Paths optimal

Application of a Stochastic Path Integral Approach to the Computations of an Optimal Path and Ensembles of Trajectories ... 

Figure 14.9 Illustration of the SPW method. Optimized path points are denoted by x...

In order to perform FEP calculations on optimized paths with a small number of images, extra images need to be added on the path between the previously optimized points. Once these extra images have been added, an optimization has to be performed to minimize them to the MEP. Here we have developed a modification to our NEB QM/MM implementation [27], This modification allows for the optimization of only selected images on the path while maintaining the points previously optimized with the parallel iterative path method or the combined procedure fixed. 

Figures 3-4 and 3-5 show the optimized paths with the added images and the original combined method [27] and parallel path optimizer method [25] calculated paths for the first and second steps of the reaction respectively. In both cases, the addition of extra images on the converged path, and subsequent optimization of these extra images produces a smoother path since the additional images allows for a better mapping of the potential energy surfaces (PESs).

Table 10.1 The optimal path of the reachability graph. The length is 15 nodes and the makespan is 10. The clock valuations of the individual clocks C are not shown, only the absolute time is presented in the second column.

The adaptive control of a batch reactor-II Optimal path control (with H.H.-Y. Chien). Automatica 2, 59-71 (1964). 

In the range of small noise intensities D, the optimal path q vi(t qf, 4>j) to the point (qf, f) is given by the condition that the action S be minimal. The variational problem for S to be extremal gives Hamiltonian equations of motion... 

Figure 6. From top to bottom action surface Lagrangian manifold (LM) and extreme paths calculated [80] for the system (17) using equations (21). The parameters for the system were A = 0.264 and

Experimentally measured ph for the system (17) for two qualitatively different situations are shown in Figs. 7 and 8. It is immediately evident (1) that the prehistory distributions are sharp and have well-defined ridges (2) that the ridges follow very closely the theoretical trajectories obtained by solving numerically the equations of motion for the optimal paths, shown by the full curves on the top planes. It is important to compare the fluctuational path bringing the system to (qf, relaxational path back towards the stable state in thermal equilibrium, Fig. 7, and away from it, Fig. 8. Figure 7 plots the distribution for the system (17) in thermal equilibrium, namely A = 0. The... 

Figure 8. Fluctuational behavior measured and calculated for an electronic model of the nonequilibrium system (17) with A = 0.264, D — 0.012. The man figure plots the prehistory probability density (pk x,t]Xf,0) and posthistory distribution to/from the remote state Xf — —0.63, t — 0.83, which lies on the switching line. In the top plane, the fluctuational (squares) and relaxational (circles) optimal paths to/from this remote state were determined by tracing the ridges of the distribution [62],...

D. Optimal Paths on a Finite Time Range, and Conclusions... 

A very good example of the usefulness of the concept of the optimal path is the idea of the logarithm susceptibility (LS) [56,87,141]. 

It is evident from the preceding discussion that the theory of the optimal paths provides a deep physical insight into the dynamics of fluctuations and is in good agreement both with the results of analog and numerical simulations and with the results of the experiments in optical systems. It has now become possible to use the prehistory formulation [60] as a basis for experiments on fluctuational... 

In this section the application of the optimal path approach to the problem of escape from a nonhyperbolic and from a quasihyperbolic attractor is examined. We discuss these two different types of chaotic attractor because it is known [160] that noise does not change very much the structure and properties of quasi-hyperbolic attractors, but that the structure of non-hyperbolic attractors is abruptly changed in the presence of noise, with a strong dependence on noise intensity. Note that for optical systems both types of chaotic attractor [161-163] (nonhyperbolic and quasihyperbolic) are observed, but a nonhyperbolic attractor is much more typical. 

A statistical analysis of the fluctuational trajectories is based on the measurements of the prehistory probability distribution [60] ph(q, t qy, tf) (see Section IIIC). By investigating the prehistory probability distribution experimentally, one can establish the area of phase space within which optimal paths are well defined, specifically, where the tube of fluctuational paths around an optimal path is narrow. The prehistory distribution thus provides information about both the optimal path and the probability that it will be followed. In practice the method essentially reduces to continuously following the dynamics of the system and constructing the distribution of all realizations of the fluctuational trajectories that transfer it from a state of equilibrium to a prescribed remote state. 

Figure 18. The most probable escape path (bottom solid curve) from S5 to the SI, found in the numerical simulations. The stable limit cycle is shown by rombs see Fig. 16 for other symbols. Parameters were h = 0.13, cty = 0.95,o>o 0.597,D = 0.0005. Top optimal force (solid line) corresponding to the optimal path after filtration [169]. The optimal path and optimal force from numerical solution of the boundary-value problem are shown by dots.

In principle, it is possible to find the optimal path by direct solution of the Pontryagin Hamiltonian (37), with appropriate boundary conditions. We must stress that even for this relatively simple system, the solution is a formidable, and almost impossible, task. First of all, in general one has no insight into the appropriate boundary conditions, in particular into those at the starting time (which belong to the strange attractor). But even if the boundaries were known, in practice the determination of the optimal path is impossible the functional R of Eq. (36) has so many local minima, that it proved impractical to attempt a (general) search for the optimal path. 

Thus, we have found that the mechanisms of escape from a nonhyperbolic attractor and a quasihyperbolic (Lorenz) attractor are quite different, and that the prehistory of the escape trajectories reflects the different structure of their chaotic attractors. The escape process for the nonhyperbolic attractor is realized via several steps, which include transitions between low-period saddle-cycles coexisting in the system phase space. The escape from the Lorenz attractor consist of two qualitatively different stages the first is defined by the stable and unstable manifolds of the saddle center point, and lies on the attractor the second is the escape itself, crossing the saddle boundary cycle surrounding the stable point attractor. Finally, we should like to point out that our main results were obtained via an experimental definition of optimal paths, confirming our experimental approach as a powerful instrument for investigating noise-induced escape from complex attractors. 

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