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Complexity of the Gaia astrometric least-squares problem and the (non-)feasibility of a direct solution method

Bombrun, A.; Lindegren, Lennart LU ; Holl, Berry LU and Jordan, S. (2010) In Astronomy & Astrophysics 516.
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 least-squares 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 (non-iterative) solution. A simplified version of the... (More)
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 least-squares 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 (non-iterative) 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 three-axis 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 fill-in 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 large-scale systematic errors in the Gaia reference frame. We briefly consider also an approximate decomposition method a la Hipparcos, but conclude that it is either sub-optimal or effectively leads to an iterative solution. (Less)
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author
organization
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type
Contribution to journal
publication status
published
subject
keywords
methods: numerical, methods: data analysis, space vehicles:, astrometry, instruments, reference systems
in
Astronomy & Astrophysics
volume
516
publisher
EDP Sciences
external identifiers
  • wos:000280275400092
  • scopus:77954650839
ISSN
0004-6361
DOI
10.1051/0004-6361/200913503
language
English
LU publication?
yes
id
dd82c679-db8a-491c-8e8f-2d055ccec0e6 (old id 1654602)
date added to LUP
2010-08-26 14:55:28
date last changed
2016-09-20 15:07:48
@article{dd82c679-db8a-491c-8e8f-2d055ccec0e6,
  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 least-squares 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 (non-iterative) 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 three-axis 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 fill-in 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 large-scale systematic errors in the Gaia reference frame. We briefly consider also an approximate decomposition method a la Hipparcos, but conclude that it is either sub-optimal or effectively leads to an iterative solution.},
  author       = {Bombrun, A. and Lindegren, Lennart and Holl, Berry and Jordan, S.},
  issn         = {0004-6361},
  keyword      = {methods: numerical,methods: data analysis,space vehicles:,astrometry,instruments,reference systems},
  language     = {eng},
  publisher    = {EDP Sciences},
  series       = {Astronomy & Astrophysics},
  title        = {Complexity of the Gaia astrometric least-squares problem and the (non-)feasibility of a direct solution method},
  url          = {http://dx.doi.org/10.1051/0004-6361/200913503},
  volume       = {516},
  year         = {2010},
}