Complexity of the Gaia astrometric leastsquares problem and the (non)feasibility of a direct solution method
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Abstract
The Gaia space astrometry mission (to be launched in 2012) will use a continuously spinning spacecraft to construct a global system of positions, proper motions and absolute parallaxes from relative position measurements made in an astrometric focal plane. This astrometric reduction can be cast as a classical leastsquares problem, and the adopted baseline method for its solution uses a simple iteration algorithm. A potential weakness of this approach, as opposed to a direct solution, is that any finite number of iterations results in truncation errors that are difficult to quantify. Thus it is of interest to investigate alternative approaches, in particular the feasibility of a direct (noniterative) solution. A simplified version of the astrometric reduction problem is studied in which the only unknowns are the astrometric parameters for a subset of the stars and the continuous threeaxis attitude, thus neglecting further calibration issues. The specific design of the Gaia spacecraft and scanning law leads to an extremely large and sparse normal equations matrix. Elimination of the star parameters leads to a much smaller but less sparse system. We try different reordering schemes and perform symbolic Cholesky decomposition of this reduced normal matrix to study the fillin for successively longer time span of simulated observations. Extrapolating to the full mission length, we conclude that a direct solution is not feasible with today's computational capabilities. Other schemes, e. g., eliminating the attitude parameters or orthogonalizing the observation equations, lead to similar or even worse problems. This negative result appears to be a consequence of the strong spatial and temporal connectivity among the unknowns achieved by two superposed fields of view and the scanning law, features that are in fact desirable and essential for minimizing largescale systematic errors in the Gaia reference frame. We briefly consider also an approximate decomposition method a la Hipparcos, but conclude that it is either suboptimal or effectively leads to an iterative solution.
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Original language  English 

Journal  Astronomy & Astrophysics 
Volume  516 
Publication status  Published  2010 
Publication category  Research 
Peerreviewed  Yes 