4.4.2 Primary source processing (AGIS)
Author(s): Uwe Lammers
The Gaia core solution aims to solve the astrometric parameters for close to 2 billion sources mainly in our Galaxy. This clearly presents an enormous computational challenge as the size of the data set, and the large number of parameters, cannot be processed sequentially. The difficulty is caused by the strong connectivity among the unknowns: each source is effectively observed relative to a large number of other sources simultaneously in the field of view, or in the complementary field of view some $106{.}^{\circ}5$ away on the sky, linked together by the attitude and calibration models. The complexity of the astrometric solution in terms of the connectivity between the sources provided by the attitude modelling was analysed by Bombrun et al. (2010), who concluded that a direct solution is infeasible, by many orders of magnitude, with today’s computational capabilities. The study neglected the additional connectivity due to the calibration model, which makes the problem even more unrealistic to attack by a direct method. However, the mathematical system of equations under consideration guarantees that a unique (apart for the six degrees of freedom discussed in the reference frame alignment in Section 4.3.2), coherent and completely independent global solution for the whole sky can be derived.
To overcome this difficulty an iterative method has been developed over a number of years using increasingly complex and efficient algorithms. This approach became known as the Astrometric Global Iterative Solution (AGIS) and now relies on a Conjugate Gradient (CG) algorithm to converge the solution efficiently (Bombrun et al. 2012). The numerical approach to AGIS is a blockiterative leastsquares solution. In its simplest form, four blocks are evaluated in a cyclic sequence until convergence. The blocks map to the four different kinds of unknowns outlined in Section 4.1.1, namely:

S:
the source (star) update, in which the astrometric parameters $\bm{s}$ of the primary sources are improved;

A:
the attitude update, in which the attitude parameters $\bm{a}$ are improved;

C:
the calibration update, in which the calibration parameters $\bm{c}$ are improved;

G:
the global update, in which the global parameters $\bm{g}$ are improved.
The G block is optional, and will perhaps only be used in some of the final solutions, since the global parameters can normally be assumed to be known a priori to high accuracy. The blocks must be iterated because each one of them needs data from the three other processes. For example, when computing the astrometric parameters in the S block, the attitude, calibration and global parameters are taken from the previous iteration. The resulting (updated) astrometric parameters are used the next time the A block is run, and so on. The mathematical description of the AGIS blockiterative leastsquares solution and the updating of each block has been outlined in detail in Sections 4 and 5 respectively of Lindegren et al. (2012). In addition to these blocks, separate processes are required for the alignment of the astrometric solution with the ICRS (see also Section 4.3.2), the selection of primary sources, and the calculation of standard uncertainties; these auxiliary processes are discussed in (Section 6 of Lindegren et al. 2012).
Additionally, it is not necessary for the AGIS solution to include all two billion sources. Instead, it is done using a selection of up to 10% of the astrometrically well behaved single sources and this is sufficient to converge the attitude and calibration solutions. The other sources can then be solved for in a secondary solution (see Section 4.4.3) using the converged parameters found in the primary AGIS solution. For Gaia EDR3 the primary solution consisted of about 14.3 million sources so the number of unknowns in the global minimization problem was about 85 million (71.5 million for the sources, 10.7 million for the attitude, 1.1 million for the calibration parameters, and 2.0 million for the global unknowns). The number of elementary observations ($l$) considered was about 6.5 billion ($6.5\times {10}^{9}$).