In [1] a regularized iterative algorithm was described which has been shown to be very suitable for solving the ill-posed image restoration problem. By incorporating deterministic constraints and adaptivity this very general algorithm is capable of achieving both noise suppression and ringing reduction in the restoration process. It consumes, however, considerable computation to obtain a (visually) stable solution due to the low convergence speed of the algorithm. The purpose of this paper is to investigate the possibilities for speeding up the convergence of this restoration method. To this end we compare the classical steepest descent algorithm (with linear convergence) with a conjugate gradients based method (superlinear convergence) and a new Q-th order converging algorithm. The latter solution method has the highest convergence rate, but is restricted in its application to space-invariant image restoration with a linear constraint. Although the actual convergence speed of the algorithms involved generally depends on the image data to be restored, it will be shown that for real-life images the constrained conjugate gradients algorithm yields a considerable convergence speed improvement.