  In a Hilbert space setting, for convex optimization, we analyze the
convergence rate of a class of first-order algorithms involving inertial
features. They can be interpreted as discrete time versions of inertial
dynamics involving both viscous and Hessian-driven dampings. The geometrical
damping driven by the Hessian intervenes in the dynamics in the form $\nabla^2
f (x(t)) \dot{x} (t)$. By treating this term as the time derivative of $ \nabla
f (x (t)) $, this gives, in discretized form, first-order algorithms in time
and space. In addition to the convergence properties attached to Nesterov-type
accelerated gradient methods, the algorithms thus obtained are new and show a
rapid convergence towards zero of the gradients. On the basis of a
regularization technique using the Moreau envelope, we extend these methods to
non-smooth convex functions with extended real values. The introduction of time
scale factors makes it possible to further accelerate these algorithms. We also
report numerical results on structured problems to support our theoretical
findings.
