Cost function simulated annealing

This criterion resumes all the a priori knowledge that we are able to convey concerning the physical aspect of the flawed region. Unfortunately, neither the weak membrane model (U2 (f)) nor the Beta law Ui (f)) energies are convex functions. Consequently, we need to implement a global optimization technique to reach the solution. Simulated annealing (SA) cannot be used here because it leads to a prohibitive cost for calculations [9]. We have adopted a continuation method like the GNC [2]. 

Simulated Annealing-based solutions [19] are conceptually the same as Genetic Algorithm-based approaches. However, the SA-based techniques, in our experience, are more sensitive to the initial settings of the parameters. Nevertheless, once the correct ones are found, the method can achieve the efficiency of GA-based solutions. We must point out that SA-based solutions have never outperformed the GA-based ones in our studies. Much of what has been mentioned regarding the GA-based solutions is also relevant for the SA technique, particularly, with respect to the cost functions. 

Alternative algorithms employ global optimization methods such as simulated annealing that can explore the set of all possible reaction pathways [35]. In the MaxFlux method it is helpful to vary the value of [3 (temperamre) that appears in the differential cost function from an initially low [3 (high temperature), where the effective surface is smooth, to a high [3 (the reaction temperature of interest), where the reaction surface is more rugged. 

The difficulty with this procedure is that simple refinement routines, such as simplex or least squares, lead only to the nearest minimum in the cost function which is unlikely to be the global minimum. The refinement procedure therefore has to be one that randomly samples different parts of configuration space so as to be able to reach different minima, ultimately selecting the global minimum. Two refinement methods have been proposed, simulated annealing and the genetic algorithm. 

Keywords Genetic algorithms Simulated annealing Structure prediction Cost function ... 

In the next section the ideas behind several methods, including the GA and a simulated annealing (SA) approach [19,20], then their implementation used to generate ionic crystal structures are reviewed. This will contain an introduction to the types of move class operators and the various types of cost functions used to modify the current trial structure(s) and to assess the quality of the trial structures, respectively. In the third section recent applications of the GA and SA approaches to closest-packed ionic systems and then to open-framework crystal structures are reviewed. 

Fig. 13 Cost function as a function of the logarithm of temperature during the annealing of NbF4. Inset structural arrangement of NbF4 at the beginning of the simulation (right) and as obtained by simulated annealing (SA) (left) [58]. Reproduced with the kind permission of the Nature Publishing Group (http //www.nature.com/)...

Since the change in squared-error is based solely on the cluster means and the sizes, (12) can be calculated very efficiently. Such efficiencies are also exploited by the simulated annealing implementation.) The use of the change in squared-error is intriguing because the sum of these cost increases along any path from a leaf node to the root of the dendrogram is constant. Hence, if one were to use (12) as a level function in the Wallace and Kanade sense, then... 

The frameworks of molecular sieves are constructed from 4-connected TO4 tetrahedra. Deem and Newsam developed an approach to optimize an initially arbitrary T-atom configuration with respect to a cost function based on the T—T distances, T—T—T angles, and number of first-neighbor T-atoms, by simulated annealing using the Monte Carlo method.147,211 This method could be used to solve 4-connected crystal structures, as well as to predict unknown hypothetical structures. 

Direct Methods Direct-Space Techniques Patterson Methods Monte Carlo Simulated Annealing Genetic Algorithm Degree of Freedom Cost Function... 

Fig. 6 Reduction of cost function with reducing temperature in simulated annealing calculations when determining the structure of siliceous nanoparticles by fitting NMR and SANS data. Short-range order occurs at higher temperatures and long-range order at lower temperatures. (View this art in color at www.dekker.com.)...

The optimization is accomplished through a simulated annealing engine with cost function given by Equation (5.1). The cost function is the weighted sum of three components total wire length, floorplan area, and total number of inter-chip contacts. 

We extended out 2.5-D floorplan design tool introduced in the previous sections to handle thermal effects. Initially, the 2.5-D/3-D floorplanning optimization is accomplished through a simulated annealing engine with a cost function defined as ... 

In order to implement the CGH, holograms are calculated by using a program based on a variation of the widely adopted simulated annealing optimization algorithm (Dames, Dowling et al., 1991), (Broomfield, Neil et al., 1992) whose cost function to niinimize the calculation error is ... 

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