Download The Lattice Boltzmann Equation for Fluid Dynamics and Beyond by Sauro Succi PDF

By Sauro Succi

In recent times, sure kinds of the Boltzmann equation--now going via the identify of "Lattice Boltzmann equation" (LBE)--have emerged which relinquish so much mathematical complexities of the genuine Boltzmann equation with no sacrificing actual constancy within the description of complicated fluid movement. This booklet offers the 1st specific survey of LBE conception and its significant functions thus far. obtainable to a huge viewers of scientists facing complicated process dynamics, the booklet additionally portrays destiny advancements in allied parts of technology the place fluid movement performs a distinctive role.

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13. 14. Initialize the size of the particle swarm n, and other parameters. Initialize the positions and the velocities for all the particles randomly. 19); Next j Next i End While. The end criteria are usually one of the following: • Maximum number of iterations: the optimization process is terminated after a fixed number of iterations, for example, 1000 iterations. • Number of iterations without improvement: the optimization process is terminated after some fixed number of iterations without any improvement.

6) j=1 where P is the total number of samples, y1j and y2j are the actual time-series and the flexible neural tree model output of j-th sample. F it(i) denotes the fitness value of i-th individual. , genetic algorithms (GA), evolution strategy (ES), evolutionary programming (EP), particle swarm optimization (PSO), estimation of distribution algorithm (EDA), and so on. In order to learn the structure and parameters of a flexible neural tree simultaneously, a tradeoff between the structure optimization and parameter learning should be taken.

1, in which the fitness function is calculated by mean square error (MSE) or root mean square error(RMSE) 03. 2. In this stage, the tree structure or architecture of flexible neural tree model is fixed, and it is the best tree taken from the end of run of the PIPE search. 5 2 y = (1 + x−2 ) , 1 + x2 1 ≤ x1 , x2 ≤ 5. 7) 50 training and 200 test samples are randomly generated within the interval [1, 5]. The static nonlinear function is approximated by using the neural tree model with the pre-defined instruction sets I = {+2 , +3 , .

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