Noise Representation in Residuals of LSQR, LSMR, and CRAIG Regularization

0477277 - ÚI 2018 RIV US eng J - Journal Article
Hnětynková, I. - Kubínová, Marie - Plešinger, M.
Noise Representation in Residuals of LSQR, LSMR, and CRAIG Regularization.
Linear Algebra and Its Applications. Roč. 533, č. 15 (2017), s. 357-379. ISSN 0024-3795. E-ISSN 1873-1856
Grant - others:GA ČR(CZ) GC17-04150J
Institutional support: RVO:67985807
Keywords : Ill-posed problems * regularization * Golub–Kahan iterative bidiagonalization * LSQR * LSMR * CRAIG
OECD category: Applied mathematics
Impact factor: 0.972, year: 2017

Golub-Kahan iterative bidiagonalization represents the core algorithm in several regularization methods for solving large linear noise-polluted ill-posed problems. We consider a general noise setting and derive explicit relations between (noise contaminated) bidiagonalization vectors and the residuals of bidiagonalization-based regularization methods LSQR, LSMR, and CRAIG. For LSQR and LSMR residuals we prove that the coefficients of the linear combination of the computed bidiagonalization vectors reflect the amount of propagated noise in each of these vectors. For CRAIG the residual is only a multiple of a particular bidiagonalization vector. We show how its size indicates the regularization effect in each iteration by expressing the CRAIG solution as the exact solution to a modified compatible problem. Validity of the results for larger two-dimensional problems and influence of the loss of orthogonality is also discussed.
Permanent Link: http://hdl.handle.net/11104/0273659