Regularization properties of range restricted LSQR method for solving large-scale linear discrete ill-posed problems

The LSQR iterative method is one of the most popular methods for solving large-scale linear discrete ill-posed problem Ax=b with an error-contaminated right-hand side.In this paper, we consider the regularization properties of range restricted LSQR (RRLSQR) method. The iteration number k always acts as the regularization parameter because of the semi-convergence. In order to verify whether or not the RRLSQR method finds a ۲-norm filtering best regularization solution for severely, moderately and mildly ill-posed problems, we present the sin \Theta theorems for the ۲-norm distances between the k dimensional left and right Krylov subspaces generated by Lanczos bidiagonalization and the k dimensional dominant left and right singular subspaces of A, and estimate the distances for the three kinds problems assuming that the singular values are simple, and develop a regularized RRLSQR method for solving linear discrete ill-posed problems. Numerical experiments confirm our theoretical results and show the efficiency of the proposed method.

کلیدواژه ها:

Linear discrete ill-posed problem ، semi-convergence ، range restricted LSQR method ، regularization property