A conjugate gradient algorithm for the astrometric core solution of Gaia
A. Bombrun1, L. Lindegren2, D. Hobbs2, B. Holl2, U. Lammers3 and U. Bastian1
1 Astronomisches Rechen-Institut, Zentrum für Astromomie der Universität Heidelberg, Mönchhofstr. 12–14, 69120 Heidelberg, Germany
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2 Lund Observatory, Lund University, Box 43, 22100 Lund, Sweden
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3 European Space Agency (ESA), European Space Astronomy Centre (ESAC), PO Box (Apdo. de Correos) 78, 28691 Villanueva de la Cañada, Madrid, Spain
Received: 17 August 2011
Accepted: 25 November 2011
Context. The ESA space astrometry mission Gaia, planned to be launched in 2013, has been designed to make angular measurements on a global scale with micro-arcsecond accuracy. A key component of the data processing for Gaia is the astrometric core solution, which must implement an efficient and accurate numerical algorithm to solve the resulting, extremely large least-squares problem. The Astrometric Global Iterative Solution (AGIS) is a framework that allows to implement a range of different iterative solution schemes suitable for a scanning astrometric satellite.
Aims. Our aim is to find a computationally efficient and numerically accurate iteration scheme for the astrometric solution, compatible with the AGIS framework, and a convergence criterion for deciding when to stop the iterations.
Methods. We study an adaptation of the classical conjugate gradient (CG) algorithm, and compare it to the so-called simple iteration (SI) scheme that was previously known to converge for this problem, although very slowly. The different schemes are implemented within a software test bed for AGIS known as AGISLab. This allows to define, simulate and study scaled astrometric core solutions with a much smaller number of unknowns than in AGIS, and therefore to perform a large number of numerical experiments in a reasonable time. After successful testing in AGISLab, the CG scheme has been implemented also in AGIS.
Results. The two algorithms CG and SI eventually converge to identical solutions, to within the numerical noise (of the order of 0.00001 micro-arcsec). These solutions are moreover independent of the starting values (initial star catalogue), and we conclude that they are equivalent to a rigorous least-squares estimation of the astrometric parameters. The CG scheme converges up to a factor four faster than SI in the tested cases, and in particular spatially correlated truncation errors are much more efficiently damped out with the CG scheme. While it appears to be difficult to define a strict and robust convergence criterion, we have found that the sizes of the updates, and possibly the correlations between the updates in successive iterations, provide useful clues.
Key words: methods: numerical / space vehicles: instruments / methods: data analysis / astrometry
© ESO, 2012