Relaxation strategy for the block Landweber method

Abstract

Abstract The block Landweber iteration is a general method for the solution of linear systems which is widely applied for image reconstructions. The convergence behavior of the block Landweber iteration is of both theoretical and practical importance. When the linear system is consistent, we derive a neat iterative representation and find an optimal relaxation coefficient by minimizing the 2-norm of the newly derived iteration matrix in the range of the conjugate transpose of the corresponding block matrix. We also establish the accelerating convergence relaxation strategies based on the maximum eigenvalue of each block matrix. When the linear system is inconsistent, we normalize each block matrix by the 2-norms of the each weighted block matrix at a uniform scale to give the relaxation strategies. The numerical simulations show that the relaxation strategies proposed for the block Landweber iteration in both consistent and inconsistent cases in this paper have the better reconstruction performances. Compared with the Landweber iteration, the block Landweber iteration has superiority in reconstruction results.