Shooting algorithm

Here, we will explain the most effective of them, the so-called shooting algorithm. For simplicity we will focus on how to do it for deterministic trajectories such as those obtained from a molecular dynamics simulation. We note, however, that very similar algorithms can be applied to stochastic trajectories [5, 7, 8]. 

Formulate the system of equations to couple the thermal-energy equation (Eq. 6.69) into the shooting algorithm. 

Steady state equations fXr the adiabatic case corresponding to (1) through (4) were solved by the parameter mapping technique combined with the Newton-Fox shooting algorithm. The steady state nonadiabatic problems were solved by the finite-difference approach. 

Fig. 16. In a transition interface sampling simulation to calculate the probability VAi i + 1 ) the shooting algorithm is used to sample an ensemble of pathways that originate in A, cross interface i and then return to A or proceed to interface i 1. Then, + 1 ) equals the fraction of the trajectories that reach interface i 1 rather than the surface of A. The dark gray pathways depicted are members of this ensemble but the light gray one is not because it does not cross interface i before returning to A...

The boundary problem for Eq. (58) was solved [117] using the Shooting Algorithm implemented with Mathematica 4 [121]. Comparison with analytical solutions in the uniform limit demonstrated agreement to within 2% for deformation profiles and to within 5% for the elastic energy. 

For the shooting algorithm we have described, acceptanee probabilities are particularly simple if phase-space modifications have a symmetric generation probability. If an asymmetry is present, that is, ... 

Scheme 1.7. Shooting algorithm with existing molecular dynamics code.

In the BzzMath library, the multiple shooting algorithms have not been implemented in a dedicated class. 

The free parameters of the model are fitted to experimental data (the 0-I-D-P-fluorescence rise of dark-adapted tobacco leaves) by means of the multiple shooting algorithm PARFIT as developed by Bock [2] for parameter identification in systems of nonlinear differential equations. We use a multiple experiment structure for measurements at different light intensities. The initial trajectory and the results are shown in Figs. 2 and 3. 

Bock, H. G., Plitt, K. J. (1984) A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems, Proc. of the 9th IFAC Worldcongress, Budapest, Hungary, Vol. IX, Colloquia 14.2, 09.2... 

H.-G. Bock and K.-J. Plitt. A multiple shooting algorithm for direct solution of optimal control problems. Preprints of the 9th IFAC World Congress, Budapest, International Federation of Automatic Control, 1984. 

The efficiency of a TPS simulation, i.e., the rate at which trajectory space is sampled, crucially depends on how in detail new trajectories are generated from old ones. While various ways to do that have been proposed [164], the so-called shooting algorithm has proven particularly useful (see Fig. 8). Since it is generally... 

Fig. 8 In the shooting algorithm for deterministic dynamics a new path blue) is generated from an old one red) by first randomly selecting one point on the old path, the shooting point. Then, the particle momenta at that point are modified by addition of a small perturbation dp. From the point with perturbed momenta the equations of motion are integrated forward and backward to obtain a complete trajectory. For small perturbations, the new trajectory wiU be close to the old one near the shooting point but wiU then rapidly diverge from it due to the chaoticity of the underlying dynamics...

In order to apply this type of analysis to pathways generated in a TPS simulation, the shooting algorithm described in Sect. 5.2 has to be slightly modified. Since in this method one would like the acceptance probability of a shooting move attempted from r to be governed by P(TP r), the new momenta at r need to be drawn from... 

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