Discrete deterministic steps

Simulation Techniques Based on Discrete Deterministic Steps... 

Cellular automata are simple mathematical idealizations of natural systems. They consist of a lattice of discrete identical sites, each site taking on a finite set of say integer values. The values of the sites evolve in discrete time steps according to deterministic rules that specijy the value of each site in terms of the values of neighboring sites. Cellular automata may thus be considered as... 

To consider evolutionary optimization as a sequence of discrete selection steps requires some explanation. In fact, it is justified only on the basis of a hierarchical order of the rate terms, in which the off-diagonal terms usually are much smaller than the diagonal ones, decreasing in binomial or Poisson-ian progression with increasing Hamming distance between template and (erroneous) replica. As a consequence, the deterministic order around an... 

This method has been devised as an effective numerical technique of computational fluid dynamics. The basic variables are the time-dependent probability distributions 7 (x, t) of a velocity class a on a lattice site x. This probability distribution is then updated in discrete time steps using a deterministic local rule. A careful choice of the lattice and the set of velocity vectors minimizes the effects of lattice anisotropy. This scheme has recently been applied to study the formation of lamellar phases in amphiphilic systems [92, 93]. 

A one-dimensional cellular automation consists of a line of sites, with each site carrying a value (0 and 1 in the simplest case). The value of each position is updated in discrete time steps by an identical deterministic rule depending on the neighbourhood of sites around it. 

These three steps constitute what I call the Langevin approach . It is widely utilized, also when the fluctuations are not due to thermal motion or to discreteness of particles, and even when they are of unspecified or unknown provenance. The applications in many branches of physics, chemistry and biology are far too numerous to list. The approach has been highly successful whenever the deterministic equations are linear, but for nonlinear cases it leads to difficulties, which are analyzed in this section. The purpose is to shield the reader from the labyrinthine literature and to convince him of the necessity of the more firmly based treatment in the next chapter. 

The basic idea of the MC approach lies in the discrete representation of the joint PDF by an ensemble of stochastic particles. Each particle carries an array of properties denoting position, velocity and scalar composition. During a fractional time stepping procedure [6] the particles are submitted to certain deterministic and stochastic processes changing each particle s set of properties in accordance with the different terms in the PDF evolution equation. Afterwards the statistical moments may be derived in the simplest case by averaging from the ensemble of particles. 

To implement the deterministic optimization approach, the sequential strategy proposed by Li et al. (1998) is used, where the whole algorithm is divided into one optimization layer with SQP as a standard NLP solver and one simulation layer, where all dependent variables are computed through an integration step. The model is a large scale DAE system which is discretized with collocation on finite elements. The whole batch time is discretized into 30 time intervals. The control variables are set as piecewise constants. The computed trajectories of the control variables for the optimal operation are illustrated in Fig. 1. 

---