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Astronomical Data Analysis Software and Systems VII
ASP Conference Series, Vol. 145, 1998
R. Albrecht, R. N. Hook and H. A. Bushouse, e
Ö Copyright 1998 Astronomical Society of the Pacific. All rights reserved.
ds.
A Parallel Procedure for the Analysis of Long­term
Sequences of Light Curves
A. F. Lanza, M. Rodon‘o 1 , U. Becciani and V. Antonuccio Delogu
Osservatorio Astrofisico di Catania, Viale A. Doria, 6 ­ I 95125
Catania, Italy
Abstract.
We present a parallel procedure which allows us to speed up the
modelling of photometric and spectroscopic observations of active binary
stars with brightness inhomogeneities on their surfaces. The procedure
has been implemented using PVM and is suitable to run on a cluster of
non­homogeneous, non­dedicated computers. It is optimized to recognize
and assign the data sets requiring the largest computational e#ort to the
most powerful CPUs of the cluster, taking into account the evolution
of their performances during the calculation in a fully dynamical way.
We report on several tests made with the workstation cluster of Catania
Astrophysical Observatory and discuss the advantages of this kind of
procedure for data analysis in Astrophysics.
1. Introduction
Binary systems belonging to the RS CVn and BY Dra classes show huge cool
spots on their photospheres which are regarded as manifestations of intense mag­
netic fields by analogy with sunspots (e.g., Tuominen et al. 1991, Strassmeier
& Linsky 1996).
The analysis of sequences of light curves spanning a long time interval (at
least one or two decades) allows us to detect activity cycles, analogous to the
solar 11­year cycle, inferring the overall properties of the stellar dynamos. More­
over, long­term data can be used to estimate the e#ects of starspots on stellar
parameters (Rodon‘o et al. 1995, Lanza et al. 1997).
We address the problem of modelling a long­term sequence of light curves
adopting recently developed tools for parallel computing and using an inhomo­
geneous cluster of CPUs.
2. Light curve modelling
The reconstruction of the surface map of an active star by using photometric data
alone is an ill­posed problem. It is possible to find a unique and stable solution
1 Istituto di Astronomia dell'Universit‘a degli Studi di Catania, Viale A. Doria, 6 ­ I 95125
Catania, Italy
67

68 Lanza, Rodon‘o, Becciani and Antonuccio Delogu
if a priori assumptions on the properties of the picture elements (pixels) of the
map are adopted such as the Maximum Entropy (hereinafter ME, e.g., Vogt et
al. 1987) and the Tikhonov criteria (hereinafter T, e.g., Piskunov et al. 1990).
The optimized map is found by minimizing the objective function Q con­
sisting of a linear combination of the # 2 and the regularizing function S: Q =
# 2 + #S. The expressions for # 2 (which gives the deviation between the fluxes
computed from the map and those observed) and the regularizing ME or T func­
tions can be found in, e.g., Cameron (1992); the Lagrange multiplier # measures
the relative weights of the a priori assumption and # 2 in constraining the solu­
tion. In our approach the best value of # is determined by the distribution of
the residuals between the observed and the computed fluxes (Lanza et al. 1997;
see also Cameron 1992). In any case several solutions are computed for di#erent
values of # in order to find the optimal value through a suitable statistical test.
In the modelling of a light curve sequence it is also of interest to determine
the physical parameters of the system components, which may be a#ected by
the presence of spots. In such a case the overall computational problem may
become much more complex and time consuming.
3. The parallel procedure
Our parallel procedure was written in standard fortran using the PVM library
version 3 and exploits a networked system to perform the analysis of a sequence
of NL light curves. The procedure starts on a main host -- the master­host --
and the single jobs are automatically spawned on all the systems on which PVM
is available. This procedure actually generates a virtual machine (hereinafter
VM). We assign the modelling of each light curve with given values of the system
parameters and # to a single system (host) which, at the end of its task, sends
the output back to the master­host.
The minimization of the function Q for all the light curves of a given se­
quence, for fixed values of the system parameters, represents a cycle of the
procedure. In general the analysis is not limited to one cycle because we can be
interested in the simultaneous determination of one or more system parameters,
such as the luminosity ratio of the system (see Rodon‘o et al. 1995, Lanza et
al. 1997). Thus, considering a typical sequence consisting of a few tens of light
curves, a few hundred modelling steps (i.e., Q minimizations) are required if #
is held fixed, and up to a few thousand if # also is varied (see also Wilson 1993).
The procedure is implemented using a master/slave paradigm. The master
is the program controlling the execution of the overall analysis and it is running
on the master­host. The master performs the following operations in sequence:
a) it identifies all available hosts and adds them to the VM; b) it assigns the
values to the system parameters and the #'s for the given cycle; c) it spawns
each analysis of the cycle on one of the hosts by choosing it according to the
number of normal points in the light curve and the current weights assigned to
the hosts; d) it periodically checks whether all the hosts of the VM are running
and, in case of a fault, restarts the lost jobs; e) it receives the output files from
the hosts which have completed their jobs and updates their current weights; f)
at the end of the cycle, according to the task assigned, it ends the execution or
starts a new cycle from step b).

