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variation of the magnitude as a function of phase angle is not constant at
different rotational phases of the object.
In order to solve this inversion problem, we have explored a fully numer-
ical option, based on techniques of “genetic” computation. This approach
seems to be particularly promising. This computational technique is based
on concepts of “survival of the fittest” which are inspired to classical stud-
ies of the evolution of living species. In particular, the possible solutions
of the problem are characterized by a set of parameters (spin period, pole
coordinates, axial ratios, etc.) which can be seen as the “genes”, or the
“DNA” of a solution. The goodness of any given solution is assessed on
the basis of its corresponding residuals with respect to a set of real or sim-
ulated observations. Better solutions give a better fit of the observational
data, and the idea of the genetic approach is to simulate an evolution of the
solution parameters, by selecting at each step only those giving the best fit.
This idea is implemented by means of a numerical algorithm, which initially
generates a large number of completely random solutions, saving in mem-
ory only a limited subset, corresponding to those producing the smallest
residuals. In general, these preliminary solutions are very bad, as one would
expect apriori from a set of completely random attempts. At this time, the
“genetic” mechanism is switched on. This consists in random coupling of
the parameters of the saved solutions, and in random variations (“genetic
mutation”) of some of the parameters constituting the “DNA” of a single
solution. If the newly born “baby” solution is better than some of those
saved until that step, it enters the “top list”, whereas the previously worst
solution is removed from the same set. In this way, after a number of the
order of one or two millions of “genetic experiments”, a very good solution
is usually found, which produces small residuals and basically solves the
inversion problem. Due to the intrinsically random nature of this “genetic”
approach, the right solution is not forcedly found always, but if the genetic
algorithm is repeatedly applied to the same set of observed data (typically
30 and 40 times) the right solution (giving the minimum residuals in dif-
ferent attempts) is usually found several times.
So far we have been able to satisfactorily invert many sets of simulated
photometric observations. The simulations indicate that the algorithm suc-
ceeds in determining with a very high accuracy the spin period, the param-
eter that would seem apriori most dicult to determine, because it should
be taken into account that accuracies of the order of 10 5 h are needed
for this parameter. In spite of this diculty, we have always been able so
far to solve the inversion problem in practically all the simulated cases, in
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