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

TL;DR

This paper analyzes how noise affects residuals in LSQR, LSMR, and CRAIG regularization methods, providing explicit relations and insights into noise propagation and regularization effects in large ill-posed problems.

Contribution

It derives explicit relations between bidiagonalization vectors and residuals, revealing how noise influences these residuals in LSQR, LSMR, and CRAIG methods.

Findings

Residual coefficients in LSQR and LSMR reflect propagated noise.

CRAIG residuals are multiples of specific bidiagonalization vectors.

CRAIG solution size indicates regularization effect per iteration.

Abstract

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.