Complexity of the Gaia astrometric leastsquares problem and the (non)feasibility of a direct solution method
(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 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... (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 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. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/1654602
 author
 Bombrun, A.; Lindegren, Lennart ^{LU} ; Holl, Berry ^{LU} and Jordan, S.
 organization
 publishing date
 2010
 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:77957875604
 ISSN
 00046361
 DOI
 10.1051/00046361/200913503
 language
 English
 LU publication?
 yes
 id
 dd82c679db8a491c8e8f2d055ccec0e6 (old id 1654602)
 date added to LUP
 20100826 14:55:28
 date last changed
 20190220 04:57:28
@article{dd82c679db8a491c8e8f2d055ccec0e6, 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.}, author = {Bombrun, A. and Lindegren, Lennart and Holl, Berry and Jordan, S.}, issn = {00046361}, 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 leastsquares problem and the (non)feasibility of a direct solution method}, url = {http://dx.doi.org/10.1051/00046361/200913503}, volume = {516}, year = {2010}, }