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In a Hilbert setting, we introduce a new dynamical system and associatedalgorithms for solving monotone inclusions by rapid methods.  Given a maximal monotone operator $A$, the evolution is governed by the timedependent operator $I -(I + \lambda(t) {A})^{-1}$, where the positive controlparameter $\lambda(t)$ tends to infinity as $t \to + \infty$. The tuning of $\lambda (\cdot) $ is done in a closed-loop way, by resolution of the algebraicequation $\lambda \norm{(I + \lambda {A})^{-1}x -x}=\theta$, where $\theta $ isa positive given constant. The existence and uniqueness of a strong globalsolution for the Cauchy problem follows from Cauchy-Lipschitz theorem. We provethe weak convergence of the trajectories to equilibria, and superlinearconvergence under an error bound condition. When $A =\partial f$ is thesubdifferential of a closed convex function $f$, we show a $\bigo(1/t^2)$convergence property of $f(x(t))$ to the infimal value of the problem. Then, weintroduce proximal-like algorithms which can be obtained by time discretizationof the continuous dynamic, and which share the same fast convergenceproperties. As distinctive features, we allow a relative error tolerance forthe solution of the proximal subproblem similar to the ones proposed in~\cite{So-Sv1, So-Sv2}, and a large step condition, as proposedin~\cite{MS1,MS2}. For general convex minimization problems, the complexity is$\bigo(1/n^2)$. In the regular case, we show the global quadratic convergenceof an associated proximal-Newton method.

A dynamic approach to a proximal-Newton method for monotone inclusions in Hilbert spaces, with complexity O(1/n^2)