplicators. Then, the equation system that must be solved here is written as ... 

The name of the distribution is due to the obtained as the vector of Lagrange multiplic for which the relative entropy,... 

Basically, two types of approaches are developed here iterative (optimization-based) approaches like the one by Sippl et al. [101] and direct approaches like the one by Kabsch [102, 103], based on Lagrange multipliers. Unfortunately, the much expedient direct methods may fail to produce a sufficiently accurate solution on some degenerate cases. Redington [104] suggested a hybrid method with an improved version of the iterative approach, which requires the computation of only two 3x3 matrix multiplications in the inner loop of the optimization. 

Confronted with a problem in which two data sets were available, Breedlove et al. (1977) chose a solution that minimizes a sum of terms not unlike expression (56). Available were two images one a blurred representation of the object, the other a superposition of sharp renderings. In this sum, the right-hand term accommodates the blurred image as in expression (56). The other term incorporates the multiple exposure via the Lagrange multiplier technique. Solutions obtained by this method illustrated the desirability of using all the available data. 

To prove this assertion, it is first useful to consider the mathematical technique of Lagrange multipliers, a method used to find the extreme (maximum or minimum) value of a function subject to constraints. Rather than develop the method in complete generality, we merely introduce it by application to the problem just considered equilibrium in a single-phase, multiple-chemical reaction system. 

An alternative method of obtaining a solution to the multiple-reaction, single-phase equilibrium problem is to use the method of Lagrange multipliers. Here one first rewrites the constraints as... 

Add a multiple of the orthonormaJity constraints using the Lagrange method. 

In the case of multiple reaction equilibria, the number of moles of all reactive compounds in chemical equilibrium can be determined with the help of nonlinear regression methods [11]. But at the same time, the element balance has to be satisfied this means the amount of carbon, hydrogen, oxygen, nitrogen has to be the same before and after the reaction. This can either be taken into account with the help of Lagrange multipliers or using penalty functions [11], as shown in Example 12.8. 

When n is described by multiple variables which are a function of more than one coordinate, each variable must satisfy the Euler-Lagrange equation as Equation (5.15). 

The same authors also presented an example of the use of the population balance equation (PBE) (distribution of biomass m) coimected to the multi-zone/CFD model. This example is in several respects relevant for the assessment of the modeling approach. The coupling of the integro-differential equation of the population balance is a numerical challenge, which can nowadays be tackled within the environment of a CFD approach, albeit without consensus on the proper closure assumptions. Still, the computational effort for the numerical solution of the population balance embedded in the multizonal model is extensive, and it is difficult to extend this approach to multiple state variables necessary for dynamic metabolic models. This is an important argument to favor the alternative method of an agent-based Lagrange-Euler approach discussed in Section 3.5. 

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