First, the system shown in the left panel of Fig. 1 will be studied:
In order to determine the corresponding orbital parameters, a run with population
size was carried out.
The number of generations () was 100, the mutation rate was
, and the number of grid cells was 49 (nx = 7, ny = 7).
For Run 1 and all other runs described in this section, the values of m1 and m2 were
constrained to lie between 0.3 and 3.0, between -50.0 and 50.0,
and and between -0.999 and 0.999. All
possible spin combinations were allowed. The total number of combinations of
the unknown variables was then approximately .Mutated chromosomes were only accepted if their values of the unknown
parameters were contained in the above intervals.
For the observational data used in Run 1, the actual values of m1, m2,
, , , s1, and s2 were
1.0, 1.0, 3.0, -0.672, 0.839, 1, and 1, respectively.
Table 1 presents the results of Run 1: The first 6 rows
show the orbital parameters of the best simulation in generations 1, 5, 10, 20, 50, and 100,
and the final row shows the actual orbital parameters of the "observed"
system. As can be seen, the GA was able to find the orbital parameters with
great accuracy. In fact, acceptable orbital parameters were obtained already
after 20 generations. After 100 generations, the image of the data (left
panel of Fig. 1) and the image obtained from the best
simulation were almost identical. This is partly due to the fact that the
same disc distributions were used both for the run and for generating the
observation. However, the method does not require perfect data of the kind
used in this run and it can, in fact, cope with quite high levels of noise,
as discussed in Sect. 6.3 below.
Not all determinations of orbits proceed as smoothly as, for example, Run 1. In order for the GA to be able to find the orbit, it is a requirement that the galaxies show some signs of interactions, i.e. distortions of some form. This was the case for Runs 1 to 4. However, in a case where the observation consisted of two more or less unperturbed discs, the GA was not able to find the orbit. The fact that signs of interactions are needed is rather obvious and does not, in practice, imply any restrictions. After all, the method is intended for interacting systems.
A more important problem is that, even for clearly interacting systems, the GA is not always able to find the correct orbit on the first attempt. Since the population size is far from infinite, there may simply never be sufficient variation in the genetic material to obtain the correct orbital parameters, at least if is small. can of course be increased, but the larger its value, the more the GA approaches a random search. Fortunately, it is always easy to distinguish between a run that fails and one that succeeds, namely through the fitness values. In a successful run, the fitness values continue to increase throughout the run, whereas in an unsuccessful run, the GA rather quickly gets stuck at low values.
For the excellent fit obtained for Run 1, the fitness of the best simulation was 0.430. The corresponding numbers for Runs 2 to 4 were 0.192, 0.339, and 0.475. In contrast, in a run that fails, fitness values above 0.1 are not reached, and usually the GA gets stuck at even lower values. Even in such situations improvements do occur, but at a very slow rate, and it is usually faster to restart the run, using different values of the random number generator seed and the mutation rate.
The results for Runs 1,2, and 4 were obtained on the first attempt, but Run 3 required two attempts. In the first attempt, with mutation rate , the GA got stuck at a suboptimal solution with fitness 0.0863. The mutation rate was then increased to 0.010 and the random number generator seed was changed, resulting in a maximum fitness of 0.339 in the second attempt.
|Figure 3: The two regions of highest density were discarded, as indicated by the black squares. The original picture, before discarding the data in the two black squares, was identical to the left panel of Fig. 1|
Thus, in this run, the deviation was computed using only the 47 (i.e. ) remaining pixels. Clearly, this problem is more difficult to solve for the GA (or anyone else!), since only partial information is accessible. The result of the run is shown in Table 3, the upper row as usual showing the orbital parameters of the best simulation and the lower row showing the observational data, which were the same as for Run 1. Even though the resulting fit is not as stunning as for Run 1, acceptable orbital parameters were obtained.
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