Constrained methods

The topics are grouped so that unconstrained methods are presented first, followed by constrained methods. The last two chapters in Part II deal with discontinuous (integer) variables, a common category of problem in chemical engineering, but one quite difficult to solve without great effort. 

If at the outset the data are very noisy and if the noise predominates in the Fourier frequency range needed to effect a restoration, constraints provide the only hope for improvement. The reason is that many of the noise values in the data would restore to physically unrealizable values by linear deconvolution. The constrained methods are inherently more robust because they must find a solution that is consistent with both data and physical reality. 

Knowing that the better nonlinear constrained methods are now available, why have researchers generally been reluctant to accept them Perhaps the linear approach has an attraction that is not related to performance. Early in a technical career the scientist-engineer is indoctrinated with the principles of linear superposition and analysis. Indeed, a rather large body of knowledge is based on linear methods. The trap that the linear methods lay for us is the existence of a beautiful and complete formalism developed over the years. Why complicate it by requiring the solution to be physically possible ... 

Perhaps the benefits of physical-realizability constraints, particularly ordinate bounds such as positivity, have not been sufficiently recognized. Surely everyone agrees in principle that such constraints are desirable. Even the early literature on this subject frequently mentions their potential advantages. For one reason or another, however, the earliest nonlinear constrained methods did not fully reveal the inherent power of constraints. 

Gold s method has been used by a number of workers, including Siska (1973), who applied it to molecular-beam scattering data, MacNeil and Dixon (1977), who applied it to photoelectron spectra, and Jones et al. (1967), who restored infrared spectra of condensed-phase samples. The author is unaware of any experimental results with this method, however, that illustrate the full potential achievable by constrained methods to be described later in this chapter. In the work of Jones et al., the resulting resolution is probably limited by the inherent breadth of spectral lines observed with condensed-phase samples. 

Like the other nonlinear constrained methods, the maximum-entropy method has proved its capacity to restore the frequency content of 6 that has not survived convolution by s and is entirely absent from the data (Frieden, 1972 Frieden and Burke, 1972). Its importance to the development of deconvolution arises from the statistical concept that it introduced. It was the first of the nonlinear methods explicitly to address the problem of selecting a preferred solution from the multiplicity of possible solutions on the basis of sound statistical arguments. 

Other constrained methods have also been applied. Beatham and Orchard (1976) experimented with Biraud s method but experienced only limited success. Vasquez et al (1981) found that maximum entropy is capable of yielding excellent results on simulated ESCA data. The authors, who used Burg s method, cite its freedom from need for trial-and-error optimization. They did, however, have to develop methods of dealing with problems of instability and lack of an order-selecting criterion. 

For the present work, we chose the constrained method described by Jansson (1968) and Jansson et al (1968, 1970). See also Section V.A of Chapter 4 and supporting material in Chapter III. This method has also been applied to ESCA spectra by McLachlan et al (1974). In our adaptation (Jansson and Davies, 1974) the procedure was identical to that used in the original application to infrared spectra except that the data were presmoothed three times instead of once, and the variable relaxation factor was modified to accommodate the lack of an upper bound. Referring to Eqs. (15) and (16) of Section V.A.2 of Chapter 4, we set k = 2o(k)K0 for 6(k) < j and k = Kq exp[3 — for o(k) > This function is seen to apply the positivity constraint in a manner similar to that previously employed but eliminates the upper bound in favor of an exponential falloff. We also experimented with k = k0 for o(k) > j, and found it to be equally effective. As in the infrared application, only 10 iterations were needed. 

As the Excel Solver is only for single objective optimization, use the e-constrained method and the Excel Solver.xls on the CD to optimize the 4-plant IE for the two objectives as in Cases A and B. For the Solver to work reliably, number of decision variables should be limited. Thus, it is recommended to set Z21 = Z32 = 0 and Z22 = 1 = 1 for Z = a, b, c and d. This would leave the capacities of the 4 plants (Xj) as the decision variables. Treat lEvP as the constraint and vary it in the range 1.213-1.419 for Case A and 1.220-1.321 for Case B, and observe the trends of the decision variables and the objective. Do they follow similar trends as the IE for 6 plants ... 

We may conclude from Fig. 4, that by implementing the density-constrained method described above, in the context of local-scaling transformations, we obtain the same potential reported by Almbladh and Pedroza [80]. An important point we would like to stress with respect to the present method is that it is based on the constrained minimization of the kinetic energy, and for this reason, it is not... 

The projected symplectic constrained method (4.20)-(4.24) is only first order accurate. We forego providing a detailed proof of this fact, but note that it could be demonstrated using standard methods [164]. Note that (4.20)-(4.24) reduces to the symplectic Euler method in the absence of constraints, and the projection of the momenta would not alter this fact. There are several constraint-preserving, second-order alternatives which generalize the Stormer-Verlet scheme. One of these is the SHAKE method [322]. The original derivation of the SHAKE method began from the position-only, two-step form of the Stormer rule for q = F(q)... 

Several useful methods have been proposed to overcome the variational coUapse problem, and a number of different schemes have been proposed for obtaining SCF wave functions for excited states [10, 16-26]. In recent years, there has been renewed interest in the orthogonality-constrained methods [14, 27] as weU as in the SCF theory for excited states [28-32]. It is clear that an experience accumulated for the HF excited state calculations can be useful to develop similar methods within density functional theory [33-36]. Some of these approaches [10, 18, 19, 23, 24, 26, 30-35] explicitly introduce orthogonality constraints to lower states. Other methods [21, 22, 25] either use this restriction implicitly or locate excited states as higher solutions of nonlinear SCF equations [29]. In latter type of scheme, the excited state SCF wave functions of interest are not necessarily orthogonal to the best SCF functions for a lower state or states of the same symmetry. 

We have presented in some detail the state of the art in available methods for the QM/MM simulations of chemical reactions in solution. We distinguish three classes of methods, each with a different approach to the diffusion of solvent molecules across the QM/MM boundary. The first class, constrained QM/MM, prevents this diffusion, and as a consequence dynamical properties are correctly described only well within the QM region. In addition, recovery of the correct structural properties at the QM/MM boundary still poses a challenge. Nonetheless, the constrained methods have the non-negligible advantage of simplicity and low computation cost, equivalent to that of a conventional QM/MM simulation. 

While the constrained method for calculating a PMF is tor-mally straightforward, the method using a biasing restraint is potentially more problematic. This is because one often encounters situations where there are large fluctuations in g f /biiu//f7 Qyg fpig trajectory for each window. This leads to inferior statistics for Another difficulty in applying... 

There are several approaches to obtaining such solutions (see Sect. 1.5.1). Basically, they are based on the conversion of the MO problem into one single objective function problem. The next section is focused on the e-constrained method which is... 

The different components are meshed and their interaction modelled by the contact algorithm or constrain method of the software [1]. The contact pair tracking methods depend on the software. Except during the assembly of the cell, which can be simplified, the relative motion of the SRU components is small, since it is mostly driven by the differences in thermal and isothermal expansion. This enables the use of a simpler small-sliding tracking method, to reduce the computation time. Similarly, the correction for nonlinear geometric effects is not required in common situations, since large deformations are not expected. 

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