An Optimally Generalized Steepest-Descent Algorithm for Solving Ill-Posed Linear Systems

It is known that the steepest-descent method converges normally at the first few iterations, and then it slows down. We modify the original steplength and descent direction by an optimization argument with the new steplength as being a merit function to be maximized. An optimal iterative algorithm with -vector descent direction in a Krylov subspace is constructed, of which the optimal weighting parameters are solved in closed-form to accelerate the convergence speed in solving ill-posed linear problems. The optimally generalized steepest-descent algorithm (OGSDA) is proven to be convergent with very fast convergence speed, accurate and robust against noisy disturbance, which is confirmed by numerical tests of some well-known ill-posed linear problems and linear inverse problems.