  Much recent attention has been devoted to gradient descent algorithms where
the steepest descent step size is replaced by a similar one from a previous
iteration or gets updated only once every second step, thus forming a {\em
faster gradient descent method}. For unconstrained convex quadratic
optimization these methods can converge much faster than steepest descent. But
the context of interest here is application to certain ill-posed inverse
problems, where the steepest descent method is known to have a smoothing,
regularizing effect, and where a strict optimization solution is not necessary.
  Specifically, in this paper we examine the effect of replacing steepest
descent by a faster gradient descent algorithm in the practical context of
image deblurring and denoising tasks. We also propose several highly efficient
schemes for carrying out these tasks independently of the step size selection,
as well as a scheme for the case where both blur and significant noise are
present.
  In the above context there are situations where many steepest descent steps
are required, thus building slowness into the solution procedure. Our general
conclusion regarding gradient descent methods is that in such cases the faster
gradient descent methods offer substantial advantages. In other situations
where no such slowness buildup arises the steepest descent method can still be
very effective.