Analysis of Long­term Sequences of Light Curves 69
Figure 1. (a) The time of execution on the reference CPU vs. the
number of data points in the analysed light curve for the linear test.
(b) The speedup vs. the number of equivalent CPUs on which the
parallel procedure is running for an ideal, a perfectly load balanced
and our real cluster of workstations, respectively.
The estimate of the weight of each host is initially based on the time it takes to
analyse a reference light curve and is updated in the course of the calculation
using the elapsed times of the previous analyses.
If a task is anomalously ended, the master detects the fault and tries to
restart it on the same host. If the fault occurs again or the host is not available,
the master deletes the host from the VM, and the analysis is re­scheduled on
the first available free host. After a user­defined period, a deleted host can be
checked again and, if it is found available, it can be added again to the VM.
Therefore the VM configuration changes dynamically and automatically during
the run with hosts being added or deleted by the master according to their
current availability status.
4. Tests and results
We have performed several tests on the workstation cluster of Catania Astro­
physical Observatory analysing a sequence of 18 light curves of the eclipsing
binary AR Lacertae (see Lanza et al. 1997).
The purpose of a first test (linear test) has been to study the variation of
the computational complexity, as a function of the number of data points in the
light curve. Only one processor has been used to run the procedure. It has been
a SuperSparc+ processor (clockspeed 50 MHz) and all the results are reported
assuming it as the reference processor. As workstation operating system we used
Solaris 2.5 and 64 MB RAM were allocated during the entire execution time.
The results are shown in Figure 1a. There is evidence that the execution time,
and thus the computational complexity, increases linearly with the number of
data points in the analysed light curve. In a second test, we analysed a sequence

70 Lanza, Rodon‘o, Becciani and Antonuccio Delogu
of 18 light curves running in parallel 4 cycles each with a di#erent value of the
luminosity ratio and a fixed # = 0.5, adopting the ME regularization. With
this test we sampled the parameter space to optimize the luminosity ratio of
the components of the system. The number of processors forming the VM has
been increased step by step to find how the speedup increases as a function of
the number of CPUs. The results of the test are reported in Figure 1b in terms
of equivalent processors, the reference processor being the CPU SuperSparc+ 50
MHz with SPECint92=76.9 and SPECfp92=80.1.
We plot in Figure 1b also the speedup expected for an ideal cluster and a
perfectly balanced application, which assumes that all the CPUs begin and end
their jobs simultaneously. This case can be realized only on a dedicated cluster.
We see that there is no significant degradation of the overall performance with
only a slight tendency toward a saturation when the number of CPUs exceeds
# 10. This is also a consequence of the fact that the processors we used are
rather similar (the maximum di#erence in their scaled powers is less than 20­
25%).
5. Conclusion
We have presented a very general and simple procedure which exploits a clus­
ter of inhomogeneous computers to significantly speed up the modelling of a
sequence of light curves. The same procedure, with only minor modifications,
can be applied also to the modelling of spectroscopic or polarimetric data (see,
e.g., Milone 1993, Vincent et al. 1993).
Our procedure can use a simple Workstation cluster, not necessarily ded­
icated, or medium­large parallel systems having a message passing software as
PVM. In the near future we plan to produce also an MPI version with an X­
based user interface to monitor processing and make it available as a public
domain software for light curve or Doppler Imaging analysis.
References
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Springer­Verlag), 33
Lanza A. F., et al., 1997, A&A, submitted
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Rodon‘o M., Lanza A. F., Catalano S., 1995, A&A, 301, 75
Strassmeier K. G., Linsky J. L. (Eds.), 1996, IAU Symp. 176, (Dordrecht:
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