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Noise representation in residuals of LSQR, LSMR, and CRAIG regularization

Research paper by Iveta Hnětynková, Marie Kubínová, Martin Plešinger

Indexed on: 16 Dec '16Published on: 16 Dec '16Published in: arXiv - Mathematics - Numerical Analysis



Abstract

Golub-Kahan iterative bidiagonalization represents the core algorithm in several regularization methods for solving large linear noise-contaminated ill-posed problems. We consider a general noise setting and derive explicit relations between noise that propagates in the bidiagonalization process 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 are proportional to 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. Influence of the loss of orthogonality is also discussed.