Open AccessEstimation of resolution and covariance for large matrix inversions
Author(s) -
Zhang Jie,
McMechan George A.
Publication year - 1995
Publication title -
geophysical journal international
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0956-540X
DOI - 10.1111/j.1365-246x.1995.tb05722.x
Subject(s) - singular value decomposition , orthonormal basis , conjugate gradient method , covariance , algorithm , mathematics , covariance matrix , resolution (logic) , matrix (chemical analysis) , eigenvalues and eigenvectors , mathematical optimization , computer science , statistics , artificial intelligence , physics , materials science , quantum mechanics , composite material
Summary Key advantages of conjugate gradient (CG) methods are that they require far less computer memory than full singular value decomposition (SVD), and that iteration may be stopped at any time to give an approximate solution; this means that they may be used to obtain solutions of problems that are too large for SVD. The disadvantage is that CG does not conveniently provide auxiliary information on the quality of the solution (resolution and covariance matrices). This may be overcome by extensions of Paige and Saunders’ LSQR algorithm, which is one of the family of CG algorithms. The extensions are produced by analogy with SVD; bidiagonalization in LSQR produces orthonormal basis vectors that can be used to construct solutions and estimates of resolution and covariance. For large problems, for which SVD can not be performed, the new method provides approximate resolution and covariance estimates that asymptotically approach those of the SVD solutions as the number of iterations increases.