Steepest descent, CG, and iterative regularization of ill-posed problems. (English) Zbl 1045.65034
Summary: The state of the art iterative method for solving large linear systems is the conjugate gradient (CG) algorithm. Theoretical convergence analysis suggests that CG converges more rapidly than steepest descent. This paper argues that steepest descent may be an attractive alternative to CG when solving linear systems arising from the discretization of ill-posed problems. Specifically, it is shown that, for ill-posed problems, steepest descent has a more stable convergence behavior than CG, which may be explained by the fact that the filter factors for steepest descent behave much less erratically than those for CG. Moreover, it is shown that, with proper preconditioning, the convergence rate of steepest descent is competitive with that of CG.
MSC:
65F22 | Ill-posedness and regularization problems in numerical linear algebra |
65F10 | Iterative numerical methods for linear systems |
65F35 | Numerical computation of matrix norms, conditioning, scaling |
94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |