Algorithm minimization

A drop of water that is placed on a hillside will roll down the slope, following the surface curvature, until it ends up in the valley at the bottom of the hill. This is a natural minimization process by which the drop minimizes its potential energy until it reaches a local minimum. Minimization algorithms are the analogous computational procedures that find minima for a given function. Because these procedures are downhill methods that are unable to cross energy barriers, they end up in local minima close to the point from which the minimization process started (Fig. 3a). It is very rare that a direct minimization method... 

The goal of all minimization algorithms is to find a local minimum of a given function. They differ in how closely they try to mimic the way a drop of water or a small ball would roll down the slope, following the surface curvature, until it ends up at the bottom. Consider a Taylor expansion around a minimum point Xq of the general one-dimensional function F(X), which can be written as... 

Order 2 minimization algorithms, which use the second derivative (curvamre) as well as the first derivative (slope) of the potential function, exhibit in many cases improved rate of convergence. For a molecule of N atoms these methods require calculating the 3N X 3N Hessian matrix of second derivatives (for the coordinate set at step k)... 

Finally, the use of the constant pressure minimization algorithm allows searching for phenomena that can be considered as precursors of pressure-induced transitions. For example, the predicted behaviour of the anatase cell constants as a function of pressure shows that the a(P) and c(P) plots are only linear for P<4 GPa, the value that is close to both the theoretical and experimental transition pressures. At higher pressures the a constant starts to grow under compression, indicating inherent structural instability. In the case of ratile there is a different precursor effect, nami y at 11 GPa the distances between the titanium atom and the two different oxygens, axial and equatorial, become equal. Once again, the pressure corresponds closely to the phase transition point. 

The first task of the estimation procedure is to quickly and efficiently screen all possible sets of interaction parameters that could be used. For example if the Trebble-Bishnoi EoS were to be employed which can utilize up to four binary interaction parameters, the number of possible combinations that should be examined is 15. The implicit LS estimation procedure provides the most efficient means to determine the best set of interaction parameters. The best set is the one that results in the smallest value of the LS objective function after convergence of the minimization algorithm has been achieved. One should not readily accept a set that... 

The elements of the covariance matrix of the parameter estimates are calculated when the minimization algorithm has converged with a zero value for the Marquardt s directional parameter. The covariance matrix of the parameters COV(k) is determined using Equation 11.1 where the degrees of freedom used... 

To offer more flexibility we adopt an approach, based on the transient simulation model TRNSYS (Klein et al., 1976), making use of the Lund DST borehole model (Hellstrom, 1989). The parameter estimation procedure is carried out using the GenOPT (Wetter, 2004) package with the Nelder and Mead Simplex minimization algorithm (Nelder and Mead, 1965) or Hooke and Jeeves minimization algorithm (Hooke and Jeeves, 1961). 

Broyden, C. G. The Convergence of a Class of Double-Rank Minimization Algorithms. J Inst Math Appl 6 76-90 (1970). 

Hiriart-Urruty, J. D. and C. Lemarechal. Convex Analysis and Minimization Algorithms. Springer-Verlag, Berlin (1993). 

Generating Inherent Structures of Liquids Comparison of Local Minimization Algorithms. 

In practice, initial guesses of the fitting parameters (e.g. pre-exponential factors and decay times in the case of a multi-exponential decay) are used to calculate the decay curve the latter is reconvoluted with the instrument response for comparison with the experimental curve. Then, a minimization algorithm (e.g. Marquardt method) is employed to search the parameters giving the best fit. At each step of the iteration procedure, the calculated decay is reconvoluted with the instrument response. Several softwares are commercially available. 

A new feature in MM3 is the full Newton-Raphson minimization algorithm. This allows for the location and verification of transition states and for the calculation of vibrational spectra. Indeed, many of the new potential functions in MM3 were included to provide a better description of the potential energy surface which is required for an accurate calculation of vibrational spectra. 

MMI/MMPI incorporates a modified Newton-Raphson energy minimization algorithm that moves atoms one by one and is quite efficient. The force field is parameterized not only for saturated hydrocarbons including cyclopropane, but also for nonconjugated olefins (17c),... 

The best-fitting set of parameters can be found by minimization of the objective function (Section 13.2.8.2). This can be performed only by iterative procedures. For this purpose several minimization algorithms can be applied, for example, Simplex, Gauss-Newton, and the Marquardt methods. It is not the aim of this chapter to deal with non-linear curve-fitting extensively. For further reference, excellent papers and books are available [18]. 

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