The Linear Method

The material here is taken mainly from the unpublished notes by Andersen [5.1] and the paper by Andersen and Woolley [5.2]. 

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We now discuss the most important theoretical methods developed thus far the augmented plane wave (APW) and the Korringa-Kolm-Rostoker (KKR) methods, as well as the linear methods (linear APW (LAPW), the linear miiflfm-tin orbital [LMTO] and the projector-augmented wave [PAW]) methods. 

Krasovskll E E and Schattke W 1997 Surface electronic structure with the linear methods of band theory Phys. Rev. B 56 12 874... 

In order to reduce the complexity of the problem, several approximation schemes have been developed. In the BGK model, the collision integral is replaced by a simple local term ensuring that the well-known Maxwell distribution is reached at thermal equilibrium [16]. The linearization method assumes that the phase space distribution is given by a small perturbation h on top of a (local) Maxwell distribu-tion/o (see, e.g., [17, 18]) ... 

For the special case for which n = 2, it can be shown that the linearization method defined above becomes identical to the Newton-Raphson method. The result may be generalized to apply to any homogeneous function of degree n. 

While it is technically erroneous to claim that the linearization method does not require any initialization (J2), it is true that the initialization procedure used appear to be quite effective. A more comprehensive discussion of initialization procedure will be given in Section III,A,5. With this initialization procedure, the linearization method appears to converge very rapidly, usually in less than 10 iterations for formulations A and B. Since the evaluation of f(x) and its partial derivatives is not required, the method is also simpler and easier to implement than the Newton-Raphson method. 

On the debit side, the linearization method is quite sensitive to the form of the network element model. Jeppson and Tavallaee (J2) reported that convergence rate was slow when the usual pump and reservoir models were incorporated, but they obtained significant improvements after the models had been suitably transformed. Although the number of iterations required is small using formulations A and B, the dimension of the matrix equation is substantial. Hence, it becomes essential to use sparse computation techniques if the method is to retain its competitive edge in larger problems. 

For formulations A and B, one general procedure is to solve the laminar flow equations which are linear and use the solution as the initial guesses for the nonlinear equations. Variations of this procedure have been used by Bending and Hutchison (B5), Wood and Charles (Wll), and Jeppson and Tavallaee (J2) in conjunction with the linearization method. 

The methodology takes the form of an MINLP, which must be linearised to find a solution. The linearization method used was the relaxation-linearization technique proposed by Quesada and Grossman (1995). During the application of the formulation to the illustrative examples it was found that only one term required linearization for a solution to be found. 

In the linearization method, the nonlinear model of Eq. (40) is linearized by a truncated Taylor expansion ... 

Experience with fitting many models indicates that the steepest descent is very stable for the initial iterations, while the linearization method is more efficient for the final iterations. A compromise method has been suggested (L5, Ml, M2) which tends to emphasize steepest descent at the outset and... 

Kelen and Tudos [1975] refined the linearization method by introducing an arbitrary positive constant a into Eq. 6-37 to spread the data more evenly so as to give equal weight to all... 

Even with the Kelen Tudos refinement there are statistical limitations inherent in the linearization method. The independent variable in any form of the linear equation is not really independent, while the dependent variable does not have a constant variance [O Driscoll and Reilly, 1987]. The most statistically sound method of analyzing composition data is the nonlinear method, which involves plotting the instantaneous copolymer composition versus comonomer feed composition for various feeds and then determining which theoretical plot best fits the data by trial-and-error selection of r and values. The pros and cons of the two methods have been discussed in detail, along with approaches for the best choice of feed compositions to maximize the accuracy of the r and r% values [Bataille and Bourassa, 1989 Habibi et al., 2003 Hautus et al., 1984 Kelen and Tudos, 1990 Leicht and Fuhrmann, 1983 Monett et al., 2002 Tudos and Kelen, 1981]. 

In spite of performance advantages in the use of nonlinear methods, it is instructive to start our deconvolution study by examining the linear methods they will give us insight into the process. The ensuing development will also define the applicability domain of linear methods and reveal their limitations. We shall see that in some circumstances a linear method is the method of choice. 

We have also learned that the point-successive methods need not demand more computer memory than point-simultaneous methods, and that the seemingly inherent asymmetry of the point-successive methods can be overcome. Furthermore, the linear methods described in this section are... 

In preceding chapters we laid a foundation for the study of deconvolution. We presented several linear methods that exemplify the groundwork available before recent developments revolutionized the deconvolution field. Why, in their simplicity and elegance, did the linear methods fail to stimulate the wide adoption of deconvolution methods in spectroscopy After all, available instrumental resolution is limiting in many applications, and the simplicity of the microcomputer makes numerical processing attractive. 

The answer lies in the relatively poor performance of linear methods, especially with band-limited data. Frequently a linear restoration reveals little true structure that could not have been seen in the original data. Even worse, noise-based artifacts often call the result into question. One might even say that the linear methods have helped to give deconvolution a bad reputation in spectroscopy. 

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 ... 

The linear methods have no provisions for bounds on 6(x). In principle, solutions are free to assume any value, as long as they satisfy the data i(x). We have shown that a problem of uniqueness results. Many functions are thus possible candidates—they all satisfy the data, or nearly so. Noise, which is always present in measurements, unfortunately provides the basis by which a linear method prefers one solution over another. 

