Downhill simplex method

Figure 4 A representative step m the downhill simplex method. The original simplex, a tetrahedron in this case, is drawn with solid lines. The point with highest energy is reflected through the opposite triangular plane (shaded) to form a new simplex. The new vertex may represent symmetrical reflection, expansion, or contractions along the same direction.

Another interesting implementation of simulated annealing for continuous minimization (like a typical parameter estimation problem) utilizes a modification of the downhill simplex method. Press et al. (1992) provide a brief overview of simulated annealing techniques accompanied with listings of computer programs that cover all the above cases. 

The choice of the exchange correlation functional in the density functional theory (DFT) calculations is not very important, so long as a reasonable double-zeta basis set is used. In general, the parameterized model will not fit the quantum mechanical calculations well enough for improved DFT calculations to actually produce better-fitted parameters. In other words, the differences between the different DFT functionals will usually be small relative to the errors inherent in the potential model. A robust way to fit parameters is to use the downhill simplex method in the parameter space. Having available an initial set of parameters, taken from an analogous ion, facilities the fitting processes. 

The simplex method given by Nelder and Mead (1965), sometimes called the downhill simplex method, is one of the few robust and efficient methods that does not use any derivative information. This greatly simplifies computational requirements and reduces the chances of errors that can crop up in the differentiation of complex rate expressions. 

The Nelder-Mead downhill simplex method is the optimization technique incorporated in the software package Matlab as fin ins or fininsearch. 

The simplex method belongs to a group of optimisation methods finding the minimum of a predefined multiparameter function (error functional). The downhill simplex method of Nelder and Mead [8] requires only function... 

Another method for finding the minimum is the so-called downhill simplex method [3, 619]. It requires only a function evaluation and does not use either function derivatives or matrix inversitMi. It may be relatively slow if one is trying to optimize many parameters and a shallow minimum, but it will always find a minimum (at least a local minimum). The problem with this technique is that it does not calculate the parameters standard deviations directly. In such cases, it is advisable, after finding the ntinimum by the simplex method, to use these parameters in the CNLS approximation, which should cmiverge quickly and provide standard deviations of the parameters. 

Commercially available software developed to process individual impedance spectra use few general algorithms such as Levenberg-Marquardt algorithm, the Nelder-Mead downhill simplex method or genetic algorithms [3-7]. The software is optimized to process only... 

The function /ext(0 fitted to the experimental real-time decay curves by means of a least-squares-fit routine, based on the downhill simplex method [308]. 

The DSC method is commonly used for the study of the cming kinetics of a UPR. The kinetic analysis of the crosslinking process initiated with MEKP was performed by means of an empirical and theoretical model based on the concept of free-radical polymerization [190]. A calculation algorithm based on the downhill simplex method and the Rimge-Kutta procedure was proposed. The non-isothermal DSC analyses allowed to obtain the concentration of the initiator and the radicals as a function of cure temperature at... 

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