A few other twists can be added, and such searches undoubtedly will never be strictly by rote. Physical systems are so different, some experience with each one is required to develop a truly efficient algorithm.
The above two Sections define this method of very fast re-annealing (VFR).
VI. Applications
Some explanation of the problems we are solving using VFR contribute a better understanding of its value.
Many large-scale nonlinear stochastic systems can be described within the framework of Gaussian-Markovian systems. For example, (deceptively) simple Langevin rate equations describe the evolution of a set of variables $M^G$. The Einstein summation convention is used, whereby repeated indices in a term are to be summed over. In the midpoint Stratonovich representation [6],
$G = 1, \dots, \Theta.$
Expanded sets of equations can represent a field $M^G(r, t)$, and the discussion below generalizes as well [7].
Another mathematically equivalent representation is given by the Fokker-Planck equation, in terms of the “drifts” $g^G$ and “diffusions” $g^{GG'}$,
where the “potential” $V$ might arise directly from the physics or by simulating the boundary conditions.
In many problems of interest, the drifts and diffusions are also parametrized. For example, these parameters can enter as expansion coefficients of polynomials describing accepted models of particular systems, e.g., modeling economic markets [8], or combat scenarios [9,10]. In combat systems, such equations appear as
where the $M^G$ are Red ($r$) and Blue ($b$) force levels, and where ${x, y, z}$ are parameters to be fit to data. Such modeling is essential to compare computer models to field data, or to qualify computer models to augment training, or to feed information from battalion-level computer scenarios into corps- and theater-level computer scenarios which must rely on highly aggregated models to run in real-time.