# On inner iterations of the joint bidiagonalization based algorithms for solving large scale linear discrete ill-posed problems

The joint bidiagonalization process of a large matrix pair A,L can be used to develop iterative regularization algorithms for large scale ill-posed problems in general-form Tikhonov regularization min_x{Ax-b_2^2+λ^2Lx_2^2} or the essentially equivalent one minLx_2 s.t. x∈𝒮 = {x|Ax-b_2≤ηe_2}, where e is a Gaussian white noise, L is a regularization matrix and η>1 slightly. A bottleneck of the algorithms is that a large scale inner least squares problem with (A^T, L^T)^T as the coefficient matrix must be solved at each outer iteration, which may be costly, especially when the solution accuracy of these problems is high. We investigate the solution accuracy requirement on the inner least squares problems and give a reliable stopping tolerance of the LSQR for iteratively solving the inner least squares problems. The results show that the solution accuracy of the inner least squares problems can be relaxed considerably while it will not reduce the accuracy of regularized solutions, thus the overall efficiency of the algorithms can be improved substantially. We also discuss some details for implementing the joint bidiagonalization based regularization algorithms efficiently and stably. Numerical experiments are made to illustrate our results and show some numerical performance of the algorithms.

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