First let us deal with deconvolution in general. We have a few admonitions to the reader of a literature report on a new method. They should ask, does the writer deal fairly with noise Even the most volatile of the linear methods can produce a reasonable restoration when noise is limited to roundoff error in the seventh significant figure of the data. A method s capability of yielding acceptable restorations in the presence of realistic noise is critical to its practicality. 

However, the linear method (i.e., a small displacement of potential) is better. Thus, going out far from Vcorr on the anodic side may run into a region in which the metal forms oxide fdms and on the cathodic side, the evolution of Hj could interfere with the anodic dissolution current, which could confusingly lead to an erroneous contribution (via H2 - 2H+ + 2e) to the anodic dissolution reaction. 

There are a number of band-structure methods that make varying approximations in the solution of the Kohn-Sham equations. They are described in detail by Godwal et al. (1983) and Srivastava and Weaire (1987), and we shall discuss them only briefly. For each method, one must eon-struct Bloch functions delocalized by symmetry over all the unit cells of the solid. The methods may be conveniently divided into (1) pesudopo-tential methods, (2) linear combination of atomic orbital (LCAO) methods (3) muffin-tin methods, and (4) linear band-structure methods. The pseudopotential method is described in detail by Yin and Cohen (1982) the linear muffin-tin orbital method (LMTO) is described by Skriver (1984) the most advanced of the linear methods, the full-potential linearized augmented-plane-wave (FLAPW) method, is described by Jansen... 

The traveltime inverse problem provides another typical example of application of the linearization method to the solution of inverse problems for wave phenomena. 

Polynomial interpolation is simply an extension of the linear method. The polynomial is formed by adding extra terms to the model to represent curved regions of the spectrum and using extra data values in the model. 

The great advantage of the methods described in this section over those described earlier is, of course, rapidity in computation. This gain in computational simplicity is, however, at the expense of theoretical rigor. It is, therefore, important to establish the accuracy of the methods described above using the exact method of Section 8.3 as a basis for comparison. The extensive numerical computations made by Smith and Taylor (1983) showed that the explicit method of Taylor and Smith ranked second overall among seven approximate methods tested (the linearized method of Section 8.4 was best). For some determinacy... 

Repeat Example 8.7.1 using the linearized method of Toor-Stewart-Prober (1964) discussed in Section 8.4. 

Even with the Kelen-Tudos refinement there are statistical limitations inherent in the linearization method. It has been shown [18] that the independent variable in any form of the linear equation is not really independent while the dependent variable does not have a constant variance. The most statistically sound method of analyzing the experimental composition data is the nonlinear method which involves direct curve fitting to the copolymer composition equation. 

Donaldson and Schnabel (1987) used Monte Carlo simulation to determine which of the variance estimators was best in constructing approximate confidence intervals. They conclude that Eq. (3.47) is best because it is easy to compute, and it gives results that are never worse and sometimes better than the other two, and is more stable numerically than the other methods. However, their simulations also show that confidence intervals obtained using even the best methods have poor coverage probabilities, as low as 75% for a 95% confidence interval. They go so far as to state confidence intervals constructed using the linearization method can be essentially meaningless (Donaldson and Schnabel, 1987). Based on their results, it is wise not to put much emphasis on confidence intervals constructed from nonlinear models. 

When condition (2) is not met, we have to deal with a stationary point of higher order, for which the linearization method also fails. It appears that there exists a great variety of types of stationary points of higher order. 

The first accurate band structure calculations with inclusion of relativistic effects were published in the mid-sixties. Loucks published [64-67] his relativistic generalization of Slaters Augmented Plane Wave (APW) method. [68] Neither the first APW, nor its relativistic version (RAPW), were linearized, and calculations used ad hoc potentials based on Slaters s Xa scheme, [69] and were thus not strictly consistent with the density-functional theory. Nevertheless (or, maybe therefore ) good descriptions of the bands, Fermi surfaces etc. of heavy-element solids like W and Au were obtained.[3,65,70,71] With this background it was a rather simple matter to include [4,31,32,72] relativistic effects in the linear methods [30] when they (LMTO, LAPW) appeared in 1975. 

The introduction continues with a brief account of the approximations usually made to arrive at a solvable electronic-structure problem, and we shall discuss several of the methods applied to calculate band structures in solids. Then there is a brief summary of the history and development of the linear methods of band-structure calculations, followed by an outline of the remaining chapters. 

Before presenting the linear method, let us briefly review how the energy-band problem has been tackled in the past. In this context we note that the traditional methods may be divided into those which express the wave functions as linear combinations of some fixed basis functions, say plane waves or atomic orbitals, and those like the cellular, APW, and KKR methods which employ matching of partial waves. As we shall see, both approaches have their strong and weak points. 

The linear methods devised by Andersen [1.19] are characterised by using fixed basis functions constructed from partial waves and their first energy derivatives obtained within the muffin-tin approximation to the potential. 

These methods therefore lead to secular equations (1.21) which are linear in energy, that is to eigenvalue equations of the form (1.19). When applied to a muffin-tin potential they use logarithmic-derivative parameters and provide solutions of arbitrary accuracy in a certain energy range. The linear methods thus combine the desirable features of the fixed-basis and partial-wave methods. 

